Geometric Biology
for the Chicago Public Schools
Robert Almgren
University of Chicago
1
I describe a geometry course that I have taught for each of the
last three summers to elementary and high school teachers from
the Chicago Public Schools. The mathematical theme of the
course is transformations of Euclidean space; we motivate this
by looking for applications to understanding the shapes of natural
objects such as starfish and seashells. The course consists of
hands-on construction exercises, theoretical discussions, and
computer exercises with Geometer’s Sketchpad.
1. Introduction
For the past three summers, I have been teaching a geometry course to
teachers from the Chicago Public Schools. This course is part of the SESAME
program—Seminars for Elementary Specialists and Mathematics Educators—
directed by Paul Sally in the Department of Mathematics at the University of
Chicago. The program is funded by fees paid by the schools of participating
teachers, who receive professional development credit towards Illinois State
Endorsement. Until this year, it was restricted to elementary teachers; in 1997
we expanded to include high school mathematics and science specialists.
The SESAME program runs throughout the academic year and for two weeks
every summer, as outlined in Table 1. The academic-year courses are a
systematic introduction to the fundamental areas of mathematics; the summer
courses are more free in their scope. An underlying principle of the program is
that we teach rigorous mathematics in order to raise the level of mathematical
understanding of the teachers. In most of the classes, though not in mine,
extensive exercises are designed and presented for them to take back to their
classrooms.
Each summer course has about half a dozen “counselors,” college students
who are employed full-time for most of the summer, for SESAME and for the
Young Scholars Program for talented high-school students which runs for the
four weeks before SESAME. The counselors help with classroom activities and
help in the computer laboratory. For my course in 1997, I also had generous
volunteer co-teaching by John Zerolis, a graduate of the Chicago Public Schools
and of the University of Chicago, who currently uses geometric visualization
methods for risk management at SwissBank Corporation.
My course is officially titled “Applications of Geometry with Computers,” and is
loosely called “Geometric Biology.” Its theme is the problem of understanding
why objects in Nature have the shapes they do. The mathematical content is
transformations of Euclidean space, and symmetry properties under these
1
Dept. of Mathematics, 5734 S. University Ave, Chicago IL 60637; almgren@math.uchicago.edu
Table 1: The SESAME Program
Academic Year Courses
10 Sessions: January to June (3 hours every two weeks)
2 Lane credits per course. Normal load: one course each academic year
• Geometry
• Number Theory
• Probability and Statistics
Summer Courses
10 Sessions: July/August (3 hours each day for two weeks)
2 Lane credits per course. Normal load: 2 courses each summer
• Applications of Geometry with Computers (“Geometric Biology”)
• Computer Science: Programming and Applications
• Mathematics for Elementary Schools
• Teaching Mathematics: Grades 6-8
• The History of Mathematics
• Introduction to Algebra
transformations. The syllabus is given in Table 2. This year for the high school
teachers, I ran the course at a slightly more advanced and more abstract level
than for the elementary teachers, but the content was more or less the same.
The summer SESAME courses meet three hours every day for two weeks at
the end of July and the beginning of August. In my course, we spend roughly
the first hour and a half of each day in the classroom, doing hands-on
manipulative exercises and discussing fundamental principles of geometry. We
spend the second hour and a half in the Macintosh computer laboratory working
with the Geometer’s Sketchpad dynamic geometry program; written exercises
are given to the students so that they may further explore the examples
demonstrated in class and construct their own sketches and demonstrations.
An essential feature of the class design is that every idea is presented in three
mutually reinforcing ways:
1. Theory: at blackboard in class with accurate mathematics;
2. Hands-on: drawing and construction exercises; and
3. Computer lab: learning to use Geometer’s Sketchpad.
In particular, the combination of hands-on manipulation and dynamic
geometry software is a very powerful one: after the students have explored both
the concreteness and the limitations of drawing exercises, they readily
appreciate the flexibility and power of good computer software. They can also
then appreciate the mathematical theory which completes and solidifies the
reasoning suggested by physical experiment.
The emphasis on transformations in elementary geometry is a rather modern
idea in elementary mathematics education, though it is at the foundation of
mathematics as it is used in practice and as it is taught at the college level. To
the elementary teachers, whose last exposure to high-school level geometry may
have been many years ago, the idea of geometry through transformation is
Table 2: Geometric Biology Syllabus
Euclidean Transformations
• Week 1: Isometries and Congruence
∗ Translations and rotations
∗ Reflections
∗ Symmetries
• Week 2: Dilations and Similarity
∗ Classification of “strip patterns”
∗ Point dilations
∗ Screw transformations
∗ The pantograph
∗ Spirals and seashells
∗ Plant growth and Fibonacci numbers
radically new. Many of the high-school teachers work with more modern texts in
their own classrooms, and are well familiar with these new ideas. Thus, this year
we have been able to push beyond the simple concepts of transformations as
independent objects worthy of study into the beginnings of group theory.
The importance of dynamic geometry software in the teaching of geometry is
probably well understood by most readers of these proceedings, and several
other articles in this volume address that issue. I would like to focus here, first
on the overall structure of my course and the sequence of ideas, and second, on
the physical manipulatives we use to convey tangibly the geometric ideas.
Space does not permit a complete account of the course but I hope to convey
some of the flavor. We have a complete set of written notes which I would be
happy to share with anyone who is interested.
2. Composition of Euclidean transformations
Our discussion of isometries of the Euclidean plane is a good example of the
three-pronged approach outlined above.
We define an isometry: a transformation of Euclidean space which preserves
distances. Clearly, if we start with a sheet of paper flat on the table, then move it
to any other position flat on the table (without stretching, tearing, or folding), the
overall motion is a plane isometry, since every pair of two points ends up exactly
as far apart as they were originally. (It is not obvious that all isometries can be
implemented as motions of a sheet of paper, but we return to this question later.)
Thus we have a physical model of the mathematical idea of an isometry. We
immediately identify the fundamental operations of translation, rotation, and (the
black sheep of the family) reflection.
Now we can ask questions about the composition of isometries: for example,
can an arbitrary combination of rotations and translations always be expressed
as a single rotation or reflection? As a particular case, we define SA to be the
operation of rotating the two-dimensional Euclidean plane (the sheet of paper)
about a fixed point A by a half-turn (180°), and ask
1. What is the effect of rotating about point A, then about point B; in
symbols, what is the operation SASB?
2. What is the effect of three successive rotations about three general points
A, B, and C, or SASBSC?
We search experimentally for the answers, using a ½” foam board,
plain drawing paper, and tracing paper, all readily available at art
stores for a few dollars. We attach a sheet of drawing paper to the
foam board using draftsman’s masking tape, and draw an arbitrary figure on the
paper. A good figure has no special symmetry, and is composed of only a few
straight lines—I like to use a simple “chair” shape. We can then copy the figure
to an overlying sheet of tracing paper, move the tracing paper in some way, and
transfer the figure back to the drawing paper by sticking a pin through the ends
and corners of the line segments. Removing the tracing paper and drawing lines
between the pin holes, we obtain the transformed copy of our original figure.
We then need to specify how to move the tracing paper so as to implement
the basic Euclidean isometries. Suppose a point A is located on the paper. We
draw a short straight line segment through A with any orientation. Then we put
the tracing paper down, copy our figure to it, and also copy the short segment
through A. By sticking a push-pin through the paper at A and spinning the
tracing paper until the segment superimposes on itself, we implement a half-turn
rotation about A. Rotations by other angles can be constructed similarly.
Given two points A and B, we implement translation by the vector AB by
drawing a line through A and B, copying A to the tracing paper, and sliding the
tracing paper along the line until A superimposes on B. Reflection in the line AB
is implemented by lifting the paper up, flipping it over, and laying it down so that
the copies of A and B superimpose on their original positions.
The foam board / tracing paper combination is thus an effective laboratory for
quantitatively exploring Euclidean transformations. We can thus carry out the
indicated half-turn rotations about points A, B, and C, and experimentally look for
the answers to the questions above.
The first answer, of course, is that rotation about A, then about B, is
equivalent to translation by twice the vector from A to B. The counselors and I
2
elicit this observation by asking appropriate questions : Because the paper
finishes “right-side up,” it is plausible that the final operation is a translation.
Since the only given points are A and B, this translation should be somehow
expressible in terms of A and B. It appears to be parallel to AB, and one can
2
Once a teacher asked me, “Do you and the counselors all take a course in asking questions?” I
said we didn’t, and asked what prompted that question. She said, “Because I notice that when
you are working with us, all of you tend to ask the same kinds of questions, and I wondered if you
learned that in a specific course.” She had very perceptively noticed the common culture of
mathematics that is transmitted by higher level study in college, and to which we are trying to
expose these teachers, if only briefly.
guess, and verify by measurement, that the distance is twice the distance from A
to B.
Similarly, one can guess that three half-turn rotations about A, then B, then C
are equivalent to a single half-turn rotation about a fourth point D, so that ABCD
is a parallelogram.
Of course, at this stage these results are only conjectures, and we are not
done until they are proved rigorously. To do this, we first show that a plane
isometry with two fixed points must be either the identity or a reflection. (A good
question is, why can’t it be “half the identity and half a reflection,” as, for
example, by folding the paper along a line. Answer: that is not an isometry.)
This theorem has an obvious physical interpretation that if two pins are stuck
through the paper, it is fixed and cannot move. We also argue that if we move
the paper in the plane without flipping it over, the result cannot be a reflection.
The proof is completed by tracing the images of the particular points A and B
under the indicated sequence of operations. A and B may be brought back to
their original positions by appending either a translation or another half-turn
rotation (here we also need the idea of the inverse of an isometry); since the
overall operation has A and B as fixed points and is not a reflection, it must be
the identity.
Following this experimentation and discussion, I
demonstrate a Sketchpad construction of the same
experiments. As the original figure and rotation points
B
A
are dragged around, and the entire picture updates
dynamically, the relationships that took an hour to
discover with pencil and paper leap out at the class and
the power of the computer is immediately apparent.
Then the class breaks for the computer lab, where we give them systematic
exercises to lead them to construct the same experiment themselves. Many
other explorations are possible using Sketchpad’s transformation facilities.
3. Reflections
We then continue to a discussion of reflections. We prove that every plane
isometry can be written as the product of at most three reflections, and hence
that two reflections suffice if the isometry is direct. Of course, two reflections are
equivalent to a translation if the lines of reflection are parellel, and to a rotation if
they intersect. To make these theorems tangible, we look for a physical
implementation.
As noted above, reflections can be implemented with tracing paper, but this is
rather cumbersome and slow. Far more effective is two mirrors, taped together
to form a dihedral wedge that stands upright on the paper surface. John Zerolis
discovered that mirrored plexiglass, for making mirrored drop ceilings, is
lightweight and sturdy, has no sharp edges, and is easily available at home
improvement superstores (for example, Home Depot in Chicago).
We verfy experimentally that two reflections can reproduce any direct isometry
by placing two identical erasers on the table in arbitrary different positions and
orientations. By appropriately positioning the pair of mirrors, the twice-reflected
image of the first one can always be made to superimpose on the second one.
This is easily tested by moving the line of sight up and down across the edge of
the mirror: the direct view is seen by looking over the top of the mirrors; the
reflected image by looking “through” the mirrors.
We close our discussion of reflections and symmetry groups by discussing the
finite groups generated by rotations and reflections passing through a common
point. We derive the structure of the cyclic groups Cn and the dihedral groups Dn,
and show that these are the only finite groups of isometries in the plane.
Examples of patterns having the cyclic and dihedral groups as their symmetry
groups can be constructed, either by using the mirrors as a kaleidoscope, or by
cutting shapes from paper.
Take a sheet of paper. Cut a slit from the outer edge to a point in the center.
Then roll the paper n times around into a cone, so that the two edges of the slit
come back to the same location (with several thicknesses of paper in between).
Then cut an arbitrary nonsymmetric pattern into the rolled-up paper cone; when
the paper is unrolled, a pattern with symmetry Cn is automatically produced.
If you press the cone flat before cutting, so as to superimpose opposite sides,
then you introduce two mirror symmetry planes, and the resulting pattern has
symmetry Dn. It is clear from this construction that it is never possible to have a
single plane of mirror symmetry if any rotational symmetry is also present.
By rolling a strip of paper into a cylinder, patterns can be made with
translation symmetry and various combinations of rotation and reflection
symmetries.
This leads naturally to a discussion of “frieze patterns,” and their classification
into seven possible combinations of symmetries, which are most simply
illustrated by patterns of letters. I also present various examples of symmetry in
decoration and art, and in the physical world of shapes of animals.
Example
Translation
… FFFF …
… SSSS …
… DDDD …
… AAAA …
…IIII…
… DWDM …
… MWMW …
X
X
X
X
X
X
X
Left-right
reflection
Up-down
reflection
Half-turn
Glide
reflection
X
X
X
X
X
X
X
X
X
Of course, all these symmetry patterns are simply illustrated using Geometer’s
Sketchpad, and I ask the students to construct examples of each.
4. Geometry theorems using transformations
There are a number of little theorems whose statements do not involve the
language of transformations, but which can most easily be proved using
transformations. I like to do a few of these to show the use of transformational
language.
R
Theorem: Take any triangle ∆ABC, and erect
squares ACQP on AC and BCRS on BC as
shown. Draw the segment PS, which is bold
Q
in the picture. Let D be the center of the
S
C
square erected on AB. D is the midpoint of
D
segment PS. Thus the midpoint of PS stays
fixed as C is moved with A and B held fixed,
P
which is easily demonstrated using Sketchpad.
A
B
Proof: Consider the transformation of the
plane consisting of a clockwise rotation of 90° about A followed by a clockwise
rotation of 90° about B. By a law of composition of rotations (discovered
experimentally), this is the same as a half-turn about D.
When we rotate about A, point P goes to point C. When we rotate about B, point
C goes to point S. Thus, a half-turn about D sends P to S, so these two points
are symmetric about D, and D is the midpoint of PS.
P
Theorem (Napoleon): Take an acute triangle ∆ABC, and erect
C
equilateral triangles on the edges, with outer corners P, Q, R.
Q
The segments AP, BQ, CR intersect at a common point F, and
F
A
B
meet at 60° angles. This point is called the “Fermat point,”
and is the point that minimizes the total distance AF+BF+CF.
Proof: The argument is based on 60° rotations about the
R
corner points, along similar lines to the previous theorem. The
proof that F minimizes the sum of lengths proceeds by rotating the candidate
point F, and showing that the minimizer must lie on the straight line between
opposite corners.
We illustrate this theorem by threading three cords through holes in a rigid
board. If the cords are tied together above the board, and weights are hung on
them underneath, then (with allowances for friction) the knot comes to rest at the
Fermat point of the holes.
Theorem: Take any triangle ∆ABC, and erect squares on
each side, whose centers are P,Q, R as shown. Then the D
C
P
segments PQ and CR are perpendicular and have equal
length.
Q
Proof: Consider the spiral transformation consisting of a
B
A
45° rotation counterclockwise about A and a dilation by
√2. This takes ∆CAR to ∆DAB. Similarly, the spiral
R
transformation consisting of a 45° rotation clockwise
about C, and dilation by √2, takes ∆PCQ to ∆BCD. The segment BD is common
to both of these “expanded” triangles. Since the expansion factors of the two
spiral transformations were the same, the original lengths PQ and CR were the
same; since the rotations were 45° in opposite directions, the angle between PQ
and CR is 90°.
5. Dilations
The theory of dilations is not difficult. But, unlike translations, rotations, and
reflections, which can be implemented on paper by sliding tracing paper or by
using mirrors, implementing dilations in the physical world is something of a
challenge. We considered two different ways that dilations can be implemented.
5.1 The pantograph
The pantograph is a simple
The
mechanical linkage which has
been known for a long time
Pantograph
(Thomas Jefferson is said to
have used one to make file
copies of his handwritten
O
P
P
letters). With point O fixed,
A
and with the rigid rods joined
B
(m AB )
= 4.000
(m CD)
by hinges at the points where
C
D
they cross, either point P0 or
point P1 can be moved, and the other will trace a scaled image. We distributed
simple plastic pantographs (available for a few dollars from educational supply
companies) and explored their properties. It takes a little practice to use such a
device: the key thing is to hold it at the point which is doing the drawing (either
P0 or P1), while watching the point which is tracing the drawing.
This figure illustrates a Sketchpad construction which we gave the students as
a directed exercise (we explained to them how to do it.) Moving points A, B, C,
and D, controls the proportions of the similar triangles, and hence the expansion
ratio. With the Sketchpad model, unless you “cheat” by using dilations, only one
of P0 or P1 can be moved, unlike with the physical object.
E
Constructing linkages is an excellent exercise for
B using Sketchpad, since one must think carefully
Peaucellier's
about the chain of logic that Sketchpad follows in
Linkage
determining your figure from the given inputs.
D
Other linkages can
A
carry out other
Lemmiscate of
D
Bernoulli
C
geometrical
M
operations of interest.
Peaucellier’s linkage implements the operation of
A
B
inversion in a circle (centered at point C in the
C
figure above). A good exercise is to figure out the
radius of the circle of inversion. As Sketchpad
exercises, we asked them to construct this
3
linkage, and also a simple linkage which draws the lemmiscate of Bernoulli.
3
A good source for this and many other interesting examples is The Penguin Dictionary of
Curious and Interesting Geometry, by David Wells, Penguin 1991.
5.2 Making a seashell from clay
Patterns similar to themselves under combinations of rotations and dilations
are spirals. We discuss the Archimedean (constant width) spiral and the
equiangular spiral, and, with the help of overhead transparancies reduced on the
photocopy machine, show that the latter but not the former has spiral symmetry.
We also (on the blackboard and with Sketchpad) stack squares together into
spirals, which leads to discussion of Fibonacci numbers and the golden ratio.
One of the most beautiful and informative examples of symmetry in Nature is
4
the shell of the Nautilus. As D’Arcy Wentworth Thompson observes,
In the growth of a shell, we can conceive no simple law than this,
namely, that it shall widen and lengthen in the same unvarying
proportions: and this simplest of laws is that which Nature tends to
follow. The shell, like the creature within it, grows in size but does
not change its shape; and the existence of this constant relativity of
growth, or constant similarity of form, is of the essence, and may be
made the basis of a definition, of the equiangular spiral.
The key point is that he gives a simple mechanical reason why one expects
the shape of the shell to be this particular mathematical form. Since living
creatures metabolize and grow at rates proportional to their current size, it is
reasonable for the animal that creates the shell to grow by a constant fraction
each time it adds a new piece of shell (rather than adding a constant mass each
time). How can we model this self-similar growth using inert materials?
We want to make a model shell from clay, by sticking together clay balls
whose sizes increase in a gradual geometric progression. Thus the question
becomes, how can we devise a geometric construction to make a sequence of
balls in which each one is, say, 20% bigger than the one before it?
Here is a fairly simple procedure. Take a chunk of modeling clay. Roll it back
and forth on the table, to shape it into a long tube of constant cross-sectional
area. With the amount of clay we use, this tube comes out about a meter or two
long, when it is between half a centimeter and a centimeter in diameter.
Then, draw an angle on a sheet of paper. On the
lower arm of the angle, mark points spaced equally
one centimeter apart. On the upper, mark points
spaced 1.2 centimeters apart. Draw straight lines
across to connect corresponding points. We can now
clip off lengths of clay with a 20% increase from one
to the next by the following scheme.
Start with any reasonably short length of the clay strip. Stretch it out along the
bottom angle of the wedge with one end at the vertex. Trace the parallel lines
across from its other end to the other side of the angle; by eye is accurate
enough. Clip off another length of the strip, whose length is the distance from
4
On Growth and Form, Cambridge University Press 1942; page 757 of the 1992 Dover reprint.
This is a wonderful book that should be on the shelf of everyone interested in the shapes of
natural objects, full of fascinating examples. One of Thompson’s main virtues is that he does not
only claim what he thinks ought to be true, but tests everything by comparison with the real world.
the vertex to that point. Now move the new strip down to the bottom of the
wedge, and repeat. When the lengths of clay are lined up parallel to each other,
they form a very nice graph of an exponential function.
Since the clay strip had uniform thickness, the volume of the pieces of clay is
in proportion to their lengths. Thus, once all the separate pieces have been
formed, they may be crumpled up into balls, with a 20% volume increment.
Since a 20% volume increase corresponds to only about a 6% increase in linear
dimension, i would have been difficult to make these balls with a precise growth
factor by attempting to work directly with the ball radius.
To make an attractive spiral seashell, the best procedure is actually to form
the balls into stubby cylinders, each one having its diameter roughly equal to its
own height. Then the cylinders can be stuck together end-to-end, forcing them
around into a circular spiral, and a very nice nautilus shell results.
6. Conclusions
We have outlined a few of the constructions we use in this two-week course,
primarily to illustrate how hands-on construction, mathematics theory, and
computer laboratory can work together to reinforce fundamental geometric ideas.
We cover a few other topics of which space does not permit discussion here: of
these, the most interesting is the prevalance of Fibonacci numbers in spirals of
plants such as broccoli, pineapple, sunflowers, and pine cones (all illustrated
with actual examples). And this year, John Zerolis gave a very interesting
presentation of the use of triangle geometry to model the relationships between
volatility and correlations of financial assets.
One area which we have not yet been able to cover as much as I would like is
geometry in three dimensions, such as solid geometry and symmetries of
polyhedra. Hands-on manipulations are more difficult than in the plane, and
adequate software does not yet exist. This year, though, I have a student
working on writing three-dimensional software, which I hope we will be able to
use in future years.
Acknowledgement
My participation in SESAME is supported by award DMS-9502059 from the National
Science Foundation under the CAREER program. This program is intended to support
academic faculty at the start of their careers; it “enhances and emphasizes the
importance the Foundation places on the development of full balanced academic
careers that include both research and education.”