Foundations Of Geometry Notes On The Film
Caleb Gattegno
Educational Solutions Worldwide Inc.
First published in the United States of America in 1981. Reprinted in 2009. Copyright Š 1981-2009 Educational Solutions Worldwide Inc. Author: Caleb Gattegno All rights reserved ISBN 000-0-00000-000-0 Educational Solutions Worldwide Inc. 2nd Floor 99 University Place, New York, N.Y. 10003-4555 www.EducationalSolutions.com
Table of Contents Foundations Of Geometry............................................ 1 1 Lines......................................................................... 5 2 Points....................................................................... 7 3 Incidence .................................................................9 4 Segments ................................................................11 5 Polygons................................................................. 15 6 Convexity ............................................................... 17 7 Angles .................................................................... 19 8 Measure Of Segments............................................. 21 9 Line-Symmetry ...................................................... 23 10 Perpendiculars ..................................................... 25 11 Right Angles .......................................................... 27 12 Parallels ...............................................................29
13 Bisectors ............................................................... 31 14 Rectangles............................................................ 33 15 Point-Symmetry ................................................... 35 16 Parallelograms......................................................37 17 Projection............................................................. 39 18 Subdivision Of Segments.......................................41 19 Measure Of Angle................................................. 43 20 Pythagoras Theorem ........................................... 45 Notes......................................................................... 49
Foundations Of Geometry
The film, Foundations of Geometry, was produced by Dr Caleb Gattegno in 1979 for use in a new geometry syllabus proposed for thirteen and fourteen year olds in France. The scenario was based on the notes on geometry prepared by Professor Gustave Choquet, of the University of Paris, for a submission to the Ministry of National Education, proposing a new syllabus for mathematics to replace that of the so-called Lichnerowitz syllabus. The submission was presented by Professors Cartan, Choquet and Leray on behalf of the AcadĂŠmie des Sciences of which they are members. The film was computer-animated by AndrĂŠ Fourrier and Serge Chicoine of the Computing Centre at the University of Montreal, Canada. It is available, for sale or hire, from Educational Solutions, 80 Fifth Avenue, New York, NY 10011, USA. The film lasts only about seventeen minutes but as an instrument for teaching it contains material that can extend over two years. Sequences sometimes only lasting for a few seconds offer viewers images that can be discussed for an hour or more.
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Foundations Of Geometry
Because the images can be comprehended in a way that words cannot be, their appearance on the screen radically transforms the teaching of geometry to abler students. Indeed, it is possible to hope that all pupils capable of seeing dynamic images and guided by alerted teachers can be led to awarenesses that are now available and easily verbalized. They may be able to learn the geometry of the curriculum and at the same time gain an autonomy of thought that is rarely accessible to the majority of pupils. The following notes on the film have two main purposes. They indicate clearly the sequence of topics being considered. They also attempt to reflect the axiomatic treatment proposed by Professor Choquet. The rigor of this treatment is guaranteed by the name of this great French mathematician who has critically developed it over more than thirty years. But it has also been chosen because it is capable of being understood and assimilated by young thirteen-to-fifteen year olds. It seems to be accessible to many more of them than any other proposed axiomatic treatment. It should be added that the notes do not specifically indicate which of the assertions that can be made about the images presented are theorems and which are axioms. Many of the assertions are presented as definitions and these enshrine awarenesses that can be developed in various ways. The deductive interlinking of assertions could be made in different ways and the isolation of those assertions which need to be taken as axioms could be an important part of the work that arises from the film. On the other hand, the film could be used
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Foundations Of Geometry
to follow a course which presupposes a certain axiomatic treatment. For the purpose of these notes the film has been conceived as yielding twenty distinct episodes. These are listed in the following pages with some — but not all — of the possible ways of formulating notions that arise. (Acknowledgement: This English version of my notes on the film owes a lot to the translator, Dick Tahta. I wish to express my thanks for his help. C.G.)
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1 Lines
The first few seconds of the film present the following notions: 1.1
The line with its property of dividing the plane into two half-planes.
1.2
The open half-plane which lies to one side of a line and does not include it — and the closed half-plane which does include the line.
1.3
There are as many lines in the plane as we please and each of these could have been chosen to divide the plane into half-planes.
1.4
The plane can be swept by a line in various ways.
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2 Points
The film then introduces, briefly, the points of the plane with the notion that there are as many of these as we please.
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3 Incidence
Points and lines are associated by certain relations — sometimes called incidence relations — indicated in the film. 3.1
Any two distinct points of the plane determine one and only one line.
3.2
There are as many lines through a point as we please.
3.3
There are as many points on a line as we please.
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4 Segments
The film now considers the notion of a line segment, referred to henceforth in these notes as a segment. Some definitions can be formulated from the various scenes in this section. 4.1
A point on a line determines two half-lines.
4.2
Any two points on a line determine a portion of the line between them called a segment, the two points being called endpoints of the segment.
4.3
A segment without its two endpoints is called open, otherwise it is closed.
4.4
There are as many points on a segment as we please; they are said to be between the end-points.
4.5
There are as many segments in the plane as we please.
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4.6
Selecting one point between the endpoints of a segment determines two contiguous segments, one on each side of this point, that are said to be end-to-end.
4.7
When the two contiguous segments determined by a point that is between the endpoints of a given segment are equal, then the point is said to be the midpoint of the given segment or of its endpoints.
4.8
Any segment of a line can be chosen as a unit and as many other segments as we please that are multiples of this unit can be formed on the line by placing end-to-end, two, three, or whatever number of segments equal to the unit. These are said to measure one, two, or whatever, units respectively.
4.9
A point may be considered as a zero-segment.
4.10
Describing a line by the continuous displacement of one of its points generates the class of all segments on this line which lie on one side of the line and have the same endpoint, called the origin.
4.11
The measures of the segments on one side of the origin are called positive, those on the other being called negative.
These definitions mean that every line can be allocated a unit measure so that the positive and negative integers can be used to describe certain relations between segments of the line. For 12
4 Segments
example, a more precise meaning can be given to such propositions as that a segment is twice, three times, etc., greater than another; or that two points are equidistant from the origin. (By defining such further notions as families of nested segments, forming increasing or decreasing sequences of segments, it could then become possible to measure any segment of the line in terms of real numbers.)
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5 Polygons
The film now considers chains of segments in the plane, these being formed by a sequence of segments each having a common endpoint with its preceding segment, but not lying on the same line as it. The chain is closed or open, according as the final endpoint does or does not coincide with the initial one. 5.1
A closed chain of segments is called a polygon. There are polygons with any number (greater than two) of segments which are usually called sides of the polygon. These polygons are usually given Greek names, with two Latin exceptions: triangle, quadrilateral, pentagon, hexagon, etc.
5.2
A polygon divides the plane into two regions called the interior and the exterior of the polygon. Points in the interior are called interior points and points in the exterior are called exterior points.
5.3
A segment joining an interior point to an exterior point meets at least one of the sides of the polygon. 15
6 Convexity
The notion of convexity is introduced in terms of polygons: 6.1
A polygon is said to be convex when a segment joining two points of its interior has all its points in the interior.
6.2
Otherwise the polygon is said to be concave; and for such a polygon there will be segments joining two interior points that contain points in the exterior as well as points in the interior.
6-3
The intersection of two convex polygons — that is the part of the plane common to both polygons — is also convex. This implies that the intersection of any number of convex polygons is convex.
6.4
A half-plane is also convex in the above sense so that the intersection of two — and therefore any number of — half-planes is also convex.
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6.5
A brief final scene of this section indicates the analogy of properties of the interior and exterior for half-lines. Thus the line joining a point on one side of a line to a point on the other is seen to contain a point of the line.
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7 Angles
As an example of the way a brief sequence of the film can generate many important notions in the development of geometry, consider the opening scene of the sequence referred to in 6.4. 7.1
The intersection of two open half-planes is called an angle. The half-lines that define the half-planes are called the sides of the angle and their point of intersection is called a vertex.
7.2
Two angles with a common vertex and one common side are called adjacent. When the other sides lie on the same line the angles are said to be supplementary. Any angle has two supplementary angles.
7.3
Two intersecting lines form two pairs of verticallyopposite (that is, vertically opposite) angles with a common vertex and no common side. Two verticallyopposite angles are equal since they have the same supplementary angle. 19
Foundations Of Geometry
7.4
Two supplementary angles together make a so-called straight angle, or angle-in-a-line. Two straight angles on either side of a line together make a so-called plane angle. An angle can vary continuously from a zero angle (when the two sides are coincident) to a straight angle, and in fact to a plane angle.
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8 Measure Of Segments
This section considers the possibility of comparing segments that do not lie on the same line. 8.1
Displacements that preserve length indicate that for two segments, one must be shorter than, equal to, or greater than the other.
8.2
A fundamental inequality arises when comparing three segments — the so-called triangle inequality asserts that the segment with endpoints, say A and B, is smaller than the sum of the segments formed by the two points A and B with a third point, say C, not on the segment AB. This may be expressed as AB≤AC + BC. The triangle inequality may be read as applying to the sides of a triangle so that the sum of two sides is always longer than the third. The inequality can also be read as asserting that the difference of two sides is smaller than the third.
8.3
If C is between A and B, i.e. on the segment AB, then there is an equality AB = AC + BC. 21
Foundations Of Geometry
(It is worth noting that inequalities refer to infinite classes of figures. Moreover, though equalities correspond to particular cases, they may also refer to infinite classes. For example, AB = AC + BC is true for all points C between A and B. 8.4
A sequence of segments on one line can be related to a sequence of segments on another. This means, in effect, that all segments of the plane can be measured in terms of one chosen unit segment. Subject to the triangle inequality this provides the plane with what is called a metric, that is to say, a way of expressing observations about figures in numerical terms.
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9 Line-Symmetry
This important section of the film begins by indicating that there are certain transformations called translations and others called rotations. But these can both be eventually defined and discussed in terms of the more fundamental reflections, or linesymmetries, and it is these that are considered in some detail. In these transformations two figures correspond in such a way that any segment of one generates a congruent segment (i.e. one on which it can be superimposed) of the other, and similarly for angles. Such a transformation is said to be an isometry. 9.1
A line-symmetry may be thought of as a folding, about a line, of one of the half-planes determined by the line onto the other half-plane, covering it exactly. It is an isometry and it is said to transform a figure into its image.
9.2
Various examples of line-symmetry are shown in the film, indicating the corresponding images of a point, a line, a segment, an open chain of segments and a polygon.
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Foundations Of Geometry
9.3
In each case, it may be observed that the image of the image is the original element. (The product of a linesymmetry with itself is the identity transformation.)
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10 Perpendiculars
The notion of perpendicularity may now be defined using linesymmetry. Basically, the join of any point not on a line to its image under line-symmetry in that line is said to be perpendicular to the line. This notion is then used in the following sections to define right-angles and parallel lines. 10.1
Consider a point P not on a line l and its symmetric image p' in the line l. The line through P and P´ is said to be perpendicular to l, meeting it at a point N called the foot of the perpendicular to l from P (or P´).
10.2
The segment PN is equal to the segment P´N.
10.3
From any point of the plane not on a particular line there is one and only one perpendicular to the line.
10.4
If l´ is perpendicular to l denoted by l┴l´, then conversely l┴l´, i.e. l is perpendicular to l´. (Perpendicularity is said to be a symmetric relation.)
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Foundations Of Geometry
10.5
The segment joining a point P to the foot of the perpendicular to a line l from P is smaller than other segments joining P to a point of l. (This is sometimes referred to as the shortest distance property.)
10.6
Consider a point P on a line l. All lines through P will have a symmetric image with respect to l. All the lines but one will be distinct from their corresponding images. The unique line that coincides with its image is said to be the perpendicular to l at P.
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11 Right Angles
The notion of a perpendicular suggests that of a right angle: 11.1
Each of the two half-lines from a point P to a line l form with any other half-line m through P an angle with vertex P The two angles so formed are adjacent and, in general, unequal. They are equal when the half-line m is perpendicular to l. The angles are then said to be right angles.
11.2
The film presents a rotation of two perpendicular lines that hints at various possibilities. For example, between each of the two half-lines from a point P on a line l is a series of half-lines from P. Passing through such positions yields a rotation of one half-line to the other. A rotation can be made in two senses, a clockwise sense and an anti-clockwise sense. The pair of angles, denoted by x and y formed at P by each position of the rotating half-line are adjacent. At first, x<y and
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Foundations Of Geometry
later x>y. The intermediate positions where x = y is that of the perpendicular to l at P. 11.3
Two right angles form a straight angle and four right angles form a plane angle.
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12 Parallels
The film now presents a sequence defining and exploring properties of parallelism. 12.1
Lines that are each perpendicular to a given line are said to have the same direction and are said to be parallel to each other.
12.2
If a line l´ is perpendicular to a line that is in turn perpendicular to a line l, then the lines l and l´ are parallel. This may be written l || l´ or l´ || l. (Parallelism is a symmetric relation.)
12.3
There is one and only one parallel to a given line through a point P not on l. (This is known as Playfair’s axiom in England. It was proved as a theorem in Euclid’s Elements — using an axiom about alternate angles, the so-called parallel postulate. Many textbooks have followed Playfair in using this equivalent form of the postulate. Other forms are 29
Foundations Of Geometry
also available from the film, for example compare item 10.3.) 12.4
A transformation that passes from a line l to a parallel line m through a series of lines all perpendicular to a line that is perpendicular to l and m is called a translation. Translations having the same direction can be compounded since if l is parallel to m and m is parallel to n, then l is also parallel to n.
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13 Bisectors
Two successive sequences indicate how to bisect a given segment and how to bisect a given angle. 13.1
If two points are chosen on the perpendicular to a line so that they are equidistant from the foot of the perpendicular, then every point of the line is equidistant from the two points.
13.2
Reciprocally, if two points A and B are chosen on a line equidistant from a point P of the line, then any point on the perpendicular to the line at P is equidistant from A and B.
13.3
The triangle inequality is invoked to show that for any point P not on the perpendicular, one of the segments PA is less than, or greater than, the other segment PB. The perpendicular is then the complete locus of points equidistant from A and B. It is called the mediator, or perpendicular bisector, of the segment AB.
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13.4
Consider an angle with vertex O and sides a and b with corresponding equal segments OA and OB. AB is a segment whose mediator passes through O. This line divides the angle into two equal parts and is called the bisector of the angle.
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14 Rectangles
The film now considers various particular types of four-sided polygons. 14.1
Two sides of a quadrilateral that do not have a common endpoint are called opposite sides. Two points of the quadrilateral that are endpoints of distinct sides are called opposite vertices and the segments joining these are called diagonals.
14.2
If each pair of consecutive sides of a quadrilateral are perpendicular to each other then it is called a rectangle.
14.3
The mediator of a side of a rectangle is the mediator of the opposite side. The two mediators of each pair of opposite sides are lines of symmetry, or axes of symmetry, for the rectangle in the sense that the rectangle is its own image under line symmetry in either of these lines. These two lines of symmetry meet in the centre of the rectangle.
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Foundations Of Geometry
14.4
Opposite sides of a rectangle are parallel and equal.
14.5
The diagonals of a rectangle meet at the centre and bisect each other at this point. This point is called a centre of symmetry for the rectangle in the sense that the rectangle is its own image under so-called point-symmetry (which is defined in the following section).
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15 Point-Symmetry
The composition of two line-symmetries in two perpendicular lines l and l´ that meet in a point O is called a point-symmetry, or central symmetry, in the point. Various compositions of symmetries of a point are indicated in this section of the film from which various observations may be made. For example: 15.1
A line-symmetry in l followed by a line-symmetry in l´ that is perpendicular to l yields the same image when these symmetries are taken in the reverse order. The result in each case is a point-symmetry.
15.2
A line-symmetry in l followed by a point-symmetry in a point O on l yields the same image when the symmetries are taken in the reverse order. The result in each case is a line-symmetry in the perpendicular to l at 0.
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16 Parallelograms
The notion of point-symmetry is particularly useful in considering the class of quadrilaterals known as parallelograms: 16.1
A quadrilateral whose diagonals intersect at their midpoints is said to be a parallelogram.
16.2
A parallelogram is its own image under point-symmetry in this intersection.
16.3
Opposite sides of a parallelogram are parallel and equal.
16.4
In general, the diagonals of a parallelogram are unequal; a parallelogram with equal diagonals is a rectangle.
16.5
A parallelogram with perpendicular diagonals is called a rhombus. The sides of a rhombus are all equal. If the rhombus is also a rectangle it is called a square.
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17 Projection
17.1
Given a point P of the plane, a line l and a direction L (denoted in the film by a dotted line), the point where the parallel to L passing through P intersects l is called the projection of P on l in this direction.
17.2
When the direction of projection is perpendicular to the line l, then the projection is called an orthogonal or perpendicular projection; otherwise it is called an oblique projection.
17.3
P has as many projections on l as there are directions in the plane, that is to say as many as we please. Any point of the line l can be considered to be a projection of P since the line joining the point to P defines a direction.
17.4
Two points A and B can be projected using the same direction to two points A´ and B´ of a line l. The segment AB is then said to be projected to the segment A´B´. Every point of the segment AB will have its unique projection in the segment A´B´. 39
Foundations Of Geometry
17.5
In particular, the above leads to a fundamental property that the centre of a segment AB projects into the centre of the projected segment A´B´.
17.6
Moreover, taking A to lie on the line l and projecting a segment AB to a segment AC, where the direction of the projection is defined by a point C on l, then the midpoint of AB projects to the midpoint of AC. In other words, the line joining the midpoints of two sides of a triangle is parallel to the third side — and conversely.
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18 Subdivision Of Segments
It has been seen that segments can be laid end-to-end and so added. Equal segments can be laid end-to-end to form multiples. The problem arises of subdividing a segment into a given number of equal segments. The method of projection leads to a solution of the problem by constructing parallels from endpoints of a constructed set of equal arbitrarily chosen segments. Thus, consider a segment AB to be subdivided into n equal segments. Take a half-line from A not lying on AB and construct n equal segments of any arbitrarily chosen length on this line. Join the endpoint X of this construction to the point B. Then XB defines the direction of the parallels from the endpoints of the constructed segments which meet AB in points forming consecutive segments all equal to AB/n. This construction plays a central role in the foundations of geometry. It is known in France as Thalesâ&#x20AC;&#x2122; theorem, a name which is sometimes reserved in England for the property of 15.5.
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19 Measure Of Angle
Thalesâ&#x20AC;&#x2122; theorem may now be invoked to establish a way of measuring projections and so of measuring angles. 19.1
A point is taken on one of the sides of an angle which defines a unit segment on that side from the vertex. The point is perpendicularly projected onto the other side. The projected segment is called the cosine of the angle.
19.2
The cosine is a function of the angle which varies from 1 to 0 as the angle varies from a zero-angle to a right-angle, and from 0 to -1 as the angle varies from a right-angle to a straight-angle.
19.3
The cosine is not dependent on the sense in which the angle is taken.
19.4
If any segment OA, that is a multiple of the unit segment and lying on one side of an angle with vertex O, is perpendicularly projected onto a segment. OA´ on the other side, then the ratio of the segments OA and OA´ is 43
Foundations Of Geometry
equal to the cosine of the angle for any position of A. (This important property may now be generalized to any segment OA.) 19.5
The projection of any segment AB on a line l â&#x20AC;&#x201D; taken perpendicular to a line m â&#x20AC;&#x201D; is measured by the product of the length of AB and the cosine of the angle between the line l and the line m.
Note: The property 19.4 permits the construction of a family of triangles which have an angle in common and whose third sides are all parallel. Such triangles are said to be homothetic. They have their three corresponding angles equal. Now consider two homothetic triangles; displace one of them by, say, a translation, followed by a rotation. The resulting pair of triangles are said to be similar to each other. Similarity includes homothety as a particular case, namely when there is no displacement from the positions in which the triangles have an angle in common.
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20 Pythagoras Theorem
The notion of similarity and that of the measure of a projection may be used to establish an important theorem traditionally ascribed to Pythagoras. This essentially characterizes the metric that can be used to measure any segment of the plane. The film indicates some classical ways of approaching the theorem. 20.1
Consider a right-angled triangle and construct the perpendicular from the vertex of the right-angle to the opposite side, known as the hypotenuse. This divides the triangle into two other right-angled triangles, each of which may be shown to be similar to the original triangle.
20.2 The construction indicates that the perpendicular projections of the two sides a and b, of the right angle together make up the third side c. Expressing this fact in terms of the cosines involved yields the familiar result that the sum of the squares of the measures of the two sides containing the right angle equals the square of the measure of the hypotenuse.
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20.3
The last scene of the film indicates the alternative Greek approach to the theorem through the notion of area which has not been mentioned so far in the film. A rectangle is constructed on the hypotenuse c with one side equal to the projection of one of the other sides, say a. The film suggests that the area of this rectangle is equal to that of the square constructed on the side a. A similar result for the other side now combines to indicate that the sum of the areas of the squares constructed on the sides a and b is equal to the area of the square constructed on the side c. (In particular, the hypotenuse of a right-angled triangle whose sides are unit segments cannot be measured by a rational number — though its area can.)
Note: The proof of the theorem could have been displayed in a few lines had writing been one of the techniques used in the making of the film. Since only a visual language was allowed, a way had to be found of conveying visually all the notions which are part of the proof as outlined in the original notes which served as a basis for these foundations. Viewers need to take the time to translate into formulas the dynamics shown in the successive scenes of the last minute or so of the film. With the notation of the accompanying diagrams, these formulas are as follows: a = c.cosB and b = c.cosA a´ = a.cosB and b´ = b.cosA
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20 Pythagoras Theorem
implying: a´= a2/c and b´ = b2/c
With the relation a´ + b´ = c the theorem of Pythagoras can now be expressed in the familiar form a2 + b2 = c2.
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Notes
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