SAR HS Math Mag 2018

M at h M ag

SA R

H

ig

h

Sc h

oo

Vo

l 1

m u l

2 1 e

M at h M ag Ed i t or s i n Ch i ef Ar t Di r ect or s

Raf i K ep ecs

L ay ou t Ed i t or s

Sh i r a M i ch ael i

Josep h Ru bi n

Rebecca Rosen zw ei g

Ay el et Ru ben st ei n

A Let t er f r om t h e Edit or s Dearest readers,

Sy l v i e W ei n st ei n Associ at e Ed i t or s

Albert Einstein once said that ?pure mathematics is, in its way, the poetry of logical ideas.? It is with this truth in mind that we present the 2016-2017 Math Mag.

Ben jam i n K l est zi ck Jon ah Bu r i an Joey Yu d el son

Our contributors explored math, aiming to share its beauty with all of you. This year?s magazine is split into four overarching sections: Phenomena, Constructs, Mathematics Meets Reality, and Investigations.

Ay el et K al f u s Ad i n Gi t i g Ar t i st s an d Ph ot ogr ap h er s

W r i t er s Ay el et K al f u s Sar ah Bol n i ck Ben K ep ecs Jon ah Bu r i an

Sar ah Bol n i ck Tam ar Gel l er

Jor d an n a K osow sk y

Ben K ep ecs

Gabr i el l a Can t or

Am i el Or bach

Rach el Coh en

M ar t i n Rosen

Sh m u el Cr i saf i

Zach Rot h st ei n

Ri t a Fed er

Ay el et Ru ben st ei n

Tam ar Gel l er

Josep h Ru bi n

Ad i n Gi t i g

Joey Yu d el son

Ay el et Ru ben st ei n Josep h Ru bi n

Jacal y n Tok ay er Tal y a K r on i sch

So go ahead. Delve into forbidden operations with zero and non-Euclidian geometry. Discover Euler?s exponentiation and fair division. Learn about math?s role in geographic profiling and the electoral college.

Tal i a K r au sz Sam m y Sok ol

The beautiful poetry of math is waiting for you.

H an n ah Sch w al be

We hope you enjoy, Facu l t y Ad v i sor s M s. An n M or h ai m e M r . Scot t Raw bon e

The 2016-2017 Editing Crew Rafi Kepecs, Joey Yudelson, Ben Klestzick, Jonah Burian, Adin Gitig, and Ayelet Kalfus

SAR H i gh Sch ool ?s An n u al M at h em at i cs Jou r n al Vol u m e 12: Fal l 2017

Math Mag 2017

3

SAR High School

Table of con t en t s

Photograph by Talya Kronisch

Sect ion 1: Phenomena 1.

2.

3.

4.

5.

Forbidden Operat ions wit h Zero Cant or Ternary Set Fibonacci Composit e Eucl id and t he Paral l el Post ul at e Non-Eucl idean Geomet ry

Sect ion 2: Const ruct s 1.

2.

3.

4.

Eul er?s Exponent iat ion Game of Lif e Fair Division Mat h, Art and Excel

Sect ion 3: Mat hemat ics meet s Real it y 1.

2.

3.

4.

Social Net works Making a Game Compet it ive How t o Make Bet t er Predict ions Geographic Prof il ing

Sect ion 4: Invest igat ions 1. El ect oral Col l ege 2. Timekeepers t hrough Time 16. Invest igat ions in Number Theory wil l be pl aced in cal l out boxes t hroughout t he magazine

Ph en om en a Ph ot ogr aph by Talya Kr on isch

Math Mag 2017

SAR High School

5

For bidden Oper ation s with th e Nu m ber Zer o

following: 0, ? , - ? , 1 (specifically for 0/0), any number you choose, and an undeter mined slope.

You can see why it is hard to do oper ations when this number can have so many possibilities for values.

By Am iel Or bach

So 1/0, is simultaneously ? and - ? . We already have inconsistencies.

But how about looking at division as the opposite of multiplication. If x=y/z, then y=xz. So if z=0, then y=0x.

Math has cer tain ground r ules that you must follow when using var ious oper ations. A number x divided by a number y (x/y) is x divided into y par ts.

X to the power of y is Xy is x multiplied by itself y times. X factor ial (x!) is a number multiplied by ever y integer below it until reaching 1. An integr al is found using the for mula 0 is a r ule- breaker. It has found a way to mess up each of these r ules, in order to create bizar re answer s, special

Math Mag 2017

exceptions, meaningless values, disagreeing limits, and gener al mayhem.

teacher would be r ight. Sor t of. Let?s see what the problem is with dividing by 0.

The problems all star t when you tr y to divide by 0. When you lear ned division in elementar y school (or at some later point when it came up), someone probably asked what happens when you divide by 0.

The teacher probably told you that you just can?t divide by 0, and that it doesn?t make any sense to split a cake between 0 people and ask how much each per son gets. And the

To use the example from before, if you divide a cake between 0 people, there is no one to have any cake. It would seem to be that each per son gets nothing. So x/0 is 0. Fine.

But now let?s look at the gr aph of y=1/x (Figure 1).

Since 0 times anything is 0, then y will be 0 regardless of what x is. So x can be any number at all! That also means that since 0=0* 1 and 0=0* 2, 0/0 equals both 1 and 2, making the statement 1=2.

This is simply false. The gr aph of y=x/x is the same as y=1 for all values except x=0, where it is undefined. Could this mean that 0/0 is 1? What about the slope of an undefined gr aph?

The slope of a gr aph (m) is equal to the change in y over the change in x (dy/dx). The change in y is infinity, since it is a line extending str aight up.

The change in x is 0, since a line has no width in the x direction when it is ver tical.

So now we have x/0 one of the

Maybe 0 0 will be easier to solve. We have a r ule that 0 x where x is an arbitr ar y real number is 0. So 0 0 is 0. But we also have a r ule that i x0 is 1 for any real number x. So 0 0 is 1.

Another contr adiction. Let?s tr y gr aphing the function y=xx (Figure 2).

As the gr aph approaches x=0, y seems to conver ge to 1.

However, this only applies to real number s. If you look at the imaginar y gr aph, it seems to conver ge to 0 (Figure 3).

Also,

to

use

a

real- life

If you take the limit of the gr aph as x approaches 0, the gr aph r apidly approaches infinity from the r ight, but negative infinity from the left.

SAR High School

Math Mag 2017

7

SAR High School

example, let?s say you invest an amount of money a with continuous interest r of 0% for 0 time. The amount of money represented would be y=ar 1 or y=a(0 0 ). Logically, you would have the same amount of money as you had initially; in other words y=a. This would make 0 0 equal to 1.

However, what if you express 0 0 as (01)* (0- 1)? This would be 0/0, which we already established as being impossible to use in equations.

This is another reason we don?t have a definite value for 0 0.

Let?s tr y some factor ials.

Taking factor ials of a positive integer x (x!) is usually ver y easy. However, what about 0!?

Usually, we would calculate the factor ial as being the number multiplied by all integer s between itself and 1.

For example, 4!=4* 3* 2* 1=24.

But 0!=1. This is because you can also calculate factor ials by the for mula x!=(x+1)!/(x+1).

Applying this to 0, 0!=(0+1)!/(0+1)=1!/1=1.

This works out fine. When you get to (- 1)!, you r un into problems.

By using the for mula, you get (- 1)!=(- 1+1)!/(- 1+1), or 0!/0=1/0.

This br ings us back to our

or iginal problem of dividing by 0.

Let?s look at one more problem of dividing by 0 by integr ating the equation f(x)=1/x. If you tr y to take the integr al, you get ?1/x dx= ?x- 1 dx=x- 1+1/(- 1+1), or x0 /0. This doesn?t work, and yet we know there must be an integr al.

After all, since one definition of the integr al is the area under a cur ve and we know there is area under f(x)=1/x (because it isn?t the x- axis), there must be an equation for the integr al. The equation is actually ?1/x dx=ln|x|+C. This may seem str ange, but there is a reason for that.

1) By the Second Fundamental Theorem of Calculus, if you take the der ivative of the integr al, you get the or iginal equation (this effectively means that we will star t with y=ln|x| (Figure 4) as our or iginal equation and tr y to get dy/dx=1/x).

a) ey=eln|x| 3) eln|x| simplifies to x. a) ey=x 4) We now take the der ivative of each side with respect to y. a) ey=dx/dy 5) But we already know that ey=x. So now we substitute that in. a)x=dx/dy 6) Finally, we take the inver se of both sides to get the answer we were looking for. a)1/x=dy/dx So those are the problems with oper ations with 0. But are there instances where it is okay to divide by 0, or per for m other oper ations that would nor mally be forbidden?

Actually, yes. Geor g Fr iedr ich Ber nhard Riemann, a Ger man

mathematician, constr ucted a hypothetical sphere extending from 0 to ? (Figure 5).

Nor mally, we can?t show infinity as a point, since it isn?t a number, but he put it at the top of his gr aph You can also per for m oper ations with the Riemann sphere, since each number, real or imaginar y, can be represented by one point on the sphere. If you dr aw a line from a point on the Car tesian plane (x- y coordinate plane) to the point ? on the top of the sphere, it will inter sect the sphere in exactly one place.

This means that each point on the sphere represents a number, and that all number s are represented on the sphere.

You are also allowed to divide

by 0 in the sphere. In the Riemann sphere, 1/0 is ? , but not because of an actual oper ation. Riemann had to define that relationship specifically. This isn?t a solution to the problem of dividing by 0; r ather, it is a special case in a single br anch of math in which you are allowed to do oper ations that are nor mally forbidden.

Have fun tr ying all of these gr aphs and equations your self!

The Riemann Sphere Reimann tamed the point of infinity by making it a point on a sphere.

Created by Leonid 2, from Wikipedia. Licensed under CC Attribution-Share Alike 3.0 Unported license.

Figure 5

a) y=ln|x| 2) We take e to the power of either side.

Math Mag 2017

SAR High School

Math Mag 2017

9

SAR High School

Mat hemat ics of The Cantor Ternary Set Paper art creat ed by Ayel et Rubenst ein and phot ographed by Josh Dorf man

By Joseph Rubin

closed inter val representing

i.e, ever y number that cannot

leftmost third of the r ightmost

the middle third of each

be represented in ter nar y

third of the leftmost third and

segment, the points border ing

except for a representation

so on, because ever y decimal 0

the removed sections will be

that contains a 1 in the fir st

represents the fir st third of

left. These endpoints are

decimal place is removed from

possible number s, (0.0- 0.1),

never removed. Additionally,

the set. In the next step, all

and a decimal 2 represents the

we will see later on that there

number s that can only be

last third of possible number s

are points left in the set that

represented in ter nar y with a 1

(0.2- 1.0).

are not on the end of any

in the second decimal place

par ticular removed section. In

are thrown out, and so on.

fact, the set has an infinite

After an infinite number of

number of points.

steps, all number s that can only be represented in ter nar y

The line star ts with l=1unit, so its endpoints are [0, 1]. In each The Cantor Ter nar y Set is a

unbroken segment as step n

To solve a geometr ic ser ies

step, we remove the open

set of points that results from

n=0, each step has 2 line

and find the remaining length

inter val that represents the

star ting with a line segment of

segments.

of the star ting unit, the

middle third of each segment.

equation that is used is

Therefore, after one iter ation,

length l=1units and repeatedly removing the open inter val representing the middle third of each successive line segment that remains, an infinite number of times. The

In the fir st step, we remove 1/3

the set of points we have left

of a unit, and in the next step,

can be represented as [0, 1/3] U

we remove 1/3 of 1/3 twice, or 2/9 of a unit. The next iter ation removes 4/27.

The r atio, r, is 2/3, and the fir st

[2/3, 1]. Because they are at the

ter m, a, is 1. Therefore, the

end of sections from which

solution is

the open inter vals were

process of removing the

To find out how much of the

removed, the points 0, 1/3, 2/3,

middle third in each iter ation

star ting unit remains after an

and 1 remain in the set and are

infinite number of iter ations,

never removed. Therefore,

is depicted in Figure 1.

In each step, two segments are for med out of the each of the previous segments. With the

the amount removed is represented by the summation :

It seems like the full unit is removed. However, if at ever y iter ation we only remove the open inter val, and not the

with a 1 in any decimal place are removed. Because there are an infinite number of steps, and in each step, points were removed, an infinite number of points were removed. However, all points that can be represented in ter nar y using only the decimal digits 0 and 2 are remaining, so there are still points in the set. In fact, the number of

representation of number s,

which points are in Cantor ?s

we can see that there are

Set, and which are not.

points in Cantor ?s Set that are not endpoints of any

be wr itten as 0.2. Ever y point

par ticular removed section.

The number 1/4 in decimal is in ter nar y. Therefore, it is in the r ightmost third of the

between those two number s,

Math

Math Mag 2017

ter nar y. This can be proven, because any decimal can be represented in binar y with only the digits 0 and 1. There are an a infinite amount of such r ational number s. Any binar y number can be taken, and after replacing all of its 1s with 2s, we can represent a new number in ter nar y for each number we star ted with in binar y.

points in Cantor ?s Ter nar y set

be able to easily figure out

and 2/3 can

decimal digits 0 and 2 in

Therefore, the number of

By using the ter nar y

be wr itten as

represented using only the

infinite, as will be shown later.

Ter nar y Set. We would like to

base 3, or ter nar y for m, 1/3 can

of number s that can be

points left is uncountably

they are present in Cantor ?s

If we wr ite our number s in

There are an infinite amount

is infinite. Fur ther more, we can prove that the Cantor Set is uncountable. In order for a set to be countable, it must be possible to list all of the points in the set, while for a set to be uncountable it must be that no matter how many points are given, a new one can always be found that exists in the set but

11

SAR High School

was unlisted. If a supposed list

the Cantor Set. Because a new

length. Since the Cantor ?s Set

This gr aph goes from 0 to 1 on

of ever y ter nar y decimal with

number can always be found

consists of closed, isolated

both axes, and it is called the

only the digits 0 and 2 were to

that exists in the Cantor ?s Set

points, it is not dense in any

Devil?s Staircase. The flat

exist, a new decimal could be

no matter how many are

inter val.

por tions of the gr aph are

created by using the following

listed, the set is uncountable.

algor ithm:

This ar gument is called Cantor 's diagonal ar gument,

Star ting with the fir st decimal digit of the fir st number in the list, then the second digit of the second number, and so on,

because star ting with the next digit in each successive decimal from the list for ms a diagonal line through the list.

inter vals in Cantor ?s Set that This is a special point of interest, because while the set contains an uncountable number of points, it has zero measure and is of zero density at ever y point.

The Cantor Set is also

0 if the digit from the list is 2,

self- similar, because in each

and 2 if the digit from the list

iter ation, the set is broken up

is 0. The new number differ s

into more exact copies of

from ever y other number on

itself. This also means that the

the list in one decimal place.

set is a fr actal.

only the ter nar y decimal digits 0 and 2, so it is an element of

of 0. Since there are no nonzero length regions in Cantor ?s Set, the only por tions

We can cr aft a function related

which are greater than 0, are

to the Cantor Set that

single points on Cantor ?s Set,

tr uncates decimals in ter nar y

and yet the gr aph goes up

at the fir st decimal place that

from 0 to 1.

contains a 1, and then replacing all decimal digits that are 2s with 1s. Now we can

This new number consists of

of the gr aph have a der ivative

of the gr aph, the der ivatives of

constr uct a number whose cor responding decimal digit is

were removed. These por tions

However, ever y point on the

inter pret the number as a

set is isolated, so the Cantor ?s

binar y decimal. A gr aph of this

Set has no inter val of nonzero

function looks like Figure 2:

The Cantor ?s Ter nar y Set is a unique set because its mathematical qualities make it uncountable, and yet of zero measure and zero density.

Even by removing an infinite number of points from the star ting line segment, the set contains just as many points as did the star ting segment.

While it may seem impossible, it is actually easy to show what points are in the set by switching our base system because the fr actal nature of the set makes it predictable.

Math Mag 2017

SAR High School

Math Mag 2017

13

SAR High School

Fi n d i n g a Fu n ct i on i n Fi bon acci

substituting this for mula into

next

the

that

backwards in time) has one

if

female. That female has one

Bn =An- 1 is tr ue then so is

male and one female parent,

Bn- 1=An- 2). This for mula only

making two bees in gener ation

gives us the number of adult

three. The male then has a

pair s, so if we add in Bn =An- 1

female parent and the female

The number sequence for med

to account for the babies we

has a parent of each gender,

by Fn=Fn- 1+Fn- 2 is known as

get An +Bn =2An- 1+An- 2. In other

giving

the Fibonacci

words,

the

gener ation four. The single

simple way to find the nth

number of pair s of r abbits at

female then has two parents,

Fibonacci number is to star t

the end of any given month by

the male has one, and the

with

adding the two totals from the

remaining female has two,

Fibonacci

month before it. Now that we

making five total bees in the

Fn=F n- 1+Fn- 2 until you get to

know that this, we can say that

fifth gener ation. The number

the nth Fibonacci

after the fir st two months in

of bees in each gener ation

Though it does work, that

which there is only one pair,

follows

Fibonacci

method is inefficient because

the next month will be the

sequence. You can der ive a

it requires many steps. There

sum of those two. So the

for mula to find the number of

is a different method that is

sequence will proceed: 1, 1, 2,

bees in any gener ation based

much more efficient and uses

If we state that An is the

3, 5, 8, 13, 21, 34, 55, 89, 144. So,

on the two before it in the

matr ices.

number of adults pair s at the

the number of pair s at the end

following way. Let the nth

rectangular ar r ay of number s

end of a month and Bn is the

of one whole year (12 months)

gener ation be Gn, and let the

with special r ules for addition

number of baby pair s at the

will be 144.

gener ation that comes after it

and multiplication. You begin

in time, a smaller number, be

with

Gn- 1. The number of males in

In order to multiply matr ices,

Gn is M n and the number of

one

females is Fn . The number of

columns of the fir st ter m by

female bees in

Fn- 1,

the rows of the second ter m to

equals the number of male

compute a cross product for

bees

each entr y in the resulting

Leonardo de Pisa was a 12th to

how exactly r abbits breed so

pair and the newbor n pair,

13th centur y mathematician

as to make the problem well

making two pair s. Then the

who is other wise known as

defined. Fir stly, none of the

fir st

filius Bonacci, or Fibonacci.

r abbits will die in the span of

second pair grows, and both

He was the author of Liber

the year. Additionally,

the

pair s mate, resulting in three

Abaci,

teach

r abbits are always bor n in a

pair s. A month later the fir st

Europe the basic computation

pair, one male and one female.

two pair s give bir th and the

that we now lear n at young

The r abbits have two ?ages?,

third grows, giving us five.

ages.

introduced

baby and adult, and, for the

This process continues to

?zero?, a concept that would

sake of simplicity, we assume

follow the Fibonacci number s.

lead to many other significant

that when they become adults

We can now use algebr a to

discover ies. The feat he is

the r abbits will mate with

deter mine the answer to the

most famous for is defining

their twin. The set of twins

or iginal question of how many

the ?Fibonacci Number s?, a set

will mate at the beginning of

r abbits there will be at the end

of number s ar r ived at

by

each month and produce a set

of the year.

preceding

of twins at the end of the

ter ms in the ser ies (0, 1, 1, 2, 3,

month, for each month until

5, 8, 13, 21, 34, 55, 89, 144, 233? ),

the end of the year. Ever y set

and

of twins bor n will do the same.

He

adding

would

also

the two

has

many

interesting

proper ties and applications. It

pair

gives bir th, the

When this process is followed

end of the month, then we can

we see that the number of

state

pair s of r abbits mir ror s the

An =An- 1+Bn- 1

Fibonacci number s. Fir st, we

number of adult pair s in a

In his book, Fibonacci posed a

have the initial pair of r abbits,

given month is going to be the

mathematical exercise about

which gives us the fir st ?one?

number of adult pair s from

r abbits. The question was: if

in the ser ies. Then they grow

the

you

of

up and mate; the female is

number of baby pair s that

opposite gender s in a closed

pregnant but has not given

grew up. We can also say that

system, how many pair s of

bir th, so we still have one pair.

Bn =An - 1 because all the adults

r abbits would there be after

Next month, the pair gives

from the month before give

one year ? Of cour se, there are

bir th to twins and mates

bir th to the babies of the

some assumptions made on

again, so we have the adult

cur rent

appear s

in

many

natur al

phenomena and br anches of math.

left

Math Mag 2017

two

r abbits

we

An =An- 1+An- 2

By M ar t i n Rosen

which

fir st,

the

last

following: because

month

plus

month.

the

the

By

SAR High School

we

Another

find

(because

can

find

example

Fibonacci

of

sequence

the

occur s

with regard to bees. A male bee is bor n from

only a

mother, while a female is bor n from two parents, a male and a female. If you star t with a male bee and deter mine the gener ations that came before it, the Fibonacci

sequence

becomes apparent. The fir st gener ation has one male. The

Math Mag 2017

gener ation

three

the

in

Gn ,

(going

bees

Gn- 1, Mn.

in

Next,

M n- 1+Fn- 1=Fn . If you substitute,

number of bees.

Finding

a

Fibonacci

Number

0

and

Sequence. A

1

and

number s

A

the must

using number.

matr ix

matr ix

add

is

a

equation:

multiply

the

matr ix.

you get Fn=Fn- 1+Fn- 2. If you plug in for Fn- 1 and Fn- 2 you can find the total number of females in any gener ation and

Now, if we plug in F2 and F1

therefore

into the equation for Fn+1 and

find

15

the

total

SAR High School

Fn, we get:

or

For the next set we have:

would be too lar ge (), so we

The method to der ive this

equal

r atio of any number in the

stop squar ing. Now, with the

for mula is called Gener ating

to the

Fibonacci

number s we found, we find

Functions. Though the proof is

preceding number approaches

the number s whose power s

somewhat

the

Phi as you go fur ther along in

add up to the power of the

application is simpler. Let us

the ser ies. Phi is not just a

number we want to find. In

say that we want to find . Just

r andom

this case 2+4=6, so it will be

plug it in and the for mula is

mathematics, but has pr actical

and . If we multiply those

easy to solve.

applications. The golden r atio

complex,

number s, we will receive the number we were looking for.

was In

most

cases

the

Binet

For mula is more efficient,

With this equation we can substitute

in

the

In the context of our equation,

though

it

does

have

we would use this method in

deficiencies. When attempting

the following way:

to calculate a lar ge Fibonacci might get a ver y long or

equation resulting in:

ir r ational number that has more digits than digit spaces on your calculator.

Or,

because

matr ix

used

dur ing

the

found

by

in

sever al

the

Renaissance,

including Leonardo Da Vinci, number s, as the number s get

number

lar ger the quotient

represented by

approaches phi. This is

Phi

because the second ter m in

(approximately

the numer ator becomes so

1.618),

small when n is lar ge that it is

unique

negligible. Therefore, you get:

mathematical

Therefore, you will lose some

multiplication is associative:

number

over

mathematicians and ar tists

Figure 1

its

number, because of the , you

or iginal

Or

Ser ies

r atio

is

a

closely

digits and not get an exact

related to the

number (Figure 1).

Fibonacci Ser ies.

The

Another way to state the Binet Therefore, you can find any

Using this method, we can

Fibonacci number with this

calculate the nth Fibonacci

the

for mula:

number efficiently.

architect,

For mula is as shown below: extr aordinar y

musician, There is another method that You can use phi in the

man

Jacques Binet

As n increases, the r atio will

This method is only efficient if

for mula, which makes it look

discovered a for mula that can

approach phi.

you calculate the power using

much neater. There are other

be used

the nth

the

connections between phi and

Fibonacci number, later called

the Fibonacci number s. Phi is

the

approximately equal to 1.618,

method

of

repeated

squar ing. Let us say that we want to find . We square a to get , and then we square it again to get . The next square

Math Mag 2017

to find

Binet

for mula

is

For mula. shown

The below:

mathematician, inventor,

and

engineer of the 16th centur y.

is also ver y efficient. In 1843, a named

ar tist,

He was an exper t in the human body, and used r atios, what he refer red to as ?divine propor tions?, to define key Fibonacci in Ar t and

measurements

Architecture

paintings. His

and if you divide two consecutive Fibonacci

SAR High School

in

his

master piece,

the Mona Lisa, depicts a The ?golden r atio,? which is

Math Mag 2017

per fect

17

face

using

SAR High School

mathematics based on the

Eu cl i d an d Th e Par al l el Post u l at e

golden r atio. Even ancient Greek

sculptures

and

buildings like the Par thenon, which

pre- dated

By Gabr i el l a Can t or

Fibonacci,

were designed in propor tions related to the number Phi.

Euclid

Fibonacci in Culture

is often

descr ibed

as

Great but before the Per sian

homes, public str uctures, and

?father of geometr y,? because he

War s. Dur ing this per iod, the

walls.

tr ansfor med

Greeks established

the

mathematics

was discovered in the 13th

geometr y in ways that had lasting

advances in philosophy, ar t, and

centur y and is pr imar ily used

effects. Unfor tunately, little is

science. Although the study of

in science and math, it has

known about his life. This is

mathematics was initiated

also

popular

because ?Euclid? was a ver y

Ancient Egypt, the Greeks greatly

culture. The movie ?The Da

common name and unlike most

advanced the subject dur ing this

author s, he never wrote about

time

himself in his books.

tr ansfor med

in

Vinci Code,? based on Dan Brown?s novel, tells the stor y of an investigation into a myster ious murder.

and

bizar re

Throughout

the

Fibonacci sequence is a ver y

rectangle,

interesting set of number s

adjacent squares of side length

with many interesting and

1. Then you dr aw a 2X2 square

intr iguing relationships and

using the sides of the fir st two

applications in math and in

squares. You continue this

nature.

process using the Fibonacci

movie, the char acter s find a ser ies of symbols and codes, including

the

As

has

been

shown,

the

dr aw

two

number s. If you then dr aw Fibonacci in Geometr y

quar ter circle spir als as in the

Fibonacci

number sequence.

you

developed

We do

into a square and a rectangle,

width

the rectangle will also be

towards the golden r atio, phi.

will

tend

histor ians

can

logically make cer tain inferences.

Proclus, who was a philosopher and mathematician from the 5th

?Euclid? put Elements,

collecting

many

math.

Eudocus?s

theorems,

perfecting

after the death of Alexander the

in size, the r atio of length to

life,

created a more abstr act for m of

geometr y, called The

the

proper ty that if they are split

Euclid?s

moder n

Classical Age,? which took place

Fibbonacci rectangle increases

the gaps in our knowledge of

for many pur poses. Thus, they

noted

Golden rectangles have the

accomplishments. Despite

which for ms the foundation of

of

As

and

and for mulas that could be used

one

relationship.

without discussing Euclid?s life

the content of his great book

Euclid lived dur ing the ?Greek

shor t- side r atio is phi to 1.

mathematics cannot be explained

proof of complex propositions

golden spir al. This is because previously

of

for

living. Most impor tant, we know

spir als. A golden rectangle is

a

study

per iod

numer ical

spir al almost identical to the

to

the

Greeks

innovative

he lived and what he did for a

of the Fibonacci sequence in side

The

in

This

mathematics from the study of

golden rectangle, you get a

long

per iod.

made great

know, however, when and where

There is another occur rence

whose

gover nments and

democr atic

Although the Fibonacci Ser ies

appeared

and

study

Elements.

calculations to the

Greek

geometr y

in

architects

used

ways

which

centur y, wrote a commentar y on Euclid?s book, The Elements, in which he said:

many

together

Theatetus?s,

and

The of also

improved the daily lives of their

bringing

citizens, in the constr uction of

demonstration the thing which were

to

irrefragable

towards

golden. If you continue this process and then dr aw quar ter circles through ever y square, you get a golden spir al. To Reproduced under CC License from Wikimedia

Math Mag 2017

make

a

One of the oldest and most complete diagrams from Euclid's Elements of Geometry is a fragment of papyrus found among the remarkable rubbish piles of Oxyrhynchus in 1896-97 by the renowned expedition of B. P. Grenfell and A. S. Hunt. It is now located at the University of Pennsylvania. The diagram accompanies Proposition 5 of Book II of the Elements, and along with other results in Book II it can be interpreted in modern terms as a geometric formulation of an algebraic identity.

Fibonacci

SAR High School

Math Mag 2017

19

SAR High School

inter ior angles will sum to

line can be dr awn par allel to a

could not prove. Er ic Temple

only somewhat loosely proved by

time, Ptolemy I founded The

Book 1 contains 5 postulates

more than 180. On the other

given line.? Although it is

Bell, a mathematician from

his predecessors. This man lived

Museum, the fir st national

that use ter ms defined earlier

side of the tr ansver sal, they

easier

the early 1900?s, provided an

in the time of the first Ptolemy.

univer sity. To ensure that the

in the book. That way, if a

will sum to less than 180. If

mathematicians agreed that it

interesting

For

univer sity had credibility, he

reader was confused, he could

extended

indefinitely,

the

couldn?t

fr ustr ation

first

hired Euclid. Euclid?s book on

refer back to the ter ms in the

lines on

the side of

the

postulate.

(Ptolemy), makes mention of

geometr y was highly regarded,

beginning of the book. Euclid

tr ansver sal

Euclid? He is then younger that

and

also

?common

inter ior angles that sum to

the

univer sity were ver y popular.

notions? about equalities and

less than 180, will eventually

inequalities. Today, we would

inter sect. The lines of the

call these ?common notions?

inter section, along with the

axioms,

tr ansver sal,

Archimedes,

immediately

who came

after

the

pupils of Plato, but older

than

Eratosthenes

lectures

at

the

and

Archimedes, the latter having been

his

contemporaries,

as

Eratosthenes somewhere says.?

[Chr is Hayhur st, Euclid the

The

Elements

was

widely

recognized at the time to be the best book available on geometr y. Unfor tunately, the or iginal copy is lost. Dur ing

Great Geometer, , 52].

Euclid?s time, if a student Proclus

self- evident

statements.

The postulates

and common notions are used to der ive the many theorems developed in the rest of the

will

for m

a

tr iangle, whose three angles sum to 180 degrees.

The

fifth

postulate

significantly

Out of all of Euclid?s

Proclus

lived

the or iginal book, and it is

postulates, the most famous

a

dur ing the ?time of the fir st

reasonable to assume that the

one is the fifth. It states, ?If a

between two points or all r ight

Ptolemy,? who

ver sion of his book that we

line segment inter sects two

angles

Egypt in 323 BCE. Therefore,

now

str aight

lines for ming two

another. The most fr ustr ating

Proclus infor ms us that Euclid

modified

the

inter ior angles on the same

aspect of the fifth postulate is

lived around 300 BCE.

or iginal. The book was ver y

side that sum to less than two

that

well or ganized, which made

r ight angles, then the two

seems provable. However, like

geometr y easier for students

lines, if extended indefinitely,

the other four, it is not. Euclid

to follow. He developed more

meet on that side on which the

could

complex ideas by building

angles sum to less than two

assumed it was tr ue in order

upon the most basic ideas. The

r ight angles.? This can be

to continue wr iting his book.

Elements contains 13 books.

explained as the following. If a

In the effor t to replace the

Book

tr ansver sal

postulate,

studied

in

Plato?s

academy but wasn?t taught by Plato, since Plato wasn?t alive by that time. After studying there, he moved to Alexandr ia, Egypt. The city was thr iving with intellectual conver sation.

Soon after

his ar r ival, he

wrote a book about geometr y entitled, The Elements. He then founded a school

in

know

1

is

a

ver sion

is

geometr y. It

about

slightly of

plane

defines many

Math Mag 2017

the same

to

for m

two

same- side

str aight are

to

can

be

equal

dr awn to

one

mathematicians, it

not

Scottish

prove

John

it,

Playfair,

of

to

the

the Fifth

Euclid?s

postulate

was

par allel

challenged

dur ing the 19th centur y. The challenge gave way to a new for m of geometr y known as

me

that

without

assumptions there is no proof.

Therefore, in any argument, examine

the

assumptions.?

[?Euclid,? The Famous People, website, http://www.thefamous

Euclidean geometr ies, people.com/profiles/eu clid- 436.php] Essentially, Bell pointed out that yes, in order

geometr y.

to prove anything some things Euclid

is

known

for

the

must

be

assumed.

The

conclusions he was able to

assumptions made must be

prove. It?s ironic, therefore,

studied,

that one of the things he is

necessar y to prove anything at

most well known for, the Fifth

all.

but

they

are

Postulate, is something he We may know ver y little about

but

a

mathematician,

inter ior inside angles that sum

proposed an alter native one in

such as ?an obtuse angle is an

to 180 degrees, then the lines

the late 18th centur y, where he

angle greater than a r ight

are par allel. If the lines are not

stated: ?through a given point

angle.?

par allel, then on one side of

in a plane, one and only one

geometr ic

adoption,

which are simple ideas such as

ter ms

common

Alexandr ia where he was a teacher. Around

lines

inter sects

widespread

postulate, such as hyperbolic

therefore, only have copies of

over

taught its

more

complicated than the other s,

insight

Postulate. He said, ?Euclid

that negates Euclid's par allel

Euclid lived. We lear n from

r uled

fifth

is

copy the or iginal by hand. We,

Euclid

the

which contains a postulate

deter mine where and when

book.

replace

Despite

non-

wanted the book they had to

that

logic

five

under stand,

same- side

to

Euclid

used

wrote

with

to

the tr ansver sal, the same- side In addition to the definitions,

SAR High School

Math Mag 2017

21

SAR High School

N on -Eu cl i d ean Geom et r y

par allel postulate. The par allel postulate

of

hyperbolic

geometr y is any statement

By Ay el et Ru ben st ei n

equivalent to the following: If

Euclidean geometr y can be

extremely fascinating.

thought of as the high school

Euclid?s axioms, the

g is any line and P is any point not on g , then there exists at least two lines through P that

geometr y that we know and

are par allel to g, as shown

love. This type of geometr y is based

on

below:

definitions,

undefined ter ms (point, line, and

plane),

and

Euclid?s

Hyperbolic geometr y can be

axioms. Euclid?s method of

thought of as the study of

There are no similar tr iangles

approaches infinity. Escher ?s

mathematical

saddle- shaped

in hyperbolic geometr y.

Circle Limit III shown below

thought

involved assuming a small set of

intuitive

postulates, theorems

axioms, and from

space.

In

hyperbolic geometr y, we work

or

on a cur ved space shaped like

deducing

a saddle instead of that of a

them.

flat plane. Working on a saddle

Euclidean geometr y can also

shaped sur face changes what

be thought of as the study of

we consider to be geometr ical

flat space. While Euclidean

basis for Euclidean geometr y,

One type of non- Euclidean

geometr y is most familiar to

were not challenged until the

geometr y

us and perhaps most pr actical,

early

centur y,

geometr y, the study of figures

non- Euclidean geometr ies are

when

non- Euclidean

on a sur face of a sphere. In

geometr ies began to emer ge.

spher ical geometr y, each line

Non- Euclidean geometr ies are

is a great circle, so any two

In

based

lines meet

tr iangles with the same angles

A depiction of Euclid?s parallel postulate.

Recreated by Ben Kepecs.

Figure 1

nineteenth

upon

one or

more

is

spher ical

in two points.

postulates that differ from

Par allel lines do not exist in

those of

Euclid, star ting

spher ical geometr y, because

from the negation of the

all lines in spher ical geometr y

Euclidean par allel postulate.

must inter sect at two points,

Euclid?s par allel postulate,

as

shown

in

Figure

tr uths, as descr ibed below. In hyperbolic geometr y, the sum

illustr ates the Poincaré disk In

hyperbolic

space,

the

concept of per pendicular to a line can be illustr ated as seen in the picture below on the left (as opposed to the concept of per pendicular to a line in Euclidean geometr y.)

model of hyperbolic geometr y.

In the Poincaré disk model of hyperbolic

geometr y,

hyperbolic

points

represented

by

points

inside

a

are

Euclidean bounding

circle. Hyperbolic lines are

of the angles of a tr iangle is

The ar tist M.C. Escher was the

represented by circular arcs

less than 180°.

fir st known ar tist to create

or thogonal to the bounding

patter ns in

the hyperbolic

circle. In Escher ?s work below,

plane. As the bounds of a

the backbone lines of the fish

hyperbolic

in Figure 3, are hyperbolic

hyperbolic

have the same areas.

geometr y,

plane

approached,

the

are plane

lines.

2.

which is one of his assumed

Math Mag 2017

axioms, as we know it today,

Hyperbolic

geometr y

states: through a point not

another type of non- Euclidean

on a line, there is no more

geometr y. The fifth postulate

than one line par allel to the

of hyperbolic geometr y is the

line, as shown in Figure 1.

negation

of

is

the Euclidean

SAR High School

Math Mag 2017

23

SAR High School

Figure 3 While Escher ?s ar twork ser ves

crocheted

as

stitches

two- dimensional

representations

of

a

mathematician

as shown in yellow in the

Similarly, hyperbolic planes

and cur rently a

model below. In the model,

approach infinity near

professor

there are three str aight lines

bounds, also demonstr ating

Cor nell

that pass through a point

negative

Univer sity,

exter nal to a given line (the

figured out how

one at the bottom). All three of

many applications to cer tain

to crochet such

the upper lines never inter sect

areas of science including the

a

the

orbit prediction of

at

model.

In

or iginal

within

intense

the exponential

Euclid?s par allel postulate on a

fields,

space

growth

hyperbolic plane. Taimina?s

astronomy. However, in order

hyperbolic

model makes the proper ties of

to apply hyperbolic geometr y

plane near the

a hyperbolic space visible to

to

bounds,

the eye and allow them to be

proper ties

must

be

Taimina

directly

under stood,

which

are

of

patter ns

a

with

that

exper ienced

in

a

tactile way.

grow

exponentially. In other words,

If one hexagon is removed at

models of a hyperbolic plane.

Daina

Taimina,

a

Latvian

Str aight

lines

r atio.

can

are

flat.

be

each ver tex, pentagons for m

sewn onto the crochet texture,

between the hexagons, and the Another

proper ty

that

char acter izes the hyperbolic plane is its constant negative cur vature. The concepts of negative

and

resulting

model

basic

eventually close, as shown below. Therefore, this model (Figure

4)

demonstr ates

constant positive cur vature.

If one hexagon is removed at

cur vature can be explained

each ver tex and is instead

using hexagon tiles. In the

replaced with a heptagon, the

image above, three hexagons

sur face will bend; however, it

come together at each ver tex,

will never close, as shown in

gener ating 360 degree angles

the

and

Therefore,

demonstr ating

zero

SAR High School

negative cur vature.

will

positive

cur vature. In other words, the

Math Mag 2017

its

crocheted models. tiles

a constant

fields,

and

Figure 4

next

in

tr avel,

Escher ?s ar twork and Tamina?s

of stitches from one row to the

tactile

other

gr adational

beautifully exhibited through

can

three- dimensional,

objects

demonstr ating the untr uth of

Taimina increased the number

to create a

has

order to imitate

hyperbolic plane, crocheting be used

cur vature.

Hyperbolic geometr y

line,

the

model

demonstr ates

Math Mag 2017

to

the

this

r ight. model

constant

25

SAR High School

Pascal ?s Tr i an gl e an d Ex p on en t i at i on

By Joey Yu d el son Pascal?s Tr iangle is a fairly well- known

mathematical

cur iosity. Though it is built using such a simple method, there are a sur pr ising number of fascinating patter ns within.

This

ar ticle

discussing ver y

will

the

cool)

not

be

(admittedly

links

between

Pascal?s Tr iangle and binomial coefficients, or power s of two, or

even

combinator ics.

Instead, we will discuss a lesser- known

patter n:

the

usage of Pascal?s Tr iangle to exponentiate. We will do this by

star ting

with

squar ing

and

then

number s

gener alizing this to arbitr ar y power s.

In

order

under stand

the

gener alization,

let?s

discuss

k- simplexes.

A

k- simplex,

for

(Reproduced from Wikipedia. Licensed under CC Attribution 3.0 License.)

to fir st

are of the for m xr /r !, where

An impor tant thing to note is For the

our

number s

to

dimensions.

higher

each

k- simplex

pur poses of this paper, we will

Pascal?s

use the function

tr iangular number s make up the

Tr iangular

Tr iangle.

third

tetr ahedr al

number s are those of the for m

Math Mag 2017

that

cor responds to a diagonal in

pur poses, can be thought of as a gener alization of tr iangular

can be represented by S4, etc.

diagonal, the

The the four th

Con st r u ct s

x(x+1)/2 The 3 dimensional

So,

diagonal, and

analogue,

tetr ahedr al

the k- simplex of tr iangular

k- simplexes inherent in the

Ph ot ogr aph by Talia Kr au sz

number, can be found by

number s can be represented

tr iangle

x(x+1)(x+2)/6. All k- simplexes

by S3, tetr ahedr al number s

exponentiation by way of it

SAR High School

Math Mag 2017

a

27

are

so on. The what

make

SAR High School

possible,

using

a

linear

combination.

Interestingly, it is in fact r ather

simple

number s

to

square

using

Pascal?s

Tr iangle. You merely have to find the xth tr iangular number (The xth number in the 3rd diagonal) and add it to the (x- 1)th tr iangular number. You can car r y out this process on Figure 1.

tr iangular number s will be

elements of the k th diagonal,

It

th

completed. Exponentiation in

is quadr atic. The tetr ahedr al

tr ailing back a total of k

Pascal?s Tr iangle, but there

Pascal?s tr iangle can be done,

for mula,

elements, each with their own

does indeed exist a way to

albeit with a ver y involved

cubic, but we r un into another

unique

that

recur sively gener ate a given

algor ithm. Simply solve the

problem. We will want to use

change with each different

element of the tr iangle based

fir st

S4(j),S4(j- 1), and S4(j- 2) to get j .

exponent. We can sum this up

on the elements above it.

Tr iangle, where

However, since the leading

with the equation on the

[x(x+1)(x+2)/6],

is

3

3

coefficient of each one is [j /6],

coefficients,

bottom.

we will need to multiply the different tetr ahedr al number s by coefficients summing to 6.

The

pur pose

for

wr iting

M k(i+1) as such shall become

The tr ick is that each element

Figure

is a linear combination of the

2

shows

Euler ?s

clear shor tly.

number x is simply to find the

for m

x

th

value in

the diagonal

cor responding sequence

of

to

ak

should

be easily

actually

indeed

oper ation, and expressing it as

defined recur sively, providing

a linear combination r ather

some ver y nice symmetr y with

The best way to think about

than

Pascal?s Tr iangle. (It should be

the coefficients is as elements

multiplication

The question then becomes:

noted that M k (0) and M k(z),

of another tr iangle, discovered

expressible as the sum of

unnecessar y, though it could

How to gener ate the next row

z>k both equal 0). The for mula

occasionally be more efficient

of the Tr iangle?

itself is M x(y)= [yM x- 1(y)]

the

to use pre- calculated values of

previous

helped a lot.

tr iangular

elements on top was the same for

predictably, is yes. However,

all the elements

will

in a given row,

require the use of higher

with the precise

order simplexes. The above

changing

S3(j)+S3(j- 1)=j 2. However, if we

Math Mag 2017

working

as

exponentiate

simple

repeated is

usually

Tr iangle

to

r ather

than

doing it out the long way. The cool thing about this, however, is not its pr acticality, but r ather the way it unites two seemingly dispar ate topics. An entirely

different

way

of

thinking about exponentiation

across

the row. Figure 3

want to gener alize to cubic equations,

r ather

is hidden in Pascal?s Tr iangle.

propor tion

equations utilize the fact that

a

Euler ?s

coefficients of the

answer,

exponentiation

+

the sum of the

squar ing number s. Can we go

higher

is

In other words,

number.

The

function

[(x- y+1)Mx- 1(y- 1)].

However, this only works for fur ther ?

your self, from

with the or iginal Tr iangle also

number s and then to add it to the

equation:

This

Tr iangle sums to x!. Symmetr y

the way to square a given

earlier

impr actical. Exponentiation is

coefficients?

that all exponentiation of the

the

Euler ?s Tr iangle (e.g. M 5(3)=66).

fact that the xth row of Euler ?s

=[x +x+x - x/2]=x . In this case,

to

proceed

Admittedly, this is extremely

deter mining the values of the

star t to get odd. We can see

and

the

number in the xth row of

previous work or the handy

2

is

some notation. M x(y) the yth

we go about

2

k

explain, let me fir st introduce

How should

2

Euler ?s

The fir st few rows are easy to

The algebr a showing this is

In gener alizing this, things

of

two elements above it. To

Tr iangle.

using

[x(x+1)/2]+[(x- 1)(x)/2]

rows

according

and called Euler ?s Tr iangle.

compile for

also fairly simple.

k

exponent,

by Euler in the 18th centur y

Now we have one task left.

tr iangular number s: x(x+1)/2.

Thus, the last step has been

exceedingly simple r ule of

are 1, 4, and 1, respectively.

for

the

star ting at the a element and

It is fairly easy to show this equation

follow

pointless because the for mula

In this case, the coefficients

the

doesn?t

below explains this in a

with

clearer fashion:

SAR High School

Math Mag 2017

29

SAR High School

Th e Gam e of L i f e

By Tam ar Gel l er Bor n from a simple set of r ules, the Game of Life has become one of

He continued to create similar

life? by tapping on it and

games including Philosopher 's

tur ning white. You can ?tur n

Football

on? any amount of cells, in any

also

known

as

"phutball" which eventually led

patter n

of

your

choosing.

him to give bir th to the Game

When you press ?star t? the

of Life.

r ules are applied and the cells either die, remain alive, or

the most

Conway's famous game grew

complex mathematical games

out of his quirky interest in

today,

cellular automata. Along with

However, despite the simple

Neumann

and insisted to inter viewer

his peer s John von Neumann

r ules, the simulations proved

cellular

with

its

endless

reproduce. descr ibed

this

patter ns and configur ations.

passionate

The game's unique qualities

backgammon as he was about

Mark

Scientific

and Stanislaw Ulam, Conway

to be ver y complex. Each cell

twenty- nine possible states

and applications star t with the

math, and worked just as hard

Amer ican that he was, ?much

engineered a new simulation

would have a value, or state

for each cell and in which

creator himself, John Conway,

to

skilled

more interested in the theor y

in which patter ns of "cells" are

and would be connected to a

ever y cell is connected to the

a ver y unique man. Conway

backgammon player as he did

behind a game than the game

simulated through a ser ies of

cer tain neighbor ing cell so

cell above, below, left, and

was bor n December 26, 1937 in

to become an accomplished

itself.? He built games off of

gener ations

that they for m a one-

r ight

Liver pool,

His

mathematician. His interest in

theor ies, r ather than theor ies

simple r ules.

passion for math and science

games led to his fascination

based

England.

arose at a young age as he was exposed

to

through

such

his

subjects

father,

a

labor ator y assistant. At just

about

become

a

with the Asian board game, Go,

in

which

he

noticed

patter ns of stones and found analogies for them in number

Alper t

off

of

games,

out the elabor ate shapes and patter ns in his games. One of games

applying

and

challenged his peer s to figure

Conway's fir st

by

was

The simulation works through a gr id, where each box is called a ?cell?. While creating the initial patter n, a cell can be ?tur ned on?, or ?brought to

multi- dimensional

or

patter n.

automation

(called

the

with

?von

Neumann? neighborhood?). He

The states of the cells change

proved

at each time- step. The new

exhibited

state of a cell is computed

automation are similar to the

from the previous states of the

biological processes of self-

neighbor ing

cells

using

reproduction. Conway set out

predefined

r ules.

Von

to simplify the simulation and

that by

the such

dynamics cellular

four year s old, he was able to

theor y. Through this game, he

called Sprouts, in which there

recite power s of four, and by

explored new definitions of a

is a piece of paper that star ts

created a new environment in

eleven year s old, he knew he

"number ", which included the

off with some spots on it. Any

which cells can have only two

wanted to be a mathematician.

"tr ansfinite"

number s

player can join any two spots

states- "on" or "living", and "off"

He

continued

to

pur sue

discovered by Geor g Cantor.

by dr awing a cur ve, given that

or "dead". Each cell would then

mathematics and received a

While

of

the cur ve does not cross a

evaluate its eight sur rounding

B.A.

number

group

pre- existing cur ve. The player

cells and respond according to the 4 following r ules.

from

Cambr idge

his

explor ations

theor y and

Univer sity in 1959. By 1964, he

theor y

become

must then mark a new spot on

had received his doctor ate and

famous in the mathematics

the cur ve that was just dr awn.

accomplished a great deal in

world, his claim to popular

The game ends when one

the mathematics community.

fame would come from the

player cannot dr aw a cur ve

1. Any live cell with fewer than

He car r ied a reputation for

way he would explain intr icate

without

two live neighbour s dies, as if

br illiance

mathematical

the

cur ve or crossing a spot that is

quirky

for ms of puzzles and games.

already connected to three

per sonality. He was just as

He saw games in new ways

cur ves.

maintaining

Math Mag 2017

while his

still

proved

to

ideas in

crossing

The Simple Rules of Life:

another

caused by under- population.

2. Any live cell with two or

SAR High School

Math Mag 2017

31

SAR High School

three live neighbour s lives on

discovered

mathematicians

began

created a glider

to the next gener ation.

that

these

questioning the Life univer se.

gun setup that

simple r ules

Was there a Life patter n that

resulted

produced

would grow to no end? Does

never

ending

an

ever y patter n eventually die

Life

patter n

out or enter a cycle?

(Figure4).

3. Any live cell with more than three live neighbour s dies, as if by over- population.

endless

number

of

4. Any dead cell with exactly

patter ns

three live neighbour s becomes

depending

a

on

live

cell,

as

if

by

One day,

the

Richard

Guy,

a

in

a

Additionally,

colleague of Conway, noticed a

Gosper was able

initial patter n of the cells. For

patter n gener ator the Game of

patter n consisting of five cells

to show that a

example, a row of three living

Life, and it was fir st publicized

that resulted in the whole

complete Game

As a result, the cell would live

cells, also known as a blinker,

in two ar ticles in Scientific

object

of Life patter n

or die and as the "gener ations"

could

between

Amer ican magazine in 1970

diagonally on the gr id. This

could be used as a computer

progressed, and the patter n

ver tical and hor izontal (Figure

and 1971. Soon people were

object became known as a

to

would

1).

mailing

glider which, in theor y, moves

addition.

reproduction.

grow,

die,

become

alter nate

their

own

discovered patter ns, and as

fixed, or shift around on the Additionally, an L-

gr id of squares.

in

shaped

moving

one

space

per for m

ar ithmetic

playing this fascinating game consisting

were easily tr anslated into

works.

computer time may already

of

cells

could

tur n

different r ules for life and

into

a four

cell

death in the search to find the

block and become

r ules that would simulate the

fixed as such on

most

patter ns.

the gr id (Figure 2).

students

Conway called his

a

var iety

interesting

Conway

and

Math Mag 2017

his

me

to

to

While clearly computationally

the App Store and I quickly did

reader s

of

significant, the Game of Life

so.

Mar tin Gardner 's columns on

can also be used to create

These fanatics were not just

mathematical

interesting patter ns.

interested in having fun; soon,

Scientific Amer ican to create a

ser ious- minded

continually

growing

horde

of

.This

inspired

challenge

patter n implement

told

download the app "Golly" from

by the

fanatics".

of

They

Conway

game's

patter n

and

asking questions about how it

"millions of dollar s in valuable

to

boxes

instantaneously and I star ted

Time magazine noted that

attempted

of

simulations. My cur iosity grew

computer progr ams. By 1974,

Conway

since. It all star ted dur ing over at my peer s who were

with computer s, the patter ns

wasted

and have been intr igued ever English class, when I looked

forever (Figure3)

people became more familiar

have been

Game of Life in eighth gr ade

three

the

games growing

and

in

At

fir st,

I

didn't

really

under stand all of the r ules to

Life

possibly

accomplish a major step in simplifying

Von

Neumann's

model for a "living machine".

William Gosper, a fan of the column, took up Conway's challenge

SAR High School

Math Mag 2017

and

successfully

I

discovered the

33

SAR High School

this fascinating game and I

more complex ver sion of the

would create r andom patter ns

beehive

on the screen to see what

below).

happened.

What

patter n

(pictured

would

happen if I wrote my name with the boxes? What would happen if I drew a flower ? suddenly came to life. One of my favor ite patter ns is the "glider "

patter n

because

despite its simplicity, it is able

these patter ns is the ?still life?

its use for

to contr ibute so much to the

categor y. A still

life is a

lecture I attended recently at

game and is the foundation of

configur ation of cells that has

Momath, I heard from Bob

many of the more complicated

the r ight

Bosch,

patter ns.

sur rounding each cell so that

mathematician

amount

of

cells

ar twork. At a

an

ar tist who

uses

I then star ted the think about

per son

a

The glider was discovered in

there are no bir ths or deaths

mathematics and patter ns to

what

each

companion to sur vive, but

1970 (mentioned above and

from one gener ation to the

create beautiful pieces of ar t.

It

can't have too many other wise

pictured in figure 3). The

next. Common examples of

He uses the game of life, more

its

it will die. The patter ns I

glider is often created from

these patter ns are the block,

specifically it's still life's to

created

r andomly

the beehive, and the boat as

create images that remain the

be

displayed in figure 5. These

same as the gener ations pass.

combined with other glider s

still life patter ns are used in a

His most popular image, an

to make new configur ations

multitude of ways including

apple, was created using a

each

patter n

box

and

represented.

represented

life,

in

simplest for m. Each box if a

who

and

needs

found

online

gener ated

configur ations and

like the glider

can

gun which

oscillator s which are patter ns that are constantly changing to other patter ns and then back to its or iginal patter n over a number of gener ations .

Spaceships, the next categor y of patter ns, are patter ns that tr avel across the board as the gener ations

go

by.

The

produces an endless supply of

simplest for m of a spaceship is

new glider s that can tr avel

the

over long distances.

discussed before. (Figure 6)

When

you

combine

glider s,

you

can

create

patter ns

like

the

blocks,

"glider "

While

two

which

still

nowhere

complicated

as

was as Von

Neumann's vir tual machine, the

beehives and boats. (Figure 5)

Math Mag 2017

The next patter ns are called

Game

expressed

of

Life

a new

field

of

There are also many other

cellular

interesting patter ns that have

"ar tificial life", as well as a

been created using the game

comprehensive

of life. One of the categor ies of

emer gent

SAR High School

Math Mag 2017

35

automation

had

theor y

and of

behavior.

SAR High School

Al l ?s Fai r i n L ov e an d M at h ; Th e L ast -Di m i n i sh er M et h od i n Asy m m et r i cal Fai r Di v i si on Gam es objective is for each player to

people str uggle and live in

improve human welfare? Fair

receive a 1/Nth share of S. We

pover ty, while other s lead

division games apply br illiant

refer to a player ?s share as s.

carefree lives of luxur y. From

mathematical

a young age, we str ive to

innovative logic to ever yday

better our selves and get our

life;

fair

methods to make life just a

share.

We

all

recall

they

use

and

scientific

dissatisfaction with a piece of

little

cake and complaining that it

Fair

was smaller than that of our

unique r ules that are defined

siblings.

the

by the specific fair division

always

method being used. The goal

No

matter

circumstances, seems

that

it

someone

more division

games

fair. have

is

of ever y fair division game is

unhappy with his/her por tion.

for ever y player to receive a

But what if there were a

fair share.

mathematical

to

number of player s N, and the

ensure that ever y division of

booty being divided S, the

method

If we call the

games

division game, both player s

at least two nonclaimants. The

and thus the fir st player does

can consider their share to be

fir st player defines a share of

not want the C- piece to be

wor th 60% of the total value.

S; thus, he/she is called a

more than (1/N)th of S. Each

Fair

are

claimant. This claimed share

consecutive player

main

is called the C- piece. What

choice to either

kind,

remains in S is called the

become the claimant, which is

division

R- piece, and the remaining

accomplished by reducing the

games, are when the set S can

player s

C- piece, or to pass. If a player

be divided

nonclaimants.

division

are

games

into

two

The fir st

continuous

that utilizes mathematics to

division

could also go to someone else,

categor ies.

Life isn't fair. Hard- working

Fair

times, there is a claimant and

divided

By Ay el et K al f u s

theor y

thus, after a two player fair

fair in

an

infinite

of such an S set would be a

equal to (1/N)th of S, and

cake. On the other hand, there

therefore it would not benefit

are

him/her

discrete

fair

division

the

games

of

become the claimant, the par t

like

of the C- piece that he/she cuts

is

composed

player s

indivisible

cooper ate, act r ationally and

paintings.

off retur ns to the R- piece, and

are unaware of their fellow

We will begin the jour ney with

the or iginal claimant retur ns

player s?

an outline of an impor tant and

successful

only

if

per sonal

value

objects,

back

to

the

group

of

fair

The fir st player claims a piece

non- claimants.

the

that is exactly (1/N)th of S, as

The or iginal claimant would

last- diminisher method, and

the player can end up with the

feel satisfied because he/she

perception of the value within

then modify that method for

C- piece. However, the C- piece

believed the piece that he/she

S. Fair division games are

asymmetr ical

fascinating

each

are games in which player s are

player places subjective value

owed different shares of the

on different par ts of S, and

over all booty.

systems. That is to say that each per son must only be aware of his or her own

because

innovative

continuous

division

method,

games, which

Banach and Bronislaw Knaster

satisfied

developed the last- diminisher

with the results? What if

method in

there was not merely one

method is a continuous fair

method, but r ather

many,

division method for three or

joined together to for m a

more player s. All player s are

complex and evolving field

put in a r andom order. At all

Math Mag 2017

reduce

C- piece. If a player chooses to

Polish mathematicians Stefan

and

to

games. The set S in such

felt

comfor table

passes, then he/she consider s the C- piece to be less than or

and

recipient

called

play and

number of ways. An example

shared goods was equitable ever y

are

has the

SAR High School

Math Mag 2017

the 1940s; the

37

SAR High School

had claimed to have been

they are allowed to remain in

exactly equal to (1/N)th of S,

the game an additional time

and since it was reduced, the

even after they have received

or iginal claimant now believes

one share wor th 20% of S. This

that

contains

method can be applied to N =

more than (N- 1/N)ths of S. All

any number of player s with

player s are given the option to

any percentage of S owed to

play or pass. The last claimant

them.

the

R- piece

M at h Desi gn s

By Sar ah Bol n i ck ?I?m a hospital administr ator,?

gr aphs

Mr. Yechiel Rosen began to

inward

explain,

outward

while

I

star ted

of

the and

flow

of

doubting how successful this

patients, or gr aphs

It is easy to look at the world

inter view would really be. How

of

and reflect that it is r iddled

did the per son before me have

weekends worked,

with injustice and inevitable

any connection to math?! An

could help him. He

unfair situations. Yet mor ally,

engineer

began

it is our obligation to see the

scientist maybe, but featur ing

cer tain

world differently. We must

a hospital administr ator in

into

the

apply logic and expend the

Math Mag seemed str ange.

computer progr am

for

extr a effor t to create har mony

However,

designed

games,

and mutual satisfaction. Life

continued,

as

a

is not about guar anteeing that

under stand his connection to

spreadsheets,

situation in which player s are

you receive your fair share,

math and his love of the

then he had the

owed different shares of S, I

r ather it is about ensur ing that

subject.

progr am gr aph it for him. But,

utilize L, a var iable I have

ever yone else does as well.

of all the player s, or the last diminisher, keeps his/her fair share. The process begins again, except that now, a fair share

is

(1/N- 1)th

of

the

remaining R piece.

When

modifying

last- diminisher

method

asymmetr ical other wise

known

divisor

of

as

computer

Mr.

I

Rosen

began

to

to

plug

number s Excel,

a for

creating and

He decided to have a little fun In

the

addition

to

being

the

Director of Administr ation at

percentages that all player s

Jamaica Hospital for the past

are owed. I begin the game as

25 year s, Mr. Rosen has a BA in

if each player were only owed

Physics, a minor in Computer

L percentage. If three player s

Science,

are playing, P1, P2, and P3, and

and

an

MBA

in

Marketing. Being a theoretical

P1 and P2 are both owed 40%

physics

of S, while P3 is only owed

major,

Mr.

Rosen

star ted to wonder how one

20%, claimants cut the cake

might

into chunks that are wor th

gr aph some of

the

things that came up in his job.

20% of S, or ? . Thus, all

Working as an administr ator

player s play as if there were 5

meant that he had to figure

player s. Except, both P1 and P2

out how to efficiently manage

have two ?lives?, meaning that

Math Mag 2017

a

the

Mr. Rosen was not done yet.

developed. L is the greatest common

or

how

the

SAR High School

hospital,

Math Mag 2017

so

creating

with his gr aphs, and after ditching the work element altogether,

he

star ted

to

exper iment with the purely aesthetic elements of these gr aphs. The ver y fir st thing he gr aphed

was

a

simple

equation? y=mx+b, but with multiple lines. The gr aph it produced was a bunch of str aight

lines

in

different

color s. Then, Mr. Rosen took these lines and star ted cur ving them, changing cer tain values,

tr anslating it across the axis to see

what

would

happen.

Before long, Mr. Rosen had created amazing gr aphs, and once he added color s and backgrounds to them, they became tr ue works of ar t. But, he was still not done. Mr. Rosen began to wonder about tr igonometr ic

cur ves,

and

then began to exper iment with tangent lines as well. By using equations such as that shown in Figure 1, he could produce eye- opening simple

images

gr aphs

and

with the

addition of some color s.

and rotating the gr aph and

39

SAR High School

can

be

made

is

the

Archimedes? spir al (Figure 5).

The designs in Figure 6 has

He then star ted to play around

had the equation of a circle

with things like hyperbolas,

super imposed

on

the

par abolas, and ellipses, as well

Archimedes? spir al,

which

as changing the thickness of

created a ?Spirogr aph? effect.

the lines or the frequencies of

The super imposition of the

sinusoids.

circle on the spir al is created

Tr anslating

the

mathematical beauty of the conic

sections

into

when

b?

is

constant.

visual

beauty, he created the piece shown

in

Figure

2.

These images Mr. Rosen could predict.

As

he

began

to

exper iment with more values and

more

complicated

equations, he began to find things that sur pr ised him: a circle made of hundreds of str aight

lines

that

would

produce swirls in the cor ner s of the gr aph, or a gr aph he thought would tur n out as a

of the var iables (Figure 3).

Although Mr. Rosen is not an ar tist, he managed to produce these stunning pieces as an outgrowth of his mathematical explor ation. Later, he was even able to incor por ate his love of Judaism by making gr aphs of Mount Sinai and of Lechem Mishnah (r itual loaves of bread), the latter displayed in Figure 4.

butter fly, but was changed into a flower by increasing one

Figure 4 Then,

adding

an

additional ter m, Ty, we can produce a tr anslation of the function in the y- direction. The challah consists of four ?br aids.?

Each br aid consists of 63 functions such that each function is tr anslated in the

y- direction

some

fr action of an amplitude

br aids are identical except each one is shifted 90 degrees out of phase of the previous set

of

?br aids?. Since the

maximum

Constr uctive

occur s when waves inter fere

functions Excel can plot in a

with each other crest- to- crest

xy- char t is 255 we now have

and destr uctive inter ference

252 functions, 63 X 4, to plot.

occur s when waves inter fere

The challah shape results from

with

multiplying

crest- to- trough. The equation

all by

of an

the ellipse.

each

other

of a circle can be wr itten as:

from the one before it.

This gives the ?br aid? its

inter ference

of

functions

number

Another interesting gr aph that

thickness. The four sets of

Math Mag 2017

SAR High School

Math Mag 2017

41

SAR High School

In Figure 7, there are two concentr ic circles, so we need 2 sets of 63 functions for the ?top? of each circle and 2 sets of

63

functions

matching

for

bottoms.

the Ty

produces a tr anslation of the functions and can be var ied to

Yet

a

This gr aph consists of two

produce

different

beautiful inter ference patter n

star s, each composed of 127

patter ns.

can be found in Figure 9.

str aight lines, where each star

many

diffr action

another

example of

The image in Figure 8 is the

is

equivalent of tapping your

distance. Var ying the distance

finger s in the middle of a pool

between the star s changes the

of water at a set rhythm. This

inter ference patter n. In Figure

would produce two sets of

9 the inter ference patter n has

circles and

as the waves

taken on the magnetic field

super impose with each other,

patter n of the two poles of a

they

and

magnet,

and

schematically in Figure 10:

constr uctively

destr uctively produce

the

inter fere

separ ated

as

by

a

small

illustr ated

Another set of remarkable, yet simple,

images,

based

on

the

are

those

equation

y=mx+b, where

inter ference

patter n. Var ying the distance between the center s of the

If we let b = 0, and var y ? from

circles

- 45째 to +45째in 255 inter vals we

changes

the

inter ference patter n.

get the adjacent plot. If we var y the ? from - 85째 to +85째 in 255 inter vals we get the plot shown in Figure 11.

When looking closely at the middle of these pictures, you can

see

spir als

some and

interesting unexpected

cur ves.

As Mr. Rosen continued to show me these gr aphs,

I

realized what people see in math.

Math Mag 2017

SAR High School

Exper imenting

Math Mag 2017

and

43

SAR High School

having fun are not usually the

Figure 12 is what I made based

an app for the iPad. So, the

things that come to mind

off of the equation M(slope)*

next time you?re bored in

when we think of math, but in

X(coordinate)

tr uth,

progr am

those

elements

are

+

made

1.

The

class, tr y out one of these

the

lines

progr ams to see what you can

essential. Mr. Rosen found ar t

multicolored, while I set the

and creativity embedded in his

background as light pink.

M at h em at ics M eet s

come up with!

Realit y

gr aphs and realized that by just playing around he could

My per sonal favor ite of Mr. Rosen?s images is the butter fly

discover new things.

When the inter view ended,

mentioned

cur iosity got the best of me,

produced

and

and

var iations

he

I

tr ied

Spreadsheets

out

Google

myself

previously

he

all

the

created

in

(a

addition. These images are so

progr am similar to Excel). It is

beautiful and at the same time

easy to use, and with some

rooted

fooling around you can end up

equations.

in

(Please note, the editor s of Math Mag do not condone the use of Desmos dur ing class.)

mathematical

with an amazing work of your own. The image shown in

Another easy progr am to tr y out is Desmos, which is also

Ph ot ogr aph by Talya Kr on isch

Math Mag 2017

SAR High School

Math Mag 2017

45

SAR High School

communities within a lar ger

Det ect i n g Ti gh t er Com m u n i t i es W i t h i n Soci al N et w or k s Usi n g t h e Gi r v an -N ew m an Al gor i t h m

network. One of the more

For example, in Figure 1, some

other

of year s. They date back to

nodes are closer connected to

other s in the network? This is

cavemen dr awing on walls of

other s

where

caves to communicate with

positioning in the gr aph and

?communities?

each

other. However, over

the number of and strength of

play. Analysts were cur ious to

time, the world?s networks

the edges connecting them.

see if there are communities

based

on

their

than

each

they are with

the

detection comes

of into

Network analysts examine the individual objects within a network (called ?nodes" and the

connections that link one with ?edges?).

Analysts study the way some are

Math Mag 2017

more

Centr ality can be identified by

centr ality of some nodes. The

how

are

Gir van?Newman algor ithm is

connected to it.This is called

a method used to divide lar ger

degree

groups of people into smaller

tightly

These studies lead to the question: are there subgroups of cer tain nodes that are more

centr ality.

Degree

centr ality can be evaluated

cluster s.

with a simple equation as

contingent

shown

assumption

in

Figure

2.

The

The algor ithm

is

upon that

the

tight- knit

Repr odu ced fr om W ik ipedia. Licen sed u n der CC Attr ibu tion Licen se..

Figu r e 3: An u n dir ected gr aph of 34 n odes an d edges r epr esen tin g th e r el ation sh ips between differ en t stu den ts?par ts of Zach ar y?s Kar ate Cl u b.

acquire the dendrogr am (a

betweenness of a cer tain edge

relatively high betweenness).

br anching

is analyzing how strong that

According

and

represents the relationships of

par ticular edge is (if a node

Newman, if you r ank edges by

similar ity among a group of

and

has a strong connection with

their betweenness, and then

entities) shown in Figure 4.

Moses

Boudour ides was intr igued by

another

edge

gr adually remove the edges

this

connecting

be

with the highest betweenness,

an

stronger than the connection

you will be left with separ ate

titled

with a node with which it's not

communities.

?Introduction to Community

as close). In connecting those

Detection in Gr aphs.? In this

two

essay, Boudour ides lays out an

centr ality is an indicator of

exposition of

a var iety of

how

mathematical

methods one

specified ver tex has, and how

could use to identify a set of

strong those connections are.

social

Figu r e 1: Som e n odes (depicted by th e col or ed dots) h ave str on ger con n ection s with oth er s based on th eir position in th e in dir ect gr aph an d th e n u m ber of an d str en gth of th e edges con n ectin g th em .

edges

have

Greek

Repr odu ced fr om W ik ipedia. Au th or is Cl au dio Roch in i.

Licen sed u n der Cr eative Com m on s Attr ibu tion -Sh ar e Al ik e 3.0 Un por ted Licen se.

many

communities

network

network and gr aph theor ies.?

nodes

based on the betweenness

other words, identifying the

relationships.

(called

is.

node

betweenness of each node, we

eventually,

str uctures through the use of

communities

two

positions within the gr aph of

another

of

br idges

letter s, then newspaper s, and

and

node is how popular that

3.

cer tain edge, or the weight. In

mathematical features of their

?ver tices")

one could detect the existence individual

Figure

algor ithm,

The centr ality of a specific and

in

algor ithm

cr ier s, then

social

Gir van?Newman

shown

each other by edges that act as

through

investigating

represent the conclusions, as Figu r e 2: wh er e k = th e degr ee of th e ver tex, i=th e ver tex, j=th e oth er n odes th at ver tex i is con n ected to, an d A= adjacen cy m atr ix A.

inter preted as the ?tr affic? of a

identified on the basis of

of

indirect gr aph is made to

is

algor ithm

the

Using

has lear ned to communicate

analysis (SNA) is ?the process

an

in

communities are connected to

of individuals that can be

Social

on

per son

betweenness of edges can be

have advanced, and society

moder n

community

This method

disper sed to

?Betweenness?-

and Mark Newman.

around for tens of thousands

technology.

network was are analyzed, an

as

invented by Michelle Gir van

closely

town

known

based

interconnected than other s.

1972. When the data of how close each

detection.

Social networks have been

Using the

prevalent methods used is Centr ality- based

By Ri t a Fed er connected

W. Zachar y from 1970 until

mathematician scientist question,

subsequently introductor y

wrote essay

and

SAR High School

node,

the

them

ter ms, many

Math Mag 2017

will

betweenness connections

a

between

the

(that

to

Gir van

the

Gir van- Newman to

r ank

diagr am

the

that

The example that is used is Zachar y?s

Kar ate

Club.

Zachar y?s Kar ate Club is a famous study network

of

done on

college

a

kar ate

clubs that was done by Wayne

47

Figu r e 4: Den dr ogr am depictin g th e specific r el ation sh ips between u n iver sity stu den ts

SAR High School

his dendrogr am can be read

one community because they

students belonged to which

by the ver tical

are

gr ade

lines. The

all

connected

by

by

using

the

height of the ver tical lines

hor izontal lines. Ver tex 26 and

Gir van- Newman Algor ithm. In

indicate the closeness of a

ver tex

relationship. The shor ter the

communities

line,

have

their

own

doing this, I

because

they

genius

hor izontal

line

communities and using this

connecting them to any other

method. By simply collecting

ver tices 32 and 33 are much

ver tices

communities.

infor mation from a var iety of

Game theor y is, put simply, a

Now, this is all ver y vague. I'd

dilemma. Let's put this all into

closer than ver tices 2 and 27.

When there is no connection

people, you can tell who is

way to win in life. Through

like to discuss a more specific

our payoff matr ix to make it

Using

by a hor izontal line, there is a

connected to whom. Social

application of game theor y,

aspect

clearer

split

networks have been around

you can consistently find the

problem

dendrogr am at a specific point

for

pr isoner 's

and subsequently identify all

infor mation

the individual communities. In

the

closer

relationship.

For

the

method,

the

example,

Gir van- Newman

we

can

cut

the

9

are

no or

between

communities.

realized

of

the

Gam e Th eor y : M ak i n g a Gam e Com p et i t i v e

identifying

By Ad i n Gi t i g of

game

theor y? a

known

as

the

2).

Keep Silent Snitch

and

the

best cour se of action (as long

extr acted

from

as you can feed the theor ies

classic example of this goes as Keep

(- 1,- 1)

(- 10,0)

them can be used in a var iety

the

follows:

(0,- 10)

(- 5,- 5)

Zachar y?s Kar ate Club, the

of

cr unch, and find the cor rect

cr ime, who are both selfish,

dendrogr am is cut at a depth

communities within a lar ger

for mula

are captured

equal to 29. The result of this

social

political,

situations, though this is not

However, the police have ver y

cut is shown in figure 5.

adver tising,

research

100 percent foolproof.) There

little evidence against either of

As I said, this is the classic

groups can design algor ithms

are really only three things

them, and know they will only

example, but in actuality a

for the best way to attr act

game

for

go to jail for a fr action of the

pr isoner s dilemma refer s to

those specific

something to be a ?game? to

time that they deser ve. A

any situation in which an

which this theor y can

smar t

individual's goals conflict with

Figu r e 6: Th ese 5 com m u n ities of th e Kar ate n etwor k ar e detected as a r esu l t of th e

In order to test the application of this method in my own life, I inter viewed five freshmen, five sophomores, five junior s, and five senior s. I collected data from all 20 students on their relationships with the other inter viewees on a scale Figu r e 5: At den dr ogr am depth equ al to 29, five differ en t com m u n ities ar e detected.

Five

communities

are

identified by this dendrogr am as shown

in

figure 6. A

community ends where there is a ver tical line that has no connection.

For

example,

ver tices 10, 4, 16, 6, and 5 are

of one to ten (one being strongest). After collecting all the data, I was able to create an

indirect

gr aph,

centur ies,

ways.

based

By

identifying

network,

on

and

communities

the

infor mation

cor rect

number s for

theor y

to

different

requires

be

dilemma.

(Figure

Two

lawyer

par tner s by

in Snitch

the law.

makes them

Figu r e 2. A pr ison er s dil em m a. Pl ayer 1 pl ays th e l eft side, wh il e Pl ayer 2 pl ays th e top.

separ ately,

a group?s goals. Univer sally,

called

without being allowed to talk

their choices are either to

one of the simplest ways to

?payoffs?). A ?payoff matr ix?

to each other. If they snitch on

cooper ate with the group, (in

identify

(Figure 1) does a ver y nice job

their

communities, and is used by

of

many

the

infor mation,

extr acted from the data. The

applied:

Gir van- Newman algor ithm is

and

world

individual groups to

subsequently

around identify appeal

player s,

results

str ategies,

(also

each

an

offer,

par tner

and

their

our example, to keep silent) or

of

this

par tner keeps silent, they will

to defect from it (in our case,

least

for

go free while their par tner

snitching). Similarly, there are

and

simple games, as it conveys

faces 10 year s in jail, and vice

univer sal payouts: T>R>P>S. A

to

the different results that can

ver sa. If they both snitch, they

Temptation

occur when different player s

will each receive five year s,

Reward

employ par ticular str ategies.

while if they both stay silent

reward (- 1), a Punishment for

the police will have no more

defecting reward (- 5) and a

evidence

Sucker

different communities.

presenting

all at

and Str ategy A

subsequently a dendrogr am, representing the infor mation I collected. Once this was done, I was able to identify which

Str ategy B

than

SAR High School

when

they

reward for

(0),

a

cooper ating

reward

(- 10).

Our

Str ategy A

(Payoff,

(Payoff,

star ted and both will be locked

example can be standardized

Str ategy B

(Payoff,

(Payoff,

up for 1 year. These cr iminals

into a univer sal

now

dilemma

face a choice, or

Math Mag 2017

a

pr isoner 's

(Figure

3).

The reason why this problem

Figu r e 1. A sim pl e gam e. Pl ayer 1 pl ays th e l eft side, wh il e Pl ayer 2 pl ays th e top.

Math Mag 2017

The

49

SAR High School

Cooper ateDefect Cooper ate(R,R)

(S,T)

Defect

(P,P)

(T,S)

regulations, expectations,

societal and

other

restr ictions are in place to tr y and

convince

people

to

cooper ate, but sometimes it's

Figu r e 3. A u n iver sal pr ison er 's dil em m a.

Pl ayer 1 pl ays th e l eft side, wh il e Pl ayer 2 is

Black

Red

introduce

Black

(6,5)

(0,9)

into the game. I'm going to

Red

(8,0)

(2,4)

introduce four ; a new player,

Figu r e 5. Ou r asym m etr ic car d gam e. Pl ayer 1 pl ays th e l eft side, wh il e Pl ayer 2 is th e top.

another

element

another str ategy, a new r ule, and

probability.

That

new

not enough. In this paper I will

down $5- a fairly lar ge sum in

player will be me, God. I can

is so famous, and remains a

explore how to make player s

this

the

either cooper ate, and reward

problem in almost ever ything

cooper ate. My job will be to

other player was either idiotic

them both if I'm satisfied with

we have issue with today

play god.

or altr uistic enough (in this

them, or punish either one of

game, both not likely) to repay

them, if they defect. The new

the favor, there is no way to

matr ices are displayed below.

(global war ming, taxes, slow WiFi,

water/food

unemployment

and

inflation/deflation, to name a few) is because, looking at figure 2, the payoffs are so obviously better if ever yone cooper ates.

However,

put

your self in Player 1?s shoes. If Player

2? your

par tner? cooper ates,

you

stand to either ser ve one year in pr ison or get away free by defecting. Of

For this task, we?re going

shor tages,

to use a different example.

This will be in the for m of a card game. Par tner s will be assigned. Each per son will receive one red and one black card. Playing your black card nets your par tner $3, while playing your red card ear ns you $2. This game is played over and over again, for n rounds.

cour se you'd

snitch! And if your par tner tells on you? You can either get revenge by forcing him to

while he goes free. You're forced to defect here too. You don't even have to know what your

par tner

is

doing? defecting is the better choice. Except it's not, because the payoffs say so and because we don't want the world to work that way. Gover nmental

Math Mag 2017

recoup your losses. So how do we

encour age

cooper ation?

Well, for star ter s, we could make this game asymmetr ic.

This will make it even less fun,

As you can see, I've punished each player in their respective ?punishment?

disadvantage, but as God, I'll be

sure

Here,

to when

change Player

Cooper ate

Cooper ate

Defect

(9,6)

(3,10)

that.

1

Defect

(11,1)

(5,5)

cooper ates he sends $5 to his

(3,3)

cooper ates

(0,5)

However, Player 2 can defect

he

pushes

$6.

Figu r e 6. God does n ot in ter ven e. Pl ayer 1 pl ays th e l eft side, wh il e Pl ayer 2 is th e top.

for a gain of $4 ($9 with the Red

str ategies.

However, this r ule requires a

especially for the player at a

Red

(5,0)

(2,2)

Figu r e 4. Ou r car d gam e. Pl ayer 1 pl ays th e l eft side, wh il e Pl ayer 2 is th e top.

Cooper ate

(2,5)

(- 4,9)

can only ear n $2 by defecting

Defect

(4,0)

(- 2,4)

($8 after adding the $6 if

make for a ver y interesting,

Player 2 cooper ates and plays

fun, or var ied game. There's

black). Although establishing

nothing to stop both player s

asymmetr ic payoffs is a star t,

from defecting ever y tur n, if

it's still ver y clear that Player 2

they want to win. Consider

will defect and have no chance

either

the

of losing. Player 1 still needs

mistake of cooper ating, even

something more to have any

for one round. They would be

chance of winning. We have to

making

Defect

push to him), while Player 1

As you can see, this wouldn't

player

Cooper ate

additional $5 Player 1 might

ser ve five year s alongside you, or get stuck with ten year s

unless

par tner, and when Player 2 Black

Black

game- and

SAR High School

Figu r e 7. God in ter ven es, an d pu n ish es Pl ayer 1. Pl ayer 1 pl ays th e l eft side, wh il e Pl ayer 2 is th e top.

Cooper ate

Defect

Cooper ate

(6,1)

(0,5)

Defect

(8,- 4)

(2,0)

condition to be filled

dilemma. The game is once

before it comes into

again

play. By default, I will

competitive.

To

find

and the

cooper ate, but if a player

per fect mix of cooper ating and

defects, there's a 50%

defecting that would make you

chance I will punish that

the most money, you would

player.

each

have to find the equilibr ium

defect, it is 50- 50 which

point. This is the point at

one will suffer. There's

which no par ticipant in the

one more catch? while I

game can gain any more than

am punishing a player,

they cur rently are; they can

my attention is on him,

only lose money by changing

and the other player is

their

free to do whatever he

this ?per fect str ategy? is my

likes without

of

next step for fur ther research.

Now,

Fir st, lets just examine how

what's the best str ategy?

our game plays out where

It's not nearly as clear

each player defects each tur n.

If

they

consequences.

Figu r e 8. God in ter ven es, an d pu n ish es Pl ayer 2. Pl ayer 1 pl ays th e l eft side, wh il e Pl ayer 2 is th e top.

Math Mag 2017

interesting

as

the

51

fear

str ategy. Investigating

pr isoner ?s

SAR High School

It?s clearly weighted towards

while the other has a six

As you can see from the end

come across in ever yday life is

Player 2. However, did he

point, which skews the

payoffs, this makes the game

astonishing,

simply get lucky with God?s

game. To make up for, that

much

implications that has for our

punishments? Let?s ver ify, by

I could change the payouts

However,

playing it through once more.

to be more equal, or I

overbalanced it in favor of

impor tant.

could impose a r ule on

Player 1, as seen in the end

inter actions with society are

player 2 in which he could

score

To

miniature pr isoner dilemmas;

only defect 4 out of five

compensate for this, I skewed

just keep in mind that the best

tur ns. Game 3: Player 2

the odds of being punished

payoffs come when ever yone

must cooper ate 20% of the

more heavily towards Player 1.

cooper ates.

time.

He now has a 60% chance of

more

of

it

balanced. seems

iter ation

1.

I

and

decision- making Many

is

the equally of

our

being punished, compared to Game 3: Iter ation 2 and

Player 2?s 40% chance. These

the odds are 60- 40 in

restr ictions tur ned this game

Player 2's favor.

Player

2 still

wins

by

into a 3 point contest (Game 3,

a

landslide. Now, what if Player 1

Game 3: Iter ation 3 and

Iter ation 2) compared to a 20

cooper ates at inter vals. Will he

the odds are 60- 40 in

point contest when we star ted,

be able to score more points?

Player 2's favor.

but Player 1 was now winning consistently.

After r unning these games, more ideas come to mind on

Although these games may not

making this game more fair.

have completely solved the

Fir st, I would have to change

consider able

the (Defect, Defect) payouts

making a two player game

when either player is being

interesting and still fair, they

punished, because cur rently

made a good star t. Fair ness is

one has a 2 point difference

of cour se a value impor tant to

problem

of

any system, and while you may not find ever ything in life interesting, that's definitely a requirement functioning

for game.

any If

you

remember, at the beginning of this ar ticle I refer red to game theor y as a way to win life. The amount

of

applicable

situations for game theor y we

Math Mag 2017

SAR High School

Math Mag 2017

53

SAR High School

Th e L at est W ay t o H u n t Ser i al Cr i m i n al s: Geogr ap h i c

By Rach el Coh en Pr of i l i n g

nthe buffer zone, which is an

If a cr ime occur s inside the

increases outside the buffer

area around the location of

buffer zone, then

zone.

cr ime

1

zero, meaning the fir st ter m

other wise. A cr iminal?s anchor

will not contr ibute to the

point is more likely to be

over all result, but the second

far ther from a cr ime within its

ter m will. The smaller the

buffer zone, and closer to the

denominator of the second

cr ime outside of the buffer

fr action, the bigger the whole

Geogr aphic profiling is the

cr iminal?s

to

For mula is used to deter mine

zone. Therefore, the points

number, resulting in a greater

method of using mathematical

commit the cr ime and their

the cr iminal?s anchor points

with the highest probability

probability that the anchor

logic

choice

are

(i.e. the locations in which an

that the cr iminal lives are on

point is indeed at (Xi , Yj ). This

This

actually greatly affected by the

offender lives or works) based

the per imeter of the buffer

denominator decreases as the

system is extremely helpful in

offender ?s mental spatial map

on past cr iminal activity. The

zone.

B,

distance from the test point to

many cases, such as those that

and daily inter actions in their

for mula

indicates the likely size of the

the cr ime site increases. This

To demonstr ate the function

contain a lar ge number of

non- cr iminal

For

probability of an anchor point

cr iminal?s buffer zone.

prevents the anchor

of

suspects

expansive

example, the offender would

at any given location based on

Then

from being too close to the

cr ime zone. Kim Rossmo, a

be most likely to commit their

past attacks and the cr iminal

whether or not the test point

cr ime site, which would be

detective inspector

fir st cr ime close to home,

routine

offender.

is in the buffer zone of cr ime n

r isky for the offender. This

Vancouver Police Depar tment,

work, or

common

Rossmo?s For mula, as follows,

by compar ing the test point?s

ter m

is credited with creating the

location. This would make the

is the centr al for mula used in

distance from the cr ime site n

probability of an anchor point

main

that

cr iminal feel comfor table and

geogr aphic

profiling.

to the size of the buffer zone.

increases inside the buffer

many

safe enough to attack, but the

The value of this for mula is

If the distance is greater than

zone away from the cr ime site.

are involved.

cr ime wouldn't be done so

calculated for each point (i, j)

B, the point is outside of the

These theor ies are each based

close that the cr iminal might

located at the coordinates (Xi,

buffer zone. The distance from

on different par ts of the logic

be identified. As the number

Yi) in a desired area based on a

the test point to the cr ime site

behind geogr aphic profiling.

of

number of past cr ime sites

is calculated using the for mula

the

which are located at (xn, yn)

for distance on a plane in

cr iminal's level of confidence,

where n takes on each cr ime

taxicab

resulting in more widespread

site

a

Xi - xn+Yj - yn .This is simply the

future

char acter istic

that

sum of the x and y distances of

these cr imes still usually don?t

retur ns zero when a test point

the two points. Distance is

r ange

is in the

calculated this way because it

and

pinpoint

probability

a

cr iminal.

in

an

for

for mula

incor por ates

the

theor ies that

to

the

of

the Rational Choice Theor y, the Cr ime Patter n Theor y, and the Theor y Behind the Jour ney of Cr ime all discuss different aspects

of

the

offender ?s

spatial behavior in relation to the cr ime committed. These theor ies

claim

appear ing

Math Mag 2017

that

arbitr ar y,

cr ime

another

so

assaults. too

site

life.

cr imes

increases, The Routine Activity Theor y,

oppor tunity

far

committed does

However, from

the

in

can

of

predict

the

succession. function

the

is

and

The

retur ns

constant,

deter mines

geometr y:

cr iminal?s home

resembles the layout of a city

or

by accounting for streets. This

place

of

model resembles an accur ate

work.

while the

n,

measure of distance to the Rossmo?s

aver age urban cr iminal.

SAR High School

Math Mag 2017

shows

equals

that

point

The constants g and f are chosen

based

on

data of

similar past cr imes. They scale the impor tance of a test point being inside and outside the buffer zone respectively. Often they are selected to be the same number below 1, but their exact values are of little impor tance.

this

for mula,

consider

figure

1:

the

The fir st ter m of the for mula is used if the suspected anchor point is outside the buffer zone.

The

lar ger

Figu r e 1

the

denominator, the smaller the

In Figure 1, the darker the cell,

probability that the possible

the

anchor

probability of an anchor point

point

anchor

is tr uly the

point.

higher

the

calculated

The

in that cell. One cr ime site was

denominator increases as the

placed in the center of the

distance from the test point to

figure. Notice how the cells get

the cr ime site increases. This

darker as they get far ther from

fir st ter m incor por ates the

the center point. The darkest

idea

area of the figure, where the

of

the

probability

decreasing as the distance

55

cr iminal?s anchor

point

is

SAR High School

most likely to be, is a diamond

gener ated

computer

murdered for ty eight female

Murderer s,?

incor por ates

anchor point also

which

the

progr am that calculated the

prostitutes, managed to stay

Cr imeStat III to find the tr avel

r ises. In addition,

boundar y of the buffer zone

result of Rossmo?s For mula for

unknown

twenty

patter n of the offender, also

with each kill, the

around the cr ime site. The

each cell in a lar ge gr id based

year s while being hunted. He

known as one?s jour ney to

murderer

buffer

a

on the locations of the cr ime

worked at a manufactur ing

cr ime. Neldner repor ts that

becomes better at

diamond because its boundar y

sites of a fictitious cr iminal.

company

Kenwor th,

Cr imeStat III found that 38

attacking

is all of the points equidistant

These values represented the

which was near East Mar ginal

female victims from year s 1982

avoiding capture.

from the cr ime site where

probabilities of the cr iminal?s

Way South, and lived at 21859

to 1983 were all murdered

distance is deter mined using

anchor point being in each

32nd Place South in King

approximately 6.80 miles from

taxicab geometr y. Past

cell. The cells were then

County, Washington, from 1981

the area around Ridgway?s

edge of the buffer zone the

colored

to 1985. Most assaults occur red

residence and 1.31 miles away

cells star t

getting br ighter

values of the cells, where a

near

common

from his work location. After

because the probability of an

darker red was placed in cells

locations.

Google

he was caught in 2001, this

anchor point being in those

with a higher probability of

Maps, coordinates for the last

method was then used for 41

cells decreases.

the cr iminal?s anchor point

location that the victim had

of the Green River Killer ?s

being located. The values of

been spotted, body recover y

cr ime sights. Cr imeStat III

the constants B, f, and g were

locations,

killer ?s

aver aged that his killings were

r apists, ar sonists, and more.

selected

anchor

were

6.59 miles away from his home

This

deter mined. The data inputted

and 1.52 miles from his work.

effective in under standing and

in

Therefore,

main

using logic and for mulas in

location where Ridgway came

anchor point was his job and

order to stop these dangerous

in contact with the victim

his second anchor point was

cr iminals.

from 1982 to 1983 and the sites

his home. In later inter views

where the bodies were found

with

from year s 1982 to 1984.

explained that his technique

represents

zone

looks

like

the

In Figure 2, three cr ime sites were placed, indicated by the green dots. Notice how the

by

a

depending on

to

produce

the

clear

images.

darkest area is between all three cr ime sites. This is

While geogr aphic profiling can

because the cr iminal is most

be ver y helpful in nar rowing

likely

lar ge

to

live

or

work

quantities

of

somewhere that is not too

infor mation, the process is not

distant from any one of the

a foolproof method for solving

cr ime

sites.

cases. One famous case that

were

highlights

Figures

1

and

Figu r e 2

Math Mag 2017

2

this

over

called

these

two

Through

and

the

points

this case included

the

Ridgway?s

police,

was basically the same for The computer system used in

Ridgway, also known as the

geogr aphic profiling cases is

Green River Killer. This case

Cr imeStat III. This progr am

While all cr ime locations are

was studied and evaluated

incor por ates

Rossmo?s

significant, the earliest acts

with Cr imeStat III by Rachel

for mula in order to deter mine

are most revealing about the

Neldner

distances

offender ?s

her

essay

and

daily

Murderer, Gar y Ridgway, to

?Geogr aphic Profiling of Ser ial

the

Assist Law Enforcement in the

Murderer, Gar y Ridgway, to

committed

increases,

the

Apprehension of Future Ser ial

Assist Law Enforcement in the

cr iminal?s

confidence

to

Murderer s.

Apprehension of Future Ser ial

venture far ther away from the

SAR High School

Over all, geogr aphic profiling

has

assisted

in

finding

and

captur ing countless ser ial killer s,

Th e Gr een River Kil l er ?s ar ea of pr obabil ity m ap, cou r tesy of Rach el Nel dn er.

system

is

extremely

Location s of im por tan ce in th e Gr een River Kil l er case, cou r tesy of Rach el Nel dn er.

routine,

Rachel

who

paper,

hunting each victim.

Geogr aphic Profiling of Ser ial

Ridgway,

Neldner ?s

statistics.

and

Ridgway

Gar y

in

is

for

behavior, and anchor point. As number

Math Mag 2017

of

cr imes

57

SAR High School

Ph ot ogr aph by Talya Kr on isch

In vest igat ion s

Ph ot ogr aph by Talya Kr on isch

Ph ot ogr aph by Sam m y Sok ol an d Han n ah Sch w albe

Math Mag 2017

SAR High School

Math Mag 2017

59

SAR High School

M at h Beh i n d El ect i on s

There are different situations

does have veto power when it

means that each state has a

that can ar ise from the fact

comes to laws. One final

minimum of 3 representatives

By Zach Rot h st ei n

that different player s can have

?problem? that can occur is

(2

different

an

that the candidate with the

Congressman) regardless of

election, or from ver y high or

major ity of the popular vote

how small the population is.

low

can lose the elector al college.

However, that minimum could

the

This has happened multiple

ar tificially be r aised to solve

quota is greater than all the

times and as recently as in

for the power imbalances in

votes combined, as no motion

2000. This can be viewed as a

the Dictator and Veto cases.

can ever pass. This is why a

problem, or it can be seen as a

quota is defined as less than

positive; the pur pose of the

the total number of votes (and

elector al college is to preser ve

greater than 50%).

states

This year is the year of a Presidential election. No matter how rough, grimy, or undesirable an election becomes, the results of this election is determined by the Electoral College.

The Electoral College is a weighted voting system in which each voter has a different amount of votes.

The amount of votes that someone has is called their weight. The concept of weight can be shown through the use of a scale; mainly, that each voter has a weight that can tip a scale, deciding who will win the election. For example, if there is an election where there are 5 voters (A, B, C, D, and E), A could have 5 votes, B could have 4 votes, C could have 3 votes, and D and E could each have 2 votes. In addition, the amount of votes that each person has does not have to be different. In fact, although equivalent to a one-vote-per-voter election, each of the 5 voters could have votes equal to 4, and it would still be considered a weighted voting system. In order to reach a victory in the weighted voting system a

Math Mag 2017

certain number of votes needs to be reached, called the quota.

Weighted voting systems, such as the electoral college, can have significant flaws relating to the imbalance or lack of power. The quota must be greater than half the total votes so that two sides don?t both win, and less than the total, so that it is possible for one side to win. W ithin a weighted voting system, each voter has power. The power of each voter is measured using the Banzhaf Index, which works by calculating the amount of times a voter is critical within all coalitions(B), and dividing that by the number of times all voters are critical(T), (B/ T), written as a percentage . This information is the most crucial information used when analyzing voting situations.

Also, A critical voter in a coalition is a voter that causes the coalition to lose when they drop out of the coalition, which is any combination of players who vote for one side of a motion (either yes or no). However, there can only be critical

voters in coalitions that win, as a critical voter changes the status of the coalition from a winning coalition to a losing coalition. This applies is coalitions that win even when one player drops out. This can be seen in Figure 1.

Figure 1 A - 3 votes B ? 2 votes C ? 2 votes Quota

-

4

weights

quotas.

anarchy

in

For

results

example, when

Also, a

A and C

BC

B and C

ABC

none

quota (greater than half of the

the ideals that founded our Coal ition nation.

It gives less A populated states which would nor mally be B/ C/ D

Cr itical Pl ayer s

overlooked, power to be BC influential and not passed BD over, thereby allowing the elector al college to do its CD

Non e (l oss)

job that it is in place to do.

Non e (l oss)

votes in

the case of

the

veto power (half of the votes in the case of college).

the elector al

The

difference

between a player with veto dictator

can

decisions

to

while

force be

someone

his

accepted, with

veto

power can cause no decision to be reached. The rest of the voter s in the previous two situations become dummies or useless. While we do not

T = (Total am ou n t of tim es voter s ar e cr itical ) = 6.

A is cr itical 2 tim es, B is cr itical 2 tim es, C is cr itical 2 tim es ,A?s Power in dex = 2/ 6 = 331/ 3% , ,B?s Power in dex = 2/ 6 = 331/ 3%, C?s Power in dex = 2/ 6 = 331/ 3%.

have a dictator state in the US, it is possible for one state to gain more than half of the votes. However, our president

SAR High School

A - 10 votes B - 4 votes C - 3 votes D -

be seen as being suppor tive of

motion from passing, you have

Cal cu l atin g Power In dices -

Figure 2

more votes alone than the

enough votes to prevent a

*ABC is a win n in g coal ition bu t sin ce n o on e wh o dr ops ou t wil l m ak e th e coal ition l ose, th er e ar e n o cr itical voter s.

the

1

?popular mob,? and this can

power and a dictator is that a AC

above

and

dictator is a voter that has

elector al college). If you have

Coalition Critical Voters A Loss (none) B Loss (none) C Loss (none) AB A and B

r ights

senator s

Math Mag 2017

2

A Non e (l oss)

Non e (l oss) Non e (l oss)

BCD

The Dictator, and Veto AB

A

Power cases can be solved AC using the same method; AD r aising the minimum number of elector s (the ABC

A

minimum weight) for each ABD state.

Cur rently, the ACD amount of representatives in each house of Congress ABCD

A

combined equals the total amount of weight that the par ticular

state

61

has.

This

votes

A A

A A

A = 8/8 = 100% B = 0/8 = 0% C = 0/8 = 0%

SAR High School

CoalitionCritical

D - 0/8=0%.

As we can see in figure 2 ?A? is needed in all the coalitions so they are the

most

elector al

Tr aditionally,

A/ B/ C/ D None BC

congress

None

100% of

the

power.

Figure 3 By r aising the minimum votes per state to 4, instead of 3, the dictator problem is solved, as shown by the power r ating compar isons. In addition, by r aising

the

minimum,

the

quota is also r aised.

Quota

Candidate A

B

based on anything other than

While this may seem unfair, if

may have changed their minds

the actual ballot. It takes into

it is done in a unifor m way

and don?t agree with congress

each

dictator/veto

anymore.) Also, as a goal, it

power state comes up, it can be

would be desir able to keep the

used fairly. By increasing the

election methods within the

minimum by a cer tain amount

elector al

(based on the situation), the

change the system a little not

total amount of votes is r aised,

overhaul the whole system.

account votes other than the

state had Tr ump with 40%,

Before under standing how this

fir st

the

Clinton With 35%, and a third

which then r aises the quota. It

problem would be solved, one

candidate. Therefore, it is not

par ty candidate with 25%, the

also adds winning coalitions without the dictator. Thus the

must under stand the Instant

time

a

college,

and

only

Runoff Voting system. Figure 4

100%

B

2/ 12 16 2/ 3%

0%

C

2/ 12 16 2/ 3%

0% 0%

using

the

?minimum r aising? method.

This

is

situation

a

where

three (or more) candidates

do

not reach the quota in the

13 20 (Eliminated)

States A, B, C, D, and E. Candidates X, Y, and Z. applied to states; each state

# Of 10 13 7

2

1 st

A

people

C

state. They would r ank their

A

2 nd B 3 rd C

B

C

C

B

A

place

A votes

would have a ?ballot? of how

B of

Candidate A

voted

within

their

candidates based on how the voter s voted. For Example: If a

state?s

1 st Place

B C

12 13 7

Or ganize into a new char t with

ballot

would

third.

Each

State

-

12

The values are deter mined by the amount of votes for each candidate

in

each

state.

would

receive as many ballots as they would have received elector s;

Eliminate Candidate with the

this is what ?weights? the IRV

This takes into account all

least fir st place votes. Then

system

votes, not just the fir st- place

college. Then, with all of the

count the 2nd place votes on all

ones.

ballots, IRV is conducted to

the ballots that candidate came in fir st on, and give those as

Essentially, all non- fir st place

the fir st place votes to those

votes are taken into account

candidates. Repeat this process

once the per son who they

until

voted for as their fir st option

one candidate has a this case,

was eliminated; as opposed to

Eliminate C and give B the 7

only counting fir st place votes.

Ex:

Quota

Tr ump in fir st, Clinton in

power to the losing minor ity.

major ity

A ? 7 Votes, B ? 5 votes, C ? 5 votes, D ? 4 votes, E ? 2 votes.

have

an unfair method, as it gives

fir st place votes per candidate:

cannot be solved

1 st Place

Figure 5 demonstr ates this.

Figure 5

second, and the third- par ty in

illustr ates this.

There is one problem with

to her.

C

was not recently elected(people

this

eliminated and all his votes go

the major ity cr iter ion.

the IRV does not discr iminate

A

In

votes because B came in 2nd on all ballots C was in fir st. The

Math Mag 2017

be

negative because this congress

Old

2/ 12 16 2/ 3%

could

glance, this may seem unfair,

Candidate New New

D

This

be viewed as being

weighted voting system that

6/ 12 50%

candidates.

their

a major ity. While, at fir st

None

Figure 3 demonstr ates this.

12

2nd place votes to help fulfill

r ank

However, this can

BD

dictator ?s power is reduced.

-

they

4) B would win because it has

Th is m eth od wou l d wor k for al l cases of dictator s

A ? 10 votes B ? 4 votes C ? 4 votes D ? 4 votes

where

situation.

in

par ty.

car r ies

would

IRV method takes into account

select the president

power ful

By definition, a dictator

college.

SAR High School

With IRV, voter s have ballots

Math Mag 2017

for

the

elector al

deter mine the winner. So if ever yone who voted for the third par ty candidate, who would most likely have the fewest elector al votes, second choice is Clinton, then Clinton would win that state since the third

par ty

63

candidate

is

SAR High School

States

A B C D E

receives. While it may seem

can be be applied in the

unfair to assume that the

situations listed. The solutions

voter s in states D and E would

that I came up with all attempt

Z Z Z X X

have voted for X had Z not

to get r id of any exter nal

3rd PlaceY X X Y Y

been in the r ace, it is ver y fair.

methods to solve problems

The entire premise of the

and

elector al

cur rent

1st Place X Y Y Z 2nd

The

amount

of

Z

weight

college

is

states

instead

modify

method

the

slightly,

of

r ights, and therefore the ballot

keeping things as close to the

ballots each state has, and

is the ballot of the states and

r aw voter as possible. Over all,

thus, the number of fir st place

not the individuals in the

These

votes.

states. Therefore, it is fair to

ever ything more fair by using

make the assumption that it is

only inter nal var iations and

deter mines the amount

Candidate # Of first place votes X

7

States with first place votes

Total ballots for those states

A

7

Y

10

B,C

10

Z

6

D, E

6

Eliminate Z because it has the least fir st place votes, and give those votes to X because on all ballots where Z is in fir st place. X will now have 13 votes, exceeding the quota of 12, and will win the election. This shows that even though Y had the plur ality, the IRV method decides the vote by analyzing all the places candidates come in. This also keeps the election ?weighted? because of

the

amount of ballots each state

Math Mag 2017

fair.

not

Candidate

solutions

exter nal

keep

methods.

Votes

Ti m ek eep er s Th r ou gh Ti m e

By Ben K ep ecs Time is the ubiquitous and

helped

univer sally

build

Sundials were the pr imar y

clocks

human

timekeeper s for many ancient

medium through which we all

histor y. However, without an

civilizations, and are still in

flow, at ever y moment. It has

under standing

dictated the daily cour ses and

mathematics

per meative

the

use today. They were used as

behind

these

early as 1500 B.C.E by the

cycles of our lives and the lives

phenomena, we could never

Egyptians, 560 B.C.E by the

of those who came before us;

have har nessed their power.

Greeks, and 293 B.C.E by the

one can even imagine the fir st

We will now explore some of

Romans. A sundial consists of

humans str iding across the

the

a rod or

plains in the light of the

through the ages, and analyze

gnomon, that casts a shadow,

setting sun, gazing up at the

the mathematical basis of each

and a dial plate, a plane onto

clock?s function.

which the shadow is cast. As

13

heavens

Y

10

guidance. Since ancient times,

for

tempor al

humanity has been on a quest

N/ A (his

to measure time?s passage; in other words, we have been

There

are

many

str uggling

more complicated

throughout of

X

Z

humankind

to

invent

the

different

timekeeper s

the Ear th rotates and the sun To function as a clock, a timekeeping device must use some

constant

or

regular

process to mark off equal increments of time, and the

per fect clock.

sum of those increments must

situations that can

wedge, called a

appear s to move across the sky, the position of the shadow changes;

sundials

timekeeper s

that

use

are the

motion of the shadow to tell time.

ar ise in the elector al college

The per fect clock must be able

be tr anslatable to a viewable

and other

weighted voting

to measure a length of time

result. For example, cur rent

One of the most useful types

systems. However, many of

accur ately and consistently.

mechanical

contain

of sundials is the equator ial

these problems der ive from

Many

have

small gear s that tick regularly,

sundial, shown in Figure 1.

the imbalance of power and

contr ibuted to this effor t by

and that ticking tr anslates to

the lack of major ity that were

dr awing upon diver se areas of

movement

discussed

knowledge in order to fur ther

around the face. One of the

be

the development of horology,

fir st

ties

the science of timekeeping.

hands;

power ful

New inventions, astronomical

swept around its face. It did

obser vations,

and

the

not have gear s; the daily r ise

problems. While the problems

discover y

of

fundamental

and fall of the sun sufficed to

are known, the solutions I

physical

and

chemical

make it functional. This clock

came up with in this ar ticle

proper ties of matter have all

in

this

ar ticle.

Similar

solutions could

applied

to

four

(minimum),

ver y

states,

and

way

many

other

SAR High School

civilizations

Math Mag 2017

clocks

of

clocks did instead,

was the sundial.

65

the

hands

not a

have

shadow

Ph otogr aph by Al i Em in ov, h ttps:/ / www.fl ick r.com / ph otos/ al iar da/ 16921882897.

CC Attr ibu tion Licen se.

Figu r e 1

SAR High School

The equator ial sundial is often

essentially a miniature model

If we took an expedition to the

little fur ther south. It would

dur ing cloudy days and the

displayed in public places as a

of the Ear th.

site of this gnomon and with

have to point up at a less steep

black night? They used a force

careful precision car ved hour

angle, cor responding to the

that

marks in the sur rounding land

lower latitude. From this we

unchanging

to use as a dial plate, we might

can see that to set the gnomon

clocks, and a substance that

be able to use that gnomon?s

of

was unifor m and abundant to

shadow

timekeeping

par allel to the axis of rotation,

fill

gnomon

we must set it at an angle

gr avity and water.

latitude of the sundial, in the

producing the shadow need

equal to the latitude. If we

Nor ther n

Hemisphere

not be protr uding from the

then

pointing towards the pole star

Ear th at the axis of rotation;

per pendicular to the gnomon,

and

indeed,

the plate will be par allel to the

decor ative

and

timekeeper,

functional

because

it

is

relatively easy to setup and use. The gnomon protr udes from the base at an angle equal

to

the

in

geogr aphical

the

As the Ear th orbits the sun, the Ear th also rotates around its axis, tilted at a 23.5 degree angle to the plane of its orbit, as

shown

in

Figure

3.

souther n

hemisphere pointing to the south celestial pole. The dial plate is set per pendicular to the

gnomon,

as

shown

schematically from the side in Figure

2.

M odified fr om W ik ipedia im age, l icen sed u n der Cr eative Com m on s CC0 1.0 Un iver sal Pu bl ic Dom ain Dedication .

Figu r e 3

One can imagine that if there were a gnomon protr uding from

the Ear th

along its

invisible axis of rotation, the gnomon might cast a shadow that would sweep around the Ear th as the Ear th rotated, as shown

in

Figure

4.

device.

as

a The

any

gnomon

set

our

equator ial

set

the

sundial

dial

par allel to the axis of rotation

equator,

will produce a shadow in

equator ial sundial (see Figure

almost

6).

direction

the

exact

(as long

same

hence

the

plate

name

as the

point on the Ear th, at what angle must we set the gnomon so that it is par allel to the axis Well,

if

our

gnomon were at the pole

Cr eated by Ben Kepecs.

Figu r e 6

depicted above, it would be pointing str aight up, or 90 degrees, from the hor izontal.

90 degrees Nor th also happens

Cr eated by Ben Kepecs.

Figu r e 2

to be the latitude at that point, The face of the dial plate is divided into 24 equal inter vals,

by geogr aphical definition (see Cr eated By Ben Kepecs

Figu r e 4a

each representing an hour.

Figure 5).Say the gnomon were a

There

to

them.

are

They

employed

(liter ally water thief in Greek), works via either inflow or outflow. In an inflow clock, time

is

measured the

that

amount

by of

Repr odu ced fr om W ik im edia Com m on s. Licen sed u n der CC l icen se.

Figu r e 7

flows into a

vessel. In an outflow clock,

water flows out of the bottom

time

by

of the funnel and is diver ted

measur ing the amount of

into a container. As the water

water

an

level in that container r ises, a

emptying vessel. In either

float is r aised and the float

case, the amount of water

pushes gear s that power a

measured and a known r ate

clock. The r ate at which the

of flow would be used to

hand moves around the face of

calculate how much time

the clock depends on the r ate

had passed. To illustr ate

at which

this point, let us consider an

filled, and by extension the

inflow clepsydr a used by the

r ate of flow from the funnel. A

Greeks around 325 B.C.E,

mathematically

such as the one shown in

effective way to deter mine the

7.

r ate of flow and thus to tell

is

tr acked

remaining

Figure

in

the container is

simple

but

The clepsydr a consists of a

time from

sundial as a pr imar y means of

funnel being constantly fed

water in the container is to

Why must the gnomon be set

telling time, most notably its

water by an input pipe and

calibr ate the mechanism. For

at

its

disappointing inefficacy in the

limited by an over flow pipe,

example, we could measure

The

absence of the sun. What were

such that the water level in the

how much water flows out of

the ancient civilizations to do

funnel remains constant. The

the clock in one minute, and

geogr aphical

to

latitude?

reason lies in the nature of the equator ial

sundial:

it

is

Cr eated by Ben Kepecs

Figu r e 4b

using

their

How does this sundial work? angle equal

to

dr ive

a

an

disadvantages

sever al

and

The water clock, or clepsydr a

water

the planet). At any par ticular

rotation?

constant

measur ing

gnomon is on the light side of

of

was

the amount

of

Repr odu ced fr om W ik im edia Com m on s. Licen sed u n der CC l icen se.

Figu r e 5

Math Mag 2017

SAR High School

Math Mag 2017

67

SAR High School

set up a propor tion to see how

sur face of the water

Therefore the r ate at which

(.0001 m 2) and a water level

defor mation (i.e, it bends).

many minutes have passed for

v1= Velocity of water flow from

the water level descends (v2) is

height of 1 m. Plugging these

Under

any future measured amount

the exit hole

0, and the ter m that contains

number s into our for mula for

quar tz will then vibr ate with a

of water :

v2= Velocity of descending

v2 can be eliminated.

volumetr ic flow r ate, we find:

known frequency, and that

water level

The height of the exit hole (h 1)

h1= height of exit hole

relative to the whole funnel is

h2= height of water level=

0, so the ter m containing h 1

amount of flow in 1 minute/1 minute = measured amount of flow/minutes passed This technique, depends

on

a

however, preexisting

timepiece (or an uncannily

=Density of water

can be deleted as well, to

g=Gr avitational

produce :

acceler ation

constant

(9.8

m/s2)

Solving for v1, we find that:

conditions,

constant frequency can be If we leave our water clock and come back to find 1 m3 of water has flowed out of the funnel,

we

amount

of

can

find

the

time that

has

har nessed

to

dr ive

an

accur ate clock. Atomic clocks are

more

accur ate

still,

tr acking time at the subatomic scale by using the resonance frequencies of atoms such as

passed:

Cesium- 133, which oscillates

accur ate human counter) to tr ack the calibr ation

cer tain

This

time.

specialized

at

a

cer tain

frequency

Must we depend upon the

for mula

from

Thus, using a timekeeping

(9,192,631,770

effor ts of our forebear s to

Ber noulli?s equation is known

mechanism that was invented

second) as it releases r adiation

calibr ate our mechanism? Can

as Tor r icelli?s Law. Since v1

thousands of year s ago, we

and

the metaphor ical fire never be

represents the velocity (m/s) at

can accur ately tr ack time.

tr ansitions.

lit without an already bur ning

which the water flows out of

While the timekeeper s of the

The necessity for

candle? Is there no escape

the exit hole, we must multiply

ancients were ingenious and

timekeeping has taken

v1 by the area (m 2) of the hole

effective,

not

great pr actical significance in

to find the volumetr ic flow

without flaw. And so, the

the past centur y with the

r ate

march

discover y of relativity theor y

from our circular conundr um,

make a clock?

Thankfully,

Figu r e 8

Since the exit hole and the

(m 3/s)

.

water sur face both lie adjacent the

laws

of

physics ser ve us well here. By using Ber noulli?s equation for fluids, we can find the r ate at

they

were

cycles

under goes

per

quantum accur ate on

Cr eated by Ben Kepecs

a conundr um that dictates that we must have a clock to

der ived

towards

that

unattainable goal, the per fect

and

timekeeper,

advent

continued.

its

implications, of

the

the

Global

to air of nor mal atmospher ic

So, knowing only the area of

pressure, P1 and P2 are equal

Eventually, mechanical clocks

Positioning System, and space

the exit hole, the height of the

and can be eliminated from

based on weights and spr ings

tr avel. However, the quest for

water

the

our equation, leaving us with:

were invented. Now, moder n

the per fect clock, a constant

gr avitational

(9.8

clocks have progressed

in

str uggle stretching back into

complexity

the annals of histor y, is more

The

level,

and constant

m/s2), we can calculate the

accur acy

r ate at which water flows out factor is found in all of the

beyond the clocks of

of the funnel. We can then use

ter ms, so it can be canceled

ancestor s.

this r ate to calculate the

out.

clocks take advantage of the

delimit the infinite. It is an

length of time that has passed

In the clepsydr a, the water

fact

a

endeavor to find that which is

8):

based on the amount of water

level in the funnel is held

piezoelectr ic

mater ial,

all around us, and seize that

P1= Air pressure just outside

that

constant by a stream of input

meaning

the exit hole

example, say our funnel has an

water and an over flow pipe.

subject to electr ic char ge, it

P2= Air pressure just above the

exit hole with an area of 1 cm 2

under goes

which the water flows out of the funnel.

Ber noulli?s equation states:

Where

(see

Math Mag 2017

Figure

has

flowed

out.

For

SAR High School

and

Cur rent

that

quar tz

quar tz that

Math Mag 2017

a

our

is

when

it

is

than that. It is an effor t to define

the

myster ious,

to

which is per petually slipping away.

mechanical

69

SAR High School

Bibliogr aph y A Review of Hidden Figu r es: Be t h e Fir st 6. Math and Musical Frequency

By Noa Kalf u s

Harkleroad, Leon.The math behind the music. Cambridge: Cambridge University Press, 2006.

https:/ / encrypted-tbn2.gstatic.com/ images?q=tbn:ANd9GcRMSG7bGz0hlCYv_LWG2SkTiE01 Hidden Figuresis a power ful

?human computer,? someone

Without them who knows where

film that brought to light the

who manually computes

we would be today? As Mar y

stor ies of three extr aordinar y

complicated equations.

Jackson, who would go on to

Afr ican Amer ican women who

However, soon her

become the fir st female

challenged not only NASA?s, but

mathematical talents were

Afr ican- Amer ican engineer at

the entire nation?s r acist and

recognized, and in 1960 she was

NASA, explains, ?I have no

sexist mindsets. The movie

working alongside one of the

choice but to be the fir st?.

focused on the stor ies of

progr am?s head engineer s, Ted

Whether black or white, male or

Kather ine Johnson, Mar y

Skopinski. Together, they

female, STEM or non- STEM,

Jackson, and Dorothy Vaughan;

coauthoredDeter mination of

each of us is blessed with a

three br illiant women who were

Azimuth Angle at Bur nout for

choice, so let?s all choose to be

told they could not reach their

Placing a Satellite Over a

the fir st.

goals for professional

Selected Ear th Position, the

advancement, yet refused to

repor t laying out the equations

accept this reality. Rather than

descr ibing the orbital spacelift

give up their dreams, these

in which the landing position a

women per sisted and made

spacecr aft was specified. Her

their own paths, establishing

work was so respected that two

career s as engineer s and

year s later, pr ior to lift off in

mathematicians.

Fr iendship 7, John Glenn said

Kather ine?s exper iences and her role in the success of Freedom 7 and the early space progr am.

7. Logic Godel?s Proofby Ernest Nagel and James R. Newman (2001) God Created the Integers: The Mathematical breakthroughs that Changed Historyedited

by Stephen Hawking (2007) pp.1255 - 1284 Godel Escher Bach: An Eternal Golden Braidby Douglas R. Hofstadter (1979)

14. The Brachistochrone https:/ / www.youtube.com/ watch?v=Cld0p3a43fU https:/ / www.youtube.com/ watch?v=skvnj67YGmw&t=941s https:/ / en.wikipedia.org/ wiki/ Brachistochrone_curve http:/ / mathworld.wolfram.com/ Cycloid.html

that he was ready to go as long The film?s pr imar y focus is on

Lk76R19Zznee66mZZ9qMyVeS

as NASA ?got the girl,? meaning Kather ine, to double check the

17. Spherical and Elliptical Geometry

number s her self using the equations she had created.

http:/ / en.wikipedia.org/ wiki/ Spherical_geometry

tr ajector y analysis for Alan

Kather ine, Mar y and Dorothy all

Shepard?sMay 1961 mission

changed the cour se of histor y,

http:/ / en.wikipedia.org/ wiki/ Elliptic_geometry

Freedom 7, Amer ica?s fir st

they faced obstacle after

human spaceflight.? When

obstacle in their attempts to be

Kather ine star ted off in the

viewed as equals within the

progr am in 1957, she was just a

mathematical community.

According to NASA, ?She did

Math Mag 2017

http:/ / mathworld.wolfram.com/ SphericalGeometry.html http:/ / noneuclidean.tripod.com/ ellipse.htmlTaimin?a, Daina.Crocheting adventures with hyperbolic planes. Wellesley, MA: A.K. Peters, 2009.

SAR High School

71