Staples STEM Journal: Issue No. 3

APRIL 2017

STAPLES

ISSUE NO. 3

STEM JOURNAL

THE MATHEMATICS OF SECRECY

CRISPR CUTS

STAPLES HIGH SCHOOL

ENGINEERING THYROID CELLS

Copyright © 2017 Staples High School STEM Journal P UBLISHED BY S TAPLES H IGH S CHOOL STEM J OURNAL Licensed under the Creative Commons Attribution-NonCommercial 3.0 Unported License (the “License”). You may not use this file except in compliance with the License. You may obtain a copy of the License at http://creativecommons.org/licenses/by-nc/3.0. Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an “AS IS ” BASIS , WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.

See the License for the specific language governing permissions and limitations under the License.

Printed for April 2017.

Contents

1

Letter from the Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A LICE S ARDARIAN ‘17

2

Living a Longer, Healthier Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A LYSSA H YMAN ‘18

3

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Science You Can Eat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

M ALINI W IMMER ‘18

5

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Chernobyl’s Legacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

J ESSICA X U ‘18

7

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Accidental Discoveries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A NGELA J I ‘19

6

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The Mathematics of Secrecy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

L EYA L UO ‘18

4

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CRISPR Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A LICE S ARDARIAN ‘17

8

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Nanomedicine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

E LECTRA S ZMUKLER ‘19

10

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Knights of the Mutually Tangent Tables . . . . . . . . . . . . . . . . . . . . . . . . . . .

CARTER T EPLICA ‘19

11

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Engineering Thyroid Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

M ARGOT M ATHER ‘17

9

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Journal Staff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Letter from the Editor

Dear Reader, As you turn the first few pages of this third issue, I urge you to consider the following question.

What are you fascinated by? Our writers have all discovered that their answers to this question are buried within, supported by, or composed entirely of STEM. This unifying blanket serves as a web of sorts, linking all of the articles within this issue. As editor, I have the distinct pleasure of reading and rereading my peers’ writing, such that every researcher mentioned, scientific method or technique described, and field of application noted, form an immediate bridge of understanding. Look out for these links to explore a broader perspective of STEM in our daily lives and achievements.

The third issue of the Journal marks a significant point in our growth as a scientific publication, especially for me. After a challenging start two years ago, I am thrilled to see how far this journal has come, and am grateful to have found a community of like-minded, STEM invested, peers and teachers. I hope you will continue to support this venture in the years to come. I still have incredible dreams and expectations for this journal, and those will come through metamorphosis, not only of the journal itself, but also of the members that make it unique. As new members join, old members depart to pursue STEM inquiries through other, collegiate venues. The Journal and its staff will continue to seek passionate and curious minds within the Staples community to leave their impact upon the journal and to aid in its development towards excellence. Perhaps you, fellow reader, are that new member!

I hope you are intrigued by the topics within this issue, as I was, but also prompted to pursue and explore the various paths of inquiry that may stem from them.

Happy discoveries and all the best for fourth quarter!

Alice Sardarian ‘17, Editor-in-Chief

2. Living a Longer, Healthier Life

By Alyssa Hyman ‘18 It is believed that our time on earth is set in stone. People are convinced that their life is their life, and that the time they spend on earth was set the moment they entered the world. However, there are regions in the world where entire populations live longer than in other regions. People wonder how this is possible, and generally believe humans cannot do anything to affect their lifespan. Quite on the contrary, human actions do play a role in an individual’s lifespan, and geneticists and other scientists have completed studies to prove it.

According to a survey conducted by the Huffington Post, nearly 75% of Americans exercise once a week [3]. These exercises are simple and short, and generally consist of running and weightlifting. Exercise is beneficial, but this kind of exercise is not very helpful for an individual.

Citizens in Ikaria, Greece, have a different take on exercise, which is one of the many reasons why a very small percentage lose their mental functions, despite old age. “The secret they teach us is the importance of engineering ‘nudges’ for physical activity into our daily life, like planting a garden, which sets up a nudge for the entire growing season to be out there watering, weeding or harvesting” [8]. This is not the most strenuous form of physical activity, but it is an unconscious type of physical activity. Citizens in Ikaria are constantly walking to a store or kneading dough for bread, rather than relying on a car or a mechanized tool [8]. By working physical activity into daily tasks, Ikaria’s citizens are living longer, healthier lives, which is something we all aspire to do.

Diet also plays an integral role in healthy aging and longevity. Many individuals attempt diets for short periods of time because they believe that they will allow them to live longer. However, it is important to consistently eat well, and to eat the right foods, as this affects telomere length.

Telomeres are located at the end of each chromosome, and “according to geneticists, telomeres prevent chromosomes from fraying and scrambling the genetic codes they contain” [5].

Telomeres naturally shorten with age, but the speed of their shortening depends on the health of the individual. Diet affects telomere length, and a study performed at the Harvard Medical School determined the best diet for individuals: the Mediterranean Diet. “Researchers found that the regimen – rich in whole grains, vegetables, fruits, legumes, nuts, fish and olive oil – appears to be associated with longer telomere length, which are indicators of slower aging” [5]. This is why

Chapter 2. Living a Longer, Healthier Life the people in Ikaria live longer: they rely on the Mediterranean Diet. However, additional research showed that the intake of individual food items in the Mediterranean diet [were] not associated with telomere length� [5]. This is why overall eating patterns are important, rather than eating one or two of these foods sporadically.

Figure 2.1: [1]

Figure 2.2: [2]

Figure 2.3: [7] However, daily activity and an improved diet are not the only way to increase longevity. Gene editing can also increase an individual’s lifespan. CRISPR, clustered regularly interspaced short palindromic repeats, edits DNA by using an enzyme called Cas9. Cas9 finds, cuts out, and replaces certain parts of DNA [6]. Oliver Medvedik, cofounder of Genspace and director of the Kanbar Center for Biomedical Research at The Cooper Union, affirms that this can increase an individual’s lifespan if the added genes protect against diseases or have the ability to increase longevity [4].

As humans, we do have the ability to improve our lives dramatically. By adding active components to our daily routine, we get the frequent exercise needed to make us healthier individuals.

Our food choices must remain consistent and healthy in order for our telomeres to grow more, or at least slow down the speed of their shortening. Although trials are still being conducted on humans, perhaps one day in the near future, we might be able to modify our genes to allow us to fight cancer and diseases, and to improve our health and longevity. For now, we can make simple changes to our daily routines, and maybe one day, we will all live like the people of Ikaria.

References 1. Colla, G. (2011). Blue Zones - Greece. Nationalgeographic.com. Retrieved from http://www.nationalgeographic.com/travel/happiest-places/blue-zones-ikaria-photos/ ?source=link_twt20111126ikaria-photos#/ikaria-herb-garden_41621_600x450.jpg 2. How the Mediterranean Diet Can Make You Smarter. (2017). The Luxury Spot. Retrieved from http://www.theluxuryspot.com/mediterranean-diet-can-make-smarter/ 3. Internicola, D. (2013). Americans’ Favorite Kind Of Exercise: Simple, Solo And Short.

The Huffington Post. Retrieved from http://www.huffingtonpost.com/2013/09/23/americansexercise-simple-solo-short_n_3975183.html 4. Powles, R. (2016). What We Learned At The DNA Conference. Longevity Reporter.org.

Retrieved from http://longevityreporter.org/blog/2016/7/19/what-we-learned-at-the-dnaconference 5. Preidt, R. (2014). Could a ’Mediterranean’ Diet Extend Your Life?. US News Health. Retrieved from http://health.usnews.com/health-news/articles/2014/12/02/could-a-mediterraneandiet-extend-your-life?src=usn_tw 6. Scientists to Begin First Human CRISPR Trial. (2016). Worldhealth.net. Retrieved from http://www.worldhealth.net/news/scientists-begin-crispr-trial-humans/ 7. What is TA-65: Telomerase Activator TA-65® Activation Supplement. T.A. Sciences®.

Retrieved from https://www.tasciences.com/what-is-ta-65/ 8. Worrall, S. (2015). Here Are the Secrets to a Long and Healthy Life. National Geographic.

Retrieved from http://news.nationalgeographic.com/2015/04/150412-longevity-health-bluezones-obesity-diet-ngbooktalk/

3. The Mathematics of Secrecy

By Leya Luo ‘18 Cryptology is the study of encoded messages, or ciphers, and how to decode them. The former branch is known as cryptography, and the latter is known as cryptanalysis. In cipher notation, P stands for plaintext, or the original message; C stands for ciphertext, or the encoded message; E stands for the encryption function, or the method where the plaintext is translated to the ciphertext; and D stands for the decryption function, or the method where the ciphertext is translated to the plaintext. Thus, the two functions can be written as E(P) = C and D(C) = P [3]. Or, the encrypted plaintext is the ciphertext and the decrypted ciphertext is the plaintext. An encrypted text is one that is written in code, and a decrypted text is one that has been decoded.

One of the most basic types of ciphers is the shift cipher. To use it, each letter in the alphabet is assigned a number from 0 to 25, and an additional number between 0 to 25 to represent ‘t.’ To encrypt a shift cipher, use the function E(P) = P + t, as each letter in the alphabet (P) is shifted up by a certain amount (t). If E(P) yields a number more than 25, subtract 25 from that number to get the letter to use. To decrypt a shift cipher, use the function D(C) = P - t, which undoes the shift (t) to yield the original message (P) [3].

Figure 3.1: [2]

Chapter 3. The Mathematics of Secrecy The shift cipher is also known as the Caesar cipher, for it was famously employed by Julius Caesar during his battles to hide information from his enemies. This cipher, as well as more complex variants, went on to conceal plots, although a major weakness was that both parties who wished to communicate without risk of interception needed to have the keys for encoding and decoding the messages. Ciphers of this nature are known as symmetric ciphers.

During WWI, ciphers became more prominent, for the advent of real-time wireless transmissions meant that all communication had to be encrypted lest it be intercepted. Britain, France, and Germany all set up decryption offices during WWI, and these offices assisted in the development of modern cryptanalysis. Britain’s Room 40 was able to intercept all conversations between America and Germany at the beginning of the war without Germany’s awareness, and thus was able to intercept and decode the Zimmerman Note in 1917, which revealed a German plot to entice Mexico into WWI by offering it Arizona, New Mexico, and Texas [5]. This led towards America’s entry into WWI. Ciphers and cryptanalysis also remained prominent during WWII. As technology advanced, machine ciphers were invented, which increased the number of possible codes a decoder could use in order to decrypt the messages, and therefore increase the message’s security. The most famous machine cipher, Enigma, used multiple rotors and a key to encrypt messages, which required codebreakers to sift through 150 million million potential permutations of encryption functions.

However, some of the codes were secretly broken by the Polish Cipher Bureau in 1932, and a machine called the Bombe, was later invented to imitate multiple Enigma machines and eventually break the Enigma code [1].

In the modern era, ciphers have shifted from using secret keys, where both the sender and the recipient of the encoded message have keys for encoding and decoding messages, to public keys, where the keys for encryption are public and the keys for decryption are private. Therefore, if someone wanted to send you an encoded message, anyone could encrypt it, but only you could decrypt it.

References 1. Buchanan, B. (2014). Codebreaking has moved on since Turing’s day, with dangerous implications. Phys.org. Retrieved from https://phys.org/news/2014-11-codebreaking-turingday-dangerous-implications.html 2. Caesar Shift Cipher. Retrieved from https://www.tutorialspoint.com/cryptography/images/ caesar_shift_cipher.jpg 3. Gunnells, P. (2004). The Mathematics of Cryptology. Presentation, Department of Mathematics and Statistics University of Massachusetts, Amherst.

4. Hoffstein, J., Pipher, J., & Silverman, J. An Introduction to Cryptography. In An Introduction to Mathematical Cryptography (1st ed., pp. 1-9). Retrieved from http://citeseerx.ist.psu.edu/ viewdoc/download?doi=10.1.1.182.9999&rep=rep1&type=pdf 5. Stupples, D. (2014). Cable snips and fake Mexican burglaries: How the WWI information battle was won. Phys.org. Retrieved from https://phys.org/news/2014-08-cable-snips-fakemexican-burglaries.html

4. Science You Can Eat

By Malini Wimmer ‘18 You sit at a table. The modern aura of the restaurant surrounds you, the ambience, the aroma, the artistry. You glance at the menu; this is not your traditional restaurant. Adorning the menu are fanciful items: flash frozen fruit, liquid nitrogen desserts, flavored foams. As you ponder which of the delicate morsels you wish to sample first, the chefs in the kitchen are using science to create your food.

Molecular gastronomy was created by Hungarian physicist, Nicholas Kurti, and French physical chemist, Hervé This, in 1992 [3]. They combined the disciplines of biology, chemistry and food science to create a fanciful eating experience unlike any up to that point. Molecular gastronomy is considered a subdiscipline of food science, and it centers around the investigation of the physical and chemical transformations of ingredients that occur in cooking. This means that it combines the two greatest topics on Earth: science and food. Molecular gastronomists study the way ingredients interact, and use biologically-based chemicals to alter the composition of food at a molecular level, to create dishes unlike anything you can find in a local restaurant.

Figure 4.1: Colloidal particles scatter light in what’s called the Tyndall Effect [5].

Chapter 4. Science You Can Eat The foundational components of molecular gastronomy are colloids. A colloid is a material composed of tiny particles of one substance that are dispersed, not dissolved, in another substance.

A mixture of the two substances is called a colloidal system, and when more than two states of matter are involved, it is called a complex disperse system. A common example of this is ice cream, where the milk, ice crystals, and sugars are in solid forms, the water is in liquid form, and the air bubbles are in gaseous form. Another popular technique is called sous vide, or under vacuum, in French. Chefs vacuum seal meat with its seasoning in a heat proof bag and submerge it in simmering water to cook. They pull the bag out afterwards and sear it on both sides to create a crispy exterior.

One of the most interesting techniques is spherification, which involves a simple gelling reaction between calcium chloride and alginate, a long chain polymer extracted from brown seaweed. They flavor the calcium chloride, remove the air bubbles by letting the mixture sit overnight, and then delicately drop the flavored calcium chloride into alginate and water, allowing it to form a bead [1].

Through these processes, one can clearly see the combination of physical and chemical processes that create the intense flavors and mouthwatering textures unique to molecular gastronomy.

Figure 4.2: [3]

Eating a full, molecular gastronomy meal is an expensive endeavor. Highly acclaimed molecular gastronomists charge a pretty penny for their magnificent molecular morsels. After all, they have to pay for both the food and the scientific equipment to transform it. To have a centrifuge in your home kitchen, for example, costs at least 500 dollars. To install one in a restaurant costs thousands [4]. This is why, despite the growing number of acclaimed chefs who practice this culinary craft, there is also a sizeable population who dislike the movement. RenĂŠ Redzepi, the executive chef of Noma - a Copenhagen restaurant that held the title of “Best Restaurant in the Worldâ€? for three consecutive years - claims molecular gastronomy has made the process of using science in the kitchen elitist, and inaccessible to the average aspiring chef [3]. The equipment necessary to execute these scientific feats is expensive, and the process is arduous and, thus, almost inaccessible. On the other hand, those who encourage molecular gastronomy emphasize the elements that can be used in all kinds of cooking, and by all kinds of chefs. One does not need to be a molecular gastronomist to enjoy a little pop of something unique in a meal.

Back in the restaurant, you are served a delicate plate of food, transformed, almost unrecognizable. You sample the science. Now it’s up to you to decide; is molecular gastronomy worth it?

Figure 4.3: [2]

References 1. Harris, W. (2017). How Molecular Gastronomy Works. HowStuffWorks. Retrieved from http://science.howstuffworks.com/innovation/edible-innovations/molecular-gastronomy2.htm 2. Jez, PhD, M. (2015). Molecular Gastronomy – The Food Science. Splice. Retrieved from http://splice-bio.com/molecular-gastronomy-the-food-science/ 3. Molecular Gastronomy | FOUR Magazine. (2015). Four-magazine.com. Retrieved from http://www.four-magazine.com/articles/518/molecular-gastronomy 4. Price Of Centrifuge, Price Of Centrifuge Suppliers and Manufacturers at Alibaba.com. (2017).

Alibaba.com. Retrieved from https://www.alibaba.com/showroom/price-of-centrifuge.html 5. Solution Suspension Gallery. Keywordsuggest. Retrieved from http://keywordsuggest.org/ gallery/562339.html

5. Accidental Discoveries

By Angela Ji ‘19 The brain is considered to be one of the most complex organs in the human body. Yet, despite all the research and studies that have explored it, scientists have just scratched the surface. Part of the reason why it is so difficult to study the brain, is because living patients cannot be examined beyond external scans, and brain cells cannot grow back if tissue samples are used. Before scanning methods such as magnetic resonance imaging (MRI) were invented and used to view the brain, many scientists discovered, or, more appropriately, happened upon the functions of the brain by accident.

Henry Molaison: “Patient H.M.” The story of Henry Molaison, known to the scientific community as “Patient H.M.,” is arguably one of the most important stories in neuroscience. He underwent an operation in 1953 for epilepsy, which, unintentionally, damaged part of his brain in the process. Afterwards, he was unable to form new memories, though he retained his old memories. The last president he remembered was Eisenhower, but he didn’t know whether Frank Sinatra was alive, and, when given the name of a close friend, he incorrectly guessed that she was a senator [2].

As tragic as this incident was, it illuminated a key component as to how memory works.

Previously, many thought that memory could not be pinpointed to a certain region of the brain, and was rather distributed across the hemispheres. This assumption led to the common belief that the extent to which a person’s memory was affected, was directly correlated to how much brain tissue was removed in procedures like Molaison’s. However, that was clearly not the case, given the severity of his amnesia [2]. During the operation, a large portion of H.M.’s hippocampus was removed, linking his subsequent amnesia to the key role of the hippocampus in memory [1].

When Molaison was later assigned to perform a simple task practicing his hand-eye coordination, he struggled initially, but began to improve, despite the fact that he couldn’t remember his previous attempts. This highlighted the presence of two memory systems in the brain: one responsible for

Chapter 5. Accidental Discoveries memories involving events, and the other, responsible for procedural and skill-requiring memories [2].

A multitude of other ‘accidental’ observations were made about Molaison and his responses.

If a dolorimeter (pain-inflicting device) was placed on his chest, he would not complain, even if his skin was burning. H.M. ate two dinners, as by the time he started eating the second meal, he had forgotten his first. After H.M.’s death, his brain was thoroughly examined, and scientists discovered a previously unreported lesion in the left hemisphere of his frontal lobe, likely caused by the surgeon who moved Molaison’s frontal lobes to operate on his medial temporal lobes [2]. Additionally, scientists observed that, although more of his hippocampus was left intact than previously assumed, it was still disconnected from structures largely involved in retaining long-term memories. Damage to the cerebellum was also noted, likely caused by the drug, phenytoin, that H.M. took all throughout his life [1].

Figure 5.1: H.M. in 1986 [2].

Phineas Gage Studies of Phineas Gage were key contributions to our understanding of the link between brain damage and changes in personality. In 1848, while Gage worked to pack explosive powder into a hole, the powder exploded and propelled the tamping iron he was using through his left cheek, brain, and skull. Quite remarkably, he survived, though he did go blind in his left eye [5].

The drastic changes that resulted from this incident became progressively more visible as time went on. Before the accident, Gage was a well-liked man, but afterwards, he had few friends.

He could no longer stick to plans, threw fits, was impatient and stubborn, used vulgar language, and was a lot less compassionate than before [4]. Though a specific region of the brain was not established as having a direct correlation with personality, the changes that resulted after Gage’s accident proved that the brain may undergo the slightest of alterations, as in personality, even after the severest of traumas. Gage died, twelve years later, after a series of seizures [5].

Figure 5.2: [5]

Louis Victor Leborgne: “Tan” The case of Louis Victor Leborgne is yet another tragic one, but a case of accidental insight into speech and language, and their elucidation of the dark mysteries of the brain. At thirty years of age, Leborgne suddenly lost his ability to speak. The only word he could say was “Tan” [3].

Unlike Molaison and Gage, there were no visible signs of external trauma for Leborgne.

He could understand speech and maintain his level of intelligence, but he could not physically pronounce any words other than “Tan” or “Tan tan” [3].

When Pierre Paul Broca, a scientist who studied language, encountered Leborgne, the singleword man had developed gangrene, could barely move, and was being rushed to surgery. Broca observed that Leborgne was still responsive in this diminished state, and could make accurate gestures, but took note of his persisting inability to articulate speech. He called Leborgne’s condition, aphémie (aphemia), known today as Broca’s Aphasia. After Leborgne’s death, a large lesion in the posterior inferior frontal gyrus was revealed, now called Broca’s Area, which is directly associated with aphasia.

Figure 5.3: [3]

Chapter 5. Accidental Discoveries Much of the knowledge accumulated about the functions of the brain have come from studying patients with brain damage. Molaison, Gage, and Leborgne are a few of those patients who have, accidentally, prompted significant advancements in the field of neuroscience. Despite all that these gentlemen allowed scientists to understand, there is still much more to be discovered. The brain is complex, and we do not even know it in its entirety.

References 1. Abbott, A. (2014). Postmortem of Famous Patient’s Brain Explains Why "H. M." Couldn’t Learn. Scientific American. Retrieved from https://www.scientificamerican.com/article/ postmortem-of-famous-patients-brain-explains-why-hm-couldnt-learn/ 2. Dittrich, L. (2016). The Brain That Couldn’t Remember. New York Times Magazine.

Retrieved from https://www.nytimes.com/2016/08/07/magazine/the-brain-that-couldntremember.html 3. Konnikova, M. (2013). The man who couldn’t speak and how he revolutionized psychology.

Scientific American Blog Network. Retrieved from https://blogs.scientificamerican.com/ literally-psyched/the-man-who-couldnt-speakand-how-he-revolutionized-psychology/ 4. Phineas Gage’s story. Uakron.edu. Retrieved from https://www.uakron.edu/gage/story.dot 5. Twomey, S. (2010). Phineas Gage: Neuroscience’s Most Famous Patient. Smithsonian.com.

Retrieved from http://www.smithsonianmag.com/history/phineas-gage-neurosciences-mostfamous-patient-11390067/

6. Chernobyl’s Legacy

By Jessica Xu ‘18 At 1:23 AM, on April 26, 1986, there was an explosion at the Chernobyl Nuclear Power Plant in the Soviet Union, creating the worst disaster ever in the history of nuclear power [3]. The calamity resulted from a lack of knowledge and carelessness of certain engineers. They ran a special test on the Number 4 reactor in order to appraise the turbine’s functionality in the event of complete power loss and, in doing so, disconnected the emergency safety systems and power regulating systems of the reactor [7]. Additionally, several design flaws in the Soviet-designed, graphite-moderated RBMK-1000 nuclear reactors resulted in a dangerously continuously increasing reactivity in the nuclear core of the reactor instead of a moderated one. When the engineers conducted their test, they inserted nuclear fuel rods into the reactor, creating a massive amount of steam and, due to the increased reactivity, causing a power surge which blew the lid off of the reactor and released radiation into the atmosphere [11].

Chernobyl is located around Pripyat, Ukraine, about sixty-five miles north of Kiev. Over fifty tons of radioactive material was released by this explosion [7]. Air currents carried the radiation over to the surrounding areas of the western Soviet Union, Eastern and Western Europe, Scandinavia, and the United Kingdom, even traveling as far as eastern North America. However, the most contaminated regions were in Ukraine, Belarus, and Russia, in which over 336,000 people were forced to evacuate and resettle [3]. Since the accident, the plant, the towns of Pripyat and Chernobyl, and their surroundings have been called the “zone of alienation”, and are off limits to humans, for the most part [11].

Though the plant has been shut down and most have been evacuated from contaminated zones, there have still been lasting, adverse health effects in the impacted areas. The immediate radiation killed twenty-eight of the reactor’s workers, and hundreds more were exposed to elevated levels of radiation. In the most contaminated regions of Ukraine, Belarus, and Russia, over 6,000 cases of thyroid cancer have been linked to the Chernobyl explosion, although the exact number is uncertain.

However, the overall adverse health effects have not increased sharply due to this accident, and the majority of the contaminated areas face radiation levels not dissimilar from natural background levels. There is also little evidence to connect Chernobyl to other cancer types. Still, the accident is

Chapter 6. Chernobyl’s Legacy projected to be responsible for a death toll of 4,000 from cancer, for those exposed to high doses of radiation, and an additional 5,000 deaths, for those who were exposed to less [10].

Figure 6.1: Radiation map of Chernobyl [5].

There have also been many negative environmental impacts. Immediately following the incident, many trees died from the high levels of radiation. This area became known as the “Red Forest� because the dead trees turned red [11]. A prevalent theory became that Chernobyl would become a dead zone, in which life would never return. It was later discovered that radiation caused dead organic matter in the surrounding areas of the accident to decompose at a slower rate due to the slowed growth of bacteria, fungi, worms, and other decomposers. Therefore, the trees that were killed immediately following the incident were still relatively intact over fifteen years later [9].

Freshwater ecosystems have also been affected by the radiation. Contained catchment soils, areas of land from which all runoff water flows to a low point and joins another body of water, can transfer radionuclides to bodies of water, causing contamination levels to remain relatively high. For example, in western Europe, the lakes demonstrate a similar contamination level to those in Ukraine and Belarus. Although there was a much lesser impact on marine ecosystems, the International Atomic Energy Agency (IAEA) discovered that the accident elevated the radionuclide levels two or three times higher than they were previously. In urban areas, the radiation also migrated from buildings and infrastructure into the soil [1]. The alterations to the surrounding ecosystems cannot

be disregarded, and neither can be the changes wrought on individual organisms.

Figure 6.2: A radioactive mushroom [12].

Although enough radioactive elements reside in many plants, animals, and mushrooms to make them unsafe for human consumption, based on theoretical studies, the prevailing theory has been that radiation has had little to no effect on plant and animal populations. However, field studies performed in early 2014 by Timothy A. Mousseau, a professor of biological sciences at the University of South Carolina-Columbia, and various other researchers have shown otherwise.

They have found that mortality, genetic anomalies, and reproductive dysfunction increased among many organisms [1]. The growth of trees are stunted, due to the radiation and reduced nutrient cycling. Species such as spiders, dragonflies, butterflies, grasshoppers, and various mammals are not present in nearly the same numbers as they are in areas not affected by radiation, and some species of birds are simply absent. Exposure to the ionizing radiation in the area also caused genetic damage in certain organisms, found to have cataracts and diminished brain sizes. There have been more incidences of tumors and developmental abnormalities in birds. Some populations of birds also had malformed sperm, and up to forty percent of birds were completely sterile in the most radioactive areas. However, this was not universal. Wolves, for example, appeared to be unaffected by radiation in terms of population density, and some species of birds had larger populations in radioactive areas. Nonetheless, an overall lack of biodiversity was found almost ubiquitously.

In spite of the harmful effects of radiation and the assumption that Chernobyl would become a dead zone, animals have started to return to Chernobyl. Due to the lack of human interference, even with the radiation, the “zone of alienation� has become a wildlife sanctuary of sorts. There has been little evidence that organisms have evolved to become resistant to radiation, but they have certainly forged their way into the less contaminated areas around Chernobyl. Certain wild animal populations, such as wild boar, deer, wolf, bear, and beaver have increased. The introduction of the endangered Przewalski wild horses into the Chernobyl area in 1998 has been mostly successful, exempting human variables such as poaching [4]. Other rare species, such as lynxs and eagle owls are also thriving. Although the wildlife may not be as healthy as their radiation-unaffected

Chapter 6. Chernobyl’s Legacy counterparts, the populations are proliferating. Trees are growing back, and the Red Forest is green once again. Additionally, a study led by Professor Jim Smith of Portsmouth University, found no evidence to support that the wildlife as a community has suffered due to radiation. Smith claims that, “It’s very likely that wildlife numbers at Chernobyl are much higher than they were before the accident,” and that human habitation had a worse effect on the animals than the radiation [6].

Although it does contrast with Mosseau’s study, the results are not entirely conflicting, as Mosseau’s focused on the effects on individual organisms, while Smith’s focuses on wildlife populations as a whole. The study provides evidence that the number of animal populations have increased, and demonstrates the resilience of wildlife, in spite of overall decreased biodiversity [2]. Another recent camera study, led by James Beasley from the University of Georgia, found that species distribution is not determined by radiation levels but rather by preferred habitat areas [13]. There are also some bacteria and fungi that flourish in elevated levels of radiation, and there have been some plant species found living unaffected by radiation. No matter the detriment the radiation caused to Chernobyl’s ecosystems, it is undeniable that life is flourishing.

The area will not be safe for human habitation for at least another 20,000 years [11]. But, at least for the moment, we can observe what the devastating Chernobyl accident has done to and for the animals in the zone of alienation.

Figure 6.3: A wolf in Chernobyl [8].

References 1. Bourguignon, D., & Scholz, N. (2016). Chernobyl 30 years on: Environmental and health effects. European Parliament Think Tank. Retrieved from http://www.europarl.europa.eu/RegData/ etudes/BRIE/2016/581972/EPRS_BRI(2016)581972_EN.pdf 2. Cell Press. (2015). Chernobyl: At site of world’s worst nuclear disaster, the animals have returned. ScienceDaily. Retrieved from https://www.sciencedaily.com/releases/2015/10/ 151005132553.htm 3. Chernobyl disaster. ScienceDaily. Retrieved from https://www.sciencedaily.com/terms/ chernobyl_disaster.htm 4. chornobyl.in.ua. Demographic parameters of a Przewalski horse population in the exclusion zone of the Chernobyl power plant. Retrieved from http://chornobyl.in.ua/en/przewalskihorse-population.html 5. CIA Factbook. (2017). Chernobyl radiation map. Retrieved from https://upload.wikimedia.org/ wikipedia/commons/thumb/2/23/Chernobyl_ radiation_map_1996.svg/727px-Chernobyl_radiation_map_1996.svg.png 6. Connor, S. (2015). Chernobyl exclusion zone becomes wildlife haven. USA TODAY. Retrieved from http://www.usatoday.com/story/news/world/2015/10/06/chernobyl-wildlifehaven/73431956/ 7. History.com Staff. (2010). Nuclear disaster at Chernobyl. History.com. Retrieved from http://www.history.com/this-day-in-history/nuclear-disaster-at-chernobyl 8. Gaschak, S. (2013). A wolf in the Chernobyl exclusion zone. Retrieved from http://www.slate.com/content/dam/slate/articles/health_and_science/nuclear_power/2013/01/ 130118_NP_wolfEX.jpg.CROP.original-original.jpg 9. Hjelmgaard, K. (2016). On edge of a human tragedy, Chernobyl also sees wildlife weirdness.

USA TODAY. Retrieved from http://www.usatoday.com/story/news/world/2016/04/17/ environment-chernobyl-30th-anniversary-biology-wildlife/82970958/ 10. Hjelmgaard, K. (2016). 30 years later: Chernobyl disaster could trigger more cancer, deaths.

USA TODAY. Retrieved from http://www.usatoday.com/story/news/world/2016/04/25/chernobyl30-year-anniversary/83220302/ 11. Lallanilla, M. (2013). Chernobyl: Facts About the Nuclear Disaster. Live Science. Retrieved from http://www.livescience.com/39961-chernobyl.html 12. Landmann, P. (2006). UKR: Chernobyl. Retrieved from https://www.usnews.com/dims4/ USNEWS/04ff2b6/2147483647/thumbnail/970x647/quality/85/?url=%2Fcmsmedia%2F54% 2F9a%2F108d10ec46f6ac8ea117c8326a41%2Fresizes%2F1500%2F160425-chernobyl-natureeditorial.jpg 13. University of Georgia. (2016). 30 years after Chernobyl, camera study reveals wildlife abundance in Chernobyl Exclusion Zone. ScienceDaily. Retrieved from https://www.sciencedaily.com/ releases/2016/04/160418161400.htm

7. CRISPR Cuts

By Alice Sardarian ‘17 Deoxyribonucleic acid, or DNA, is the maestro to our concerto of body systems. It’s unique series of base pairs (composed of A, T, G, and C), code for proteins that build all of the structures that we depend on. There exist some proteins, however, that we would rather eliminate or, perhaps, ’update,’ because of their implications in human disease or deficiencies. Genetic engineering permits these man-made alterations. Previous gene editing technologies involved homologous combination, where base pair sequences are switched between two similar strands of DNA; plasmid injection, where a circular piece of DNA can take up new genetic material and then be easily replicated within bacteria; and ‘bioballistics,’ where silver particles are covered in genetic material and used to transfer the material into a target cell [5].

The public often views science as having a strong dependency on novel, unfamiliar thoughts in order to find groundbreaking discoveries; this is not entirely true when it comes to CRISPR. This new arts and crafts tool for geneticists, was revealed through a look into the past, instead of the future, to learn from “our ancestors” - bacteria [4]. Just as we have a marvelous immune system that combats the flu or wound infections, bacteria have their own immune system to fight off and train an army of genes against pathogens and viruses. Bacteria use their own gene-editing systems, like CRISPR, to cut out invading pieces of ‘bad’ DNA, but leaving behind a segment. This remnant is used by the bacteria defense army to flag the enemy, should it return, and to be prepared for attack. After learning from these extraordinary, 3.7-4.2 billion year old role models, researchers at the Broad Institute of MIT and Harvard, as well as UC Berkeley, developed CRISPR, which is a combination of a Cas9 cutting structure and a select, guiding RNA (gRNA) [8].

So, how does it work? The gRNA is a pre-designed, 20 base-long sequence that is enclosed within a longer strand of RNA necessary to bind to the target DNA. The gRNA guides (explained by its title) the Cas9 structure, which is an enzyme, to the specific location in the DNA, coded by the 20 base-long sequence of As, Ts, Gs, and Cs, also known as adenine, thymine, guanine, and cytosine. Once the location is pinpointed, Cas9 makes a ‘cut’ through both strands of the double-helix DNA. As soon as this cut is made, an automatic alarm is signaled within the cell to begin repairing what has just been damaged. It is at this point, that we can alter the genome.

Chapter 7. CRISPR Cuts Various cells have different DNA repair mechanisms, consequently, there exist different means of editing depending on the mechanism.

Figure 7.1: General process of genetic alteration through Cas9-gRNA mechanism [7].

Two major mechanisms work well with CRISPR: non-homologous end-joining and homology directed repair. Non-homologous end-joining allows CRISPR to cut and remove a section of DNA completely, so that the leftover strands fuse together. The mechanism also allows CRISPR to cut and revise a single base pair. The second mechanism, known as homology directed repair, allows CRISPR to completely cut out and replace a DNA sequence with homologous (similar) DNA endings as the original segment, but with an edit hidden in-between, that has the potential of replacing a faulty sequence with one that can possibly eliminate the sequence’s expression altogether [6].

Figure 7.2: [2]

CRISPR itself can be manipulated to function in other ways. The Cas9 enzyme can be broken down so that it does not cut, like “a broken scissor,” but is still guided to the specified, DNA location [2]. This results in a block in gene expression, as that segment of DNA cannot be used to form proteins. After blocking gene expression, we can introduce a Cas9 activator to allow the expression again, in a controlled manner. Activators can include natural triggers like chemicals or light. Gene manipulation in this way might allow us to determine exactly what certain gene sequences may code for, prior to damaging them with Cas9.

All of these edits may seem simple and easy, however, greater care must be taken when working with CRISPR, especially since a slight change in DNA can induce grand and possibly unfavorable results. The National Institutes of Health has been so cautious with allowing CRISPR work on human subjects, that they just recently, in June of 2016, allowed a team of researchers to remove T cells from patients, and edit them to host proteins that recognize pathogens, and others that hide the T cells from recognition by the pathogens themselves to prevent their destruction [3]. The NIH is, altogether, valid in its unease, as CRISPR is not as accurate as it seems. Part of the problem involves the gRNA, whose specificity and accuracy determines where the DNA gets cut. Even though 20 base pairs make up the gRNA, not all 20 need to match in order for the Cas9 to attach and proceed with its destruction. This leaves a large potential for error that still needs to be amended.

Some have suggested developing a Cas9 that only cuts through one strand of the DNA instead of two, such that two Cas9-gRNA structures must work together and match to the exact same location, in order to make a complete cut [2].

Figure 7.3: Blue, T-cell, attacking a pink, cancer cell [3].

As CRISPR becomes ever-popular in labs for its ease of operation and lower costs in comparison to other genome-editing techniques, individuals, like biologist Ellen Jorgensen, warn that “this science is moving much faster than the regulatory mechanisms that govern it” [1]. So, before seeking to recreate a wooly mammoth, eliminate a harmful trait, or edit an embryo, we must consider the potential for error, without hindering progress.

Chapter 7. CRISPR Cuts

References 1. Jorgensen, E. (2016). What you need to know about CRISPR. TED. Retrieved from https://www.ted.com/talks/ellen_jorgensen_what_you_need_to_know_about_crispr 2. Ledford, H. (2016). CRISPR: gene editing is just the beginning. Nature. Retrieved from http://www.nature.com/news/crispr-gene-editing-is-just-the-beginning-1.19510 3. Reardon, S. (2016). First CRISPR clinical trial gets green light from US panel. Nature.

Retrieved from http://www.nature.com/news/first-crispr-clinical-trial-gets-green-light-fromus-panel-1.20137 4. Richard Dawkins Quotes. BrainyQuote. Retrieved from https://www.brainyquote.com/quotes/ quotes/r/richarddaw363331.html 5. Techniques of Genetic Engineering. (2010). Biotechnology Forums. Retrieved from http://www.biotechnologyforums.com/thread-43.html 6. What is CRISPR?. Editas Medicine. Retrieved from http://www.editasmedicine.com/crispr 7. What is CRISPR-Cas9?. (2016). Yourgenome.org. Retrieved from http://www.yourgenome.org/ facts/what-is-crispr-cas9 8. Zimmer, C. (2017). Scientists Say Canadian Bacteria Fossils May Be Earth’s Oldest. The New York Times. Retrieved from https://www.nytimes.com/2017/03/01/science/earthsoldest-bacteria-fossils.html?_r=0

8. Engineering Thyroid Cells

By Margot Mather ‘17 The long awaited cure for thyroid disease, which afflicts about 20 million Americans, may be in the hands (or paws) of a few lab rats [3]. Researchers have found a way to coax genetically modified, embryonic stem cells from mice, to develop into thyroid cells. Through careful experimentation with the Nkx2-1 gene, which codes for a thyroid transcription factor (protein for transcribing DNA into RNA), as well as a thyroid-specific enhancer-binding protein (protein for tissue specific gene expression), the researchers have also found the gene to be capable of converting stem cells into thyroid cells [1]. This can prove useful in treating thyroid disease by allowing stem cells to be predetermined with the proper structure and function of a thyroid cell and, thus, avoid potentially harmful mutations.

The key to this breakthrough discovery is in a specific protein. The NKX2-1 gene provides instructions for making a sequence of amino acids, protein building blocks, that join to form what is called homeobox protein Nkx-2.1, a distinct member of the homeobox protein family. This protein is responsible for the development and function of the brain, lungs, and thyroid gland, and can be used to reconstruct damaged areas of the thyroid gland that may cause or be causing disease [2].

The thyroid is a large, ductless gland in the neck that regulates growth and development by secreting hormones at a metabolic rate. The gland is a part of the endocrine system, a collection of glands that produce hormones to regulate metabolism, growth and development, tissue function, sexual function, reproduction, sleep, and mood, among other functions. These regulating hormones are manufactured from iodine, derived from the foods you eat. Vast regions of the body are affected by the thyroid gland, making it a dangerous location for disease [4].

Most thyroid complications result in a condition in which the gland is either overactive and produces too much hormone (hyperthyroidism), or underactive and produces too little (hypothyroidism) [3]. In the case of hyperthyroidism, the excessive production of thyroid hormone causes an increased metabolism, and consumes too much energy. This explains why the main symptom of this disease is exhaustion. Other symptoms include stress, nervousness, weight loss, and a rapid heart rate. These are just a few of the damaging effects of thyroid disease that can be eliminated with new stem cells.

Chapter 8. Engineering Thyroid Cells

Figure 8.1: Locations where this homeobox protein is active [6].

The severity of thyroid disease can range from a localized, harmless goiter in which the gland is severely enlarged, to a life-threatening cancer. Researchers believe that taking advantage of the Nkx2-1 gene in order to create new thyroid cells, as well as to ensure hormones are manufactured and travel through the bloodstream normally, is the first step toward an effective, human stem cell protocol, for new treatments of thyroid disease [3].

References 1. Brady MD, B. (2017). Thyroid Gland, How it Functions, Symptoms of Hyperthyroidism and Hypothyroidism. EndocrineWeb. Retrieved from https://www.endocrineweb.com/conditions/ thyroid-nodules/thyroid-gland-controls-bodys-metabolism-how-it-works-symptomshyperthyroi 2. NKX2-1 gene. (2017). Genetics Home Reference. Retrieved from https://ghr.nlm.nih.gov/ gene/NKX2-1 3. Paddock PhD, C. (2017). Engineering thyroid cells from stem cells may lead to new therapies.

Medical News Today. Retrieved from http://www.medicalnewstoday.com/articles/315653.php 4. Thyroid Problems – Symptoms, Causes, and Diagnosis. WebMD. Retrieved from http://www.webmd.com/women/guide/understanding-thyroid-problems-basics#1 5. Truong MD, A. (2017). Papillary & Follicular (Well-Differentiated) Thyroid Cancer Patient Information. Sites.google.com. Retrieved from https://sites.google.com/site/dranhtruong/ bellevue-thyroid-cancer 6. Wilbertz, T., Maier, S., & Perner, S. (2010). NKX2-1 (NK2 homeobox 1). Atlas of Genetics and Cytogenetics in Oncology and Haematology. Retrieved from http://atlasgeneticsoncology. org/Genes/NKX2-1ID44015ch14q13.html

9. Nanomedicine

By Electra Szmukler ‘19 Nanotechnology has the power to transform our society. It revolves around the use of devices that are 100 nanometers or smaller, allowing manipulation of molecular-sized substances. A nanometer is a billionth of a meter and is around three to five atoms wide. A human hair is around 75,000 nanometers, or nm [3]. Beyond computer chips, nanotechnology is also at work in daily products like sunscreen, to improve light ray absorption, or in winter boots, to make them water-proof [2]. The forefront of nanotechnology advancement, however, is occurring in medicine.

Just imagine a bot that might traverse all of your blood vessels and smallest capillaries to reach and dismantle a cancerous growth.

Throughout history, we have been limited by skill and machinery when treating ailments within the body. Medications haven’t been able to reach their targets, and many areas of the body can only be reached by surgeons in risky procedures. Nanotechnology is being used as the basis for a new, more effective drug delivery system. Many pathogenic culprits like bacteria and viruses, are nanosized, and can evade our larger efforts. Nanomedicine, thus, becomes very important to bridge this divide, and save many more lives than previously possible.

Nanotechnology can help collect medical data and provide immediate input upon treatment application. “Lab-on-a-chip” is being developed to use biosensing and microfluidics (the science of manipulating fluids) to actively record body response data by reading genetic material from single-cells “with high spatio-temporal resolution” [4]. The chip has also actively been supported by the Defence Advanced Research Products Agency (DARPA) with the hopes that it could detect toxic chemical uptake within hostile, foreign regions. Chips such as these can be placed within the body upon degradable, 3D-printed scaffolding, that contain only organic elements and that do not upset the body’s equilibrium. The chip is still in the research and trial stage of development, but the potential for its applications are quite unbelievable.

Other nanomedicine research has shown that nanotechnology could be used to help beat cancer through the use of DNA nanobots that alter DNA strands. Within ten to twenty years, it may be possible to construct these bots on the micrometer scale using nanometer scaled parts; surgeons could then piece together or break apart proteins at a molecular level to regenerate a damaged heart

Chapter 9. Nanomedicine or liver [1]. Though not at this advanced junction yet, nanotechnology is already being applied in pharmaceutical products to allow for more effective drug delivery and uptake. Not only does nanotechnology give us the means to help more patients in more efficient ways, but new drug therapies have also been shown to have fewer side effects than traditional therapies.

With every advancement, we must also consider the potential for misuse and error. The more research and information we acquire on the subject of nanotechnology, the easier it becomes for someone to hack into the programs and alter how they work. There is also the issue of control and regulations; since nanotechnology is being used in many fields, there are questions as to who should be allocated to inforce policies. There are also ethical issues as some individuals believe that the tools will give us more ‘god - like’ powers, thought it may conversely solve other ethical issues, such as being able to construct cells instead of acquiring them from stem cells. Overall, the benefits of nanotechnology in medicine would far outweigh its risks.

Though still in its infantile development, nanomedicine has and continues to advance to improve our society. Look for it, as medicine trends away from invasive surgical procedures and other archaic methods, and towards smart, nanobots.

Figure 9.1: A nanorobot in action [5].

References 1. Chen, A. (2002). The Ethics of Nanotechnology. SCU.edu. Retrieved from https://www.scu.edu/ ethics/focus-areas/technology-ethics/resources/the-ethics-of-nanotechnology/ 2. Crawford, M. (2016). 10 Ways Nanotechnology Impacts Our Lives. ASME.org. Retrieved from https://www.asme.org/engineering-topics/articles/technology-and-society/10ways-nanotechnology-impacts-lives 3. Handy, J. (2011). How Big is a Nanometer?. Forbes.com. Retrieved from https://www.forbes.com/ sites/jimhandy/2011/12/14/how-big-is-a-nanometer/#7ef2e6786fb0 4. Kou, S., Cheng, D., Sun, F., & Hsing, I. (2016). Microfluidics and microbial engineering.

RSC.org. Retrieved from http://pubs.rsc.org/en/content/articlelanding/ 2016/lc/c5lc01039j#!divAbstract 5. What Nanobots Are Made Out Of. (2009). Nanogloss.com. Retrieved from http://nanogloss.com/ nanobots/what-nanobots-are-made-out-of/

10. Knights of the Mutually Tangent Tables

By Carter Teplica ‘19 Once upon a time there lived a clever princess and a sharp-witted prince, both avid students of mathematics. One day, the King (who was the Princess’ father) came to them, his brow furrowed in thought.

“I have the most terrible problem. As you know, last week I named several new Knights of the Multiple Round Tables, and there’s not enough room for them at the next meeting. I shall have to order a new table, but there’s no space for another in the Hall. Whatever shall I do?"

This was, indeed, a grievous difficulty. Once the original Round Table had filled up, another of the same size had been ordered.1 But due to the populace’s penchant for doing noble deeds, and the King’s penchant for rewarding them with knighthoods, even the two Tables in place had now been filled.2

“I consulted the royal prophet this morning, and he gave me this riddle. I can’t make head or tail of it, so can you two try?"

The King handed the princess a scroll, on which was written a short poem.

For pairs of lips to kiss maybe Involves no trigonometry.

‘Tis not so when four circles kiss Each one the other three.

1 And

the name of the order had been emended. to a recent dieting fad, all the knights had zero mass, so they fit in the infinitely small space between the tables and could be ignored when laying the tables out.

2 Due

Chapter 10. Knights of the Mutually Tangent Tables To bring this off the four must be As three in one or one in three.

If one in three, beyond a doubt Each gets three kisses from without.

If three in one, then is that one Thrice kissed internally.

They pored over the poem for a while, until the prince had an idea.

“I know! By ‘kiss’ he means ‘touch’—that is, have a point of tangency.3 The solution involves a new table being tangent with the original two, as well as the wall. The three tables will each ‘get three kisses from without,’ one from each of the other tables and one from the wall, while the outer circle—the room—will be ‘thrice kissed internally’ by the three tables."

“While you’re at it, you should put in a fourth table on the other side," noted the princess.

“They’ll be the same size anyway."

“Perfect!" exclaimed the king. He rang the bell to summon the Royal Carpenter into the room.

“We shall need two more round table for the Grand Hall. Mahogany, please, and by Monday."

“How large, my liege?" Each of the original tables was half a fathom4 in radius, but the new ones would clearly have to be smaller.

This stumped them all. The prince and princess made a noble effort5 to solve it using geometry, but they couldn’t arrive at a sensible answer.6 Eventually, they called for the Royal Prophet again, only to be given another riddle.

Four circles to the kissing come.

The smaller are the benter.

The bend is just the inverse of The distance from the center.

Though their intrigue left Euclid dumb There’s now no need for rule of thumb.

Since zero bend’s a dead straight line And concave bends have minus sign, The sum of the squares of all four bends Is half the square of their sum. “‘The sum of the squares of all four bends is half the square of the sum,’" the prince mused.

“That has a nice ring to it. Written out, that would look like this, where a, b, c, and d are the curvatures, or ‘bends,’ of the circles."7 3 Formally,

for two circles to be tangent, there must exist exactly one point on both circles; the circles have the same slope at this point.

4 91 modern centimeters, or three feet for those of us who insist upon using a measuring system based upon the length of a 483-year-old queen’s arm.

5 Pun intended.

6 At one point, after attempting a solution using differential geometry, octonions, and a French-English dictionary, they came up with a value of six hundred forty-seven miles and realized they had made an unfixable error back on page 47.

7 The curvature of a circle is one divided by its radius.

1 a2 + b2 + c2 + d 2 = (a + b + c + d)2 2 Try determining the radius d1 of each of the new tables before you read on. Remember, a = b = 1 and c = 12 .

The radii of the new tables can also be found purely geometrically. Try doing it this way too.

The next week, the king named even more knights, and decided he needed four more tables between the original tables and the wall.

How big must they be?

Eventually, after the king named even more knights and got more tables, the room looked like this. Legend has it that upon seeing the room full of tables, the king exclaimed, “By Apollo, that’s pretty!" and as such a shape like this is called an Apollonian gasket.8

In gratitude for the knighthood, one of the knights, Sir Phredericke, had given the king four large spherical gemstones (each one inch in radius). The king decided to set them around his scepter, touching each other in a pyramid. The original sceptre had consisted of a slender pole topped with a large gold sphere, but he realized he would have to replace the gold sphere with a smaller one so as to fit it among the four gemstones. (In the final sceptre, each of the five spheres would touch each of the four others.)

Vexed by the problem, he came again in great concern to the prince and princess. "Whatever shall I do? I must set the gemstones around my scepter, or Sir Phredericke will be terribly offended, but I do not know how large to make the golden sphere that will fit between the four gemstones!

All is lost!"

They all agreed that this was a most terrible problem. What they needed was a generalization to the sums-of-squares rule, but for three dimensions. In desperation, they decided once more to consult the prophet. As usual, they received a riddle.

To spy out spherical affairs An oscular surveyor Might find the task laborious, The sphere is much the gayer, 8 Apollonian gaskets are actually named after the ancient Greek mathematician Apollonius, who found a more complex geometric solution to the problem, but we find our explanation more interesting.

Chapter 10. Knights of the Mutually Tangent Tables And now besides the pair of pairs A fifth sphere in the kissing shares.

Yet, signs and zero as before, For each to kiss the other four The square of the sum of all five bends Is thrice the sum of their squares.

“I see!" said the princess. “Instead of

1 2

in the original equation, we use 13 ."

1 a2 + b2 + c2 + d 2 + e2 = (a + b + c + d + e)2 3 “Then the radius of the inner sphere is..."

What is the radius of the gold sphere?

A week later, the prince and princess took the RTMC9 12A. Being very good at math, they got all the way to the last question in the allotted time.

1015 RTMC 12A, Question 2510 A collection of circles in the upper half-plane, all tangent to the x-axis, is constructed in layers as follows. Layer L0 consists of two circles of radii 702 and 732 that are externally tangent. For k 1, S the circles in kj=01 L j are ordered according to their points of tangency with the x-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer Lk consists of the 2k 1 circles constructed in this way. Let S S = 6j=0 L j , and for every circle C denote by r(C) its radius. What is 1

 pr(C) ?

C2S

Solve the problem! (While an AMC Problem 25 is usually very hard, this one is quite doable with our theorem. You’ll want to get an equation for the square root of the bend of a circle given the square roots of the bends of the two above it, and then make a table for all the circles. There are only 65 circles, so you can write them all out if you want, but there is a pattern in the table that can save you time. Also, you should multiply all the bends by 702 ⇤ 732 at the start for simplicity.)

Addendum The theorem discussed in this problem was discovered by Descartes for two dimensions and is referred to as Descartes’ Circle Theorem or simply Descartes’ Theorem. The poem, titled “The Kiss Precise," is by the nineteenth-century Scottish mathematician Frederick Soddy, who discovered the extension of the theorem to spheres. The theorem also applies to four or more dimensions, as discovered and expressed in another verse by mathematician Thorold Gosset: 9 Round

Table Mathematics Competition.

1015 RTMC coincidentally happened to be exactly the same as the 2015 AMC, except that the former was written in a mixture of Old English, Old French, and Medieval Latin and the test-takers were allowed to use abaci.

10 The

And let us not confine our cares To simple circles, planes and spheres, But rise to hyper flats and bends Where kissing multiple appears, In n-ic space the kissing pairs Are hyperspheres, and Truth declares As n + 2 such osculate Each with an n + 1 fold mate The square of the sum of all the bends Is n times the sum of their squares.

References 1. Soddy. (2017). Pballew.net. Retrieved from http://www.pballew.net/soddy.html 2. Stedl, Tod. File:ApollonianGasket-1 2 2 3.svg (2008). Commons.wikimedia.org. Retrieved from https://commons.wikimedia.org/wiki/File:ApollonianGasket-1_2_2_3.svg

11. Journal Staff

Faculty Advisor Ms. Karen Thompson

Editor-in-Chief Alice Sardarian ‘17

Layout Editors Evan Feder ‘17 Jonathan Alter ‘17

Writers Alice Sardarian ‘17 Alyssa Hyman ‘18 Angela Ji ‘19 Carter Teplica ‘19 Electra Szmukler ‘19 Jessica Xu ‘18 Leya Luo ‘18 Malini Wimmer ‘18 Margot Mather ‘17 Special thanks to PTA Wrecker Grants for sponsoring this issue.

To contribute to our next issue, kindly email as1005819@students.westport.k12.ct.us