Teaching Mathematics and its Applications Advance Access published May 18, 2006
TEACHING MATHEMATICS AND ITS APPLICATIONS, 2006
1 of11
doi:10.1093/teamat/hrl004a
Experimental mathematics
in the curriculum (part 1)
J. P. Ward
Submitted March 2006
Abstract
Much of modern mathematical research requires a serious level of commitment and
ability on the part of the researcher and, as a natural consequence, precludes any
real involvement by undergraduate mathematicians. However, to put it crudely,
Computer Algebra Systems, can make experts of us all. With the use of such systems
even the average undergraduate mathematician can aspire to discover interesting
yet still unexplained behaviour in many areas of mathematics. Of course, interesting
results still need a true expert to furnish proofs. This article explores the area of
experimental mathematics and, for a particular problem, the so-called Bu¡on
puzzle, demonstrates how it can be made accessible to undergraduates. We give
a detailed lecture by lecture account of the description of the puzzle, its solution,
including rigorous proofs. We argue that mathematics research, from discovery,
through proof to prediction, should play a prominent role in the undergraduate
curriculum. In experimental mathematics this is certainly feasible.
In part 2 of this work, I consider possible extensions of the basic Bu¡on puzzle
continuing with a lecture by lecture delivery at a level accessible to ¢rst year
undergraduate mathematicians.
1. Introduction
It is rare that a research problem can be successfully reworked (some would say watered down) so
as to be accessible to undergraduate mathematics students. Either the original concept is too deep
requiring a sophisticated knowledge of mathematics to appreciate its importance and meaning
or else the analysis used to solve the problem is beyond the scope of most undergraduates. Yet,
it is important that undergraduates get exposure to real research not only because it should be
a bona fide part of their undergraduate degree but also because they can then make informed
choices as to their career paths on graduation. It is also surely beyond doubt that students are
better able to absorb abstract techniques/methods if they are employed on a real problem.
The following topic arose out of real research, albeit in the area of experimental mathematics.
(Using music as an analogy, experimental mathematics is akin to folk music, you can do it almost
anywhere, you do not need to be particularly gifted, you can teach yourself, the down-side is that,
often, passers-by throw coins at you). As readers will be able to judge for themselves, the
‘problem’ described (the so-called Buffon puzzle) is interesting in its own right, and the method of
The Author 2006. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications.
All rights reserved. For permissions, please email: [email protected]
2 of11
TEACHING MATHEMATICS AND ITS APPLICATIONS, 2006
solution covers many areas of mathematics all of which are well within the compass of a typical
first year university student. The areas involved are:
.
.
.
.
.
.
.
.
history of mathematics;
regular and irregular polygons, polyhedra;
simple graph plotting and interrogation using a computer algebra system (CAS);
matrices: permutation and circulant matrices;
eigenvalues and eigenvectors;
complex numbers and vectors;
linear algebra, linear independence, dimension and spanning sets; and
computer algebra.
The solution to the Buffon puzzle makes no significant references to either calculus or analysis
and it has the advantage of having a reasonably obvious step by step approach to its solution.
The problem is easy to visualize and understand and, if students are allowed a hands-on
approach, has a genuine wow factor. At the moment, this puzzle and its proof appears not to
have any genuine applications—it simply sits there and basks in its own glory. It is intrinsically
interesting—this is mathematics research at its most basic level.
Of course, in common with many research topics, the complete problem could be described in a
1 h seminar. But this is missing the point. The purpose is to slow the presentation down so that an
average undergraduate cannot only follow each step but also take part. When slowed down, the
1-h seminar becomes a full module of 24 contact hours.
Although we are trying to introduce experimental mathematics into the curriculum, it cannot
be done in an unstructured way (to return to the music analogy, this is not jazz). Experimental
mathematics can often drift into areas that are too sophisticated and difficult for the average
student to deal with. If this is to be a successful exercise the student should always feel in control
and comfortable with the mathematics s/he is expected to use. Much of the work described here
will be carried out by students though the lecturer (dare we refer to them as facilitators?) must be
on hand to help with some of the more taxing parts.
2. The Buffon puzzle
Like many puzzles the Buffon puzzle is easy to describe and appreciate but less easy to analyse
mathematically. The puzzle, attributed to Count Buffon, the French naturalist, is as follows.
Consider an arbitrary polygon. Generate a second (called a descendant) by joining the centres of
consecutive edges together (see Fig. 1A). It is found that on repeating the construction over
and over, a certain regularization takes place: as I show (3), irregular polygons are transformed,
in the limit, to affine regular polygons. This is a truly remarkable result and will fascinate
students as order will appear to come out of chaos. To give a rigorous proof of this result (and
its generalizations), the student will encounter many areas of mathematics, but the level of
mathematics that is required to understand the proof is not high.
The puzzle first appeared in Edward Riddles’ edition (1840) of the Recreations in Mathematics
and Natural Philosophy of Jacques Ozanam. The behaviour observed in the puzzle has been
noticed since Roman times. When creating mosaics Roman craftsmen observed that original
designs of squares within squares would become more and more regular even though there were
initial errors associated with the first outer square (1). The puzzle has also been described in some
detail (but with no hint of a solution) in the popular mathematics book ‘You are a
Mathematician’ (2). A partial solution was reported by Berlekamp, Gilbert and Sinden (4)
TEACHING MATHEMATICS AND ITS APPLICATIONS, 2006
3 of11
Fig 1.
who were more interested in the occurrence of convex descendents from a given initial
polygon. We note that there are fairly obvious extensions of the puzzle that can be considered
(as in Fig. 1B: division by a fixed ratio 0 p 1, or as in Fig. 1C in which the new vertices are the
centroid of centroids of adjoining edges).
In the following section, we present a step by step approach to the analysis of this puzzle and
we envisage that this is delivered to students over a period of 12 weeks.
3. Lecture plan, Lectures 1^12
3.1. Lecture 1: The Buffon puzzle
Start with a general (convex or concave) quadrilateral: show that the first and hence all
subsequent descendants are parallelograms. Ask for a proof using vectors.
Show it only works for side divisions which are 1:1. Show that it does not work for higherorder polygons.
Show the regularization effect with a pentagon (convex and concave). Get students involved.
Give out a blank paper with only the outline of the outer pentagons shown in Fig 2. Get them to
join the mid-points of consecutive sides by eye and to repeat this construction at least five times.
Get students to conjecture what kind of regularization is observed.
3.2. Lecture 2: stars, degenerates and convex regular polygons
Since the discussion we are about to undertake involves innumerable references to polygons, it is
wise to set up these constructs using a decent notation. This material is standard though not often
seen by undergraduates.
Define a regular polygon with n sides and let the length of one side be 2‘1. Let R be the
radius of the circumscribing circle. Vertices are numbered as shown in Fig 3A. We label this
polygon Pn1 .
Clearly ‘1/R ¼ sin(/2) ¼ sin(/n). Now, on n vertices we can construct other polygons.
The polygon formed by joining vertices 1 to 3 to 5 . . . and back to 1 is labelled Pn2 . In general
4 of11
TEACHING MATHEMATICS AND ITS APPLICATIONS, 2006
Fig 2.
Fig 3.
(Fig. 3B), the polygon Pnq has side of length 2‘q and joins vertices
1 to ðqÞmodn þ 1 to ð2qÞmodn þ 1 to ð3qÞmodn þ 1 . . . to 1 q ¼ 1; 2; . . . ; n 1
This is a polygon with side 2‘q where ‘q/R ¼ sin(q/n).
Note that Pnq may not have n distinct sides. Produced, in this way, there are always (n 1)
polygonal versions. See Fig. 4 for examples in which n ¼ 6, 7.
A polygon, constructed in this way, is called degenerate if it contains fewer than n vertices.
If orientation is ignored, some of the polygons in the sequence may be copies of others (in Fig. 4
we have omitted to include copies). Some (if n is large enough) may be degenerate forms of
degenerate forms. For example, a hexagon has two degenerate forms and no star versions.
A pentagon has no degenerates (like all prime-numbered polygons) but one star. The 12-gon
has a degenerate hexagon (which has a degenerate triangle and digon), a degenerate square and
a single star version.
For a given n the various polygonal versions can be placed in a definite order: Pn1 ; Pn2 ; . . .. If we
omit to include copies and if n is even, this sequence ends with a degenerate digon Pnn=2 whereas if
n is odd it ends with a star Pnðn1Þ=2 .
Get students to validate these formulae for particular values of n.
Introduce the idea of semi-regular polygons (i.e. those with equal sides or with equal angles;
see Fig. 5 in which the inner hexagon is equiangular whereas the outer hexagon has sides of equal
length). Get them to suggest semi-regular quadrilaterals.
TEACHING MATHEMATICS AND ITS APPLICATIONS, 2006
5 of11
Fig 4.
Fig 5.
3.2.1. Further work
(a) Consider Euler’s function and its relation to the number of star versions of a polygon [(n)
is the number of numbers less than n and co-prime (no factors in common) to it].
(b) Look at the ratios of sides to parallel diagonals in regular polygons: derive general formulae
in terms of the order of the polygon and the length of one side. Obtain the golden ratio for
pentagons.
(c) Look at the history of polygons and polyhedra. Get students to find out about the Egyptian/
Persian/Indian/Greek/Roman interest in this area, Platonic solids, Archimedian solids,
Kepler, Poinsot, etc., and Gauss and the 17-gon.
3.3. Lecture 3: use of a computer algebra system
Help students write a simple program to draw an n-sided polygon with vertices placed in random
positions and to create descendants. In the simple program which follows, each side is divided
6 of11
TEACHING MATHEMATICS AND ITS APPLICATIONS, 2006
into two equal parts. If the original polygon is left on screen the final iterated polygon will hardly
be seen (it shrinks to the centroid of the original polygon). If the user wishes to see both the
original and the iterated polygon, they should uncomment the final line.
The following is a Matlab program:
1: clg;
2: noiterations ¼ 20;
3: nvertices ¼ 5;
4: counter ¼ nvertices þ 1;
5: x ¼ rand(counter,1).*10;
6: y ¼ rand(counter,1).*10;
7: x(counter) ¼ x(1);y(counter) ¼ y(1);
8: xx ¼ x;
9: yy ¼ y;
10: for k ¼ 1:noiterations
11: for m ¼ 1:nvertices
12: X(m) ¼ (1/2).*(x(m þ 1) þ x(m));
13: Y(m) ¼ (1/2).*(y(m þ 1) þ y(m));
14: end
15: X(counter) ¼ X(1);
16: Y(counter) ¼ Y(1);
17: for m ¼ 1:nside
18: x(m) ¼ X(m);
19: y(m) ¼ Y(m);
20: end
21: end
22: plot(x,y,’k’);
23: hold on;
24: %below is original polygon (expanded in size)
25: xxx ¼ xx
26: yyy ¼ yy
27: %plot(xxx,yyy, ’g’)
3.3.1. Further Work
(a) Get students to interrogate the final polygon to find the ratios of sides to parallel diagonals.
Get them to compare these with ratios obtained from regular polygons.
(b) Get students to extend the program to describe the division process other than 1:1 (Fig. 1B).
(c) Get students to consider division according to Fig. 1C and to note their observations.
(d) Encourage students to investigate the regularization that occurs.
(e) Encourage students to look at external division of a side and even to contemplate division of a
side by a complex number.
TEACHING MATHEMATICS AND ITS APPLICATIONS, 2006
7 of11
3.4. Lecture 4: introduce affine geometry
Definition: An affine transformation of the plane is a function of the form
Tðx; yÞ ¼ ðax þ by þ e; cx þ dy þ f Þ
ad bc 6¼ 0:
where a, b, c, d, e, f are constants.
Two geometrical entities are affinely equivalent if one can be obtained from the other by an
affine transformation.
Derive or simply state the two main properties:
(1) T takes lines to lines and parallel lines to parallel lines; that is, if L is a line, then so is T(L).
If lines L1 and L2 have the same slope then the lines T(L1) and T(L2) also have the same slope
(though not necessarily the same as the original).
(2) Ratios between corresponding segments on parallel lines are preserved.
Explain that the concept of angle does not exist in affine geometry.
Encourage students to imagine the effect of an affine transformation on a square, on other
polygons, and on a circle. Give them time to see a possible connection between affine geometry
and the Buffon puzzle. Show that an affine transformation transforms x2 þ y2 ¼ r2 into an
ellipse—prove that an ellipse is always obtained as long as ad bc 6¼ 0. Encourage more interested
students to define what an affine transformation in three dimensions might be. Show that
an affine version of the sphere x2 þ y2 þ z2 ¼ r2 is an ellipsoid (subject to a similar constraint).
For both of these they will need to familiarize themselves with quadratic forms, quadric surfaces
and descriminants.
3.4.1. Important stage
Conjecture that polygons obtained by iteration are affine regular polygons (i.e. obtained from a
regular polygon by an affine transformation). Dwell on how remarkable this result is.
3.5. Lecture 5: the buffon transformation matrix
Use of vectors to describe the transformation taking a polygon with vertices placed in arbitrary
positions to a descendant (Fig. 6).
Fig 6.
8 of11
TEACHING MATHEMATICS AND ITS APPLICATIONS, 2006
From the figure:
1
r01 ¼ ðr1 þ r2 Þ;
2
1
r02 ¼ ðr2 þ r3 Þ; etc:
2
Defining column vectors:
r ¼ ½r1 ; r2 ; . . . ; rn T
r0 ¼ ½r01 ; r02 ; . . . ; r0n T
then the transformation can be written in matrix form:
2
3
1 1
62 2 0 ... 07
6
7
6
7
60 1 1 ... 07
6
7
0
2
2
r ¼ Tr T ¼ 6
7
6. . .
7
.
.
6 .. .. ..
.. 7
..
6
7
41
15
0 0 ...
2
2
This describes a single transformation. After k such transformations:
rðkÞ ¼ Tk r
Students should be asked to show (using these relations) that the centroid of the new polygon is
the same as the centroid of the old one.
Students should be asked to produce an appropriate transformation matrix to describe the
hybrid Buffon transformations described in Fig. 1B and C.
3.6. Lectures 6, 7: eigenvalues and eigenvectors
Here, the lecturer could possibly digress from the main discussion and consider revising (or
introducing if necessary) eigenvalues and eigenvectors. A discussion involving the characteristic
equation, repeated eigenvalues, linear independence of eigenvectors would be useful.
3.7. Lecture 8: Introduce the n n permutation matrix
2
0
0 0
... 0
1
3
61 0 0 ... 0 07
6
7
6
7
6
7
0
1
0
.
.
.
0
0
P¼6
7
.. .. 7
..
6 .. .. ..
4. . .
. .5
.
0 0 0 ... 1 0
Get students to derive its properties:
Pei ¼ eiþ1
and Pen ¼ e1
where ei is an n 1 column vector with zero components except that the ith component is 1.
Get students to show that the characteristic equation for P is
ln 1 ¼ 0
TEACHING MATHEMATICS AND ITS APPLICATIONS, 2006
9 of11
(This looks daunting but drops out quite quickly. It is quite a decent exercise on manipulating
n n determinants).
Obtain the eigenvalues in complex exponential form:
lk ¼ e2ik=n
k ¼ 1; 2; . . . n
Get students to confirm that the eigenvectors corresponding to eigenvalue ¼ e2ik/n take the
form
Xk ¼ ½k ; 2k ; . . . ; nk ¼ 1T
k ¼ 1; 2; . . . ; n
3.8. Lecture 9: introduce the circulant matrix
The n n circulant matrix C has the form (5):
2
c0 cn1
6 c
c0
6 1
C¼6
.
6 .
4 .
cn1
cn2
...
c1
3
cn1 . . . c2 7
7
7
7
5
...
c0
Explain that the transformation matrix T is a special case of a circulant.
Obtain the relation between the permutation matrix and the circulant:
C ¼ fðPÞ where fðxÞ ¼ c0 þ c1 x þ . . . þ cn1 xn1
Show that the eigenvalues of C are f(nr), r ¼ 1, 2, . . . , n with the same eigenvectors as that of
P as described earlier.
3.9. Lectures 10, 11: results from linear algebra
This would be a good time to revise linear algebra, linear independence, spanning sets, etc.
Show that T is a circulant matrix with eigenvalues:
1 1 2im=n
þ e
m ¼ 1; 2; . . . ; n
2 2
These are positioned symmetrically on the circle z ¼ (1/2) þ (1/2)eit in the Argand plane.
See Fig. 7. The eigenvalues of maximum modulus (other than ln ¼ 1) are
1 1
l1 ¼ þ e2i=n ¼ cos ei=n and ln1 ¼ cos ei=n ¼ l1
2 2
n
n
in which * denotes the complex conjugate. The corresponding eigenvectors are
T
and Xn1 ¼ X1
X1 ¼ e2ði=nÞ ; e2ð2i=nÞ ; . . . ; 1
lm ¼
All the eigenvalues are distinct. Hence the corresponding eigenvectors are linearly independent
and therefore, an arbitrary n 1 column vector can be expressed as a linear sum of the
eigenvectors
r ¼ 1 X1 þ 2 X2 þ . . . þ n Xn
10 of11
TEACHING MATHEMATICS AND ITS APPLICATIONS, 2006
Fig 7.
therefore,
Tkr r ¼ 1 lk1 þ . . . n T n Xn
in which T n Xn ¼ Xn
hence,
Tkr r n Xn ¼ 1 lk1 X1 þ . . . þ n1 lkn1 Xn1
Ignoring the eigenvalue l ¼ 1 (which, as we have seen, can be effectively eliminated by a change of
origin to the centroid), we see that T has two eigenvalues of maximum modulus, l1 and ln1 ¼ l1
with corresponding eigenvectors X1 ; X1 ; so as n ! 1
Tk Y n Xn ! ð1 lk1 X1 þ n1 lk
1 X1 Þ
where 1 and n1 are complex and X1 ¼ [e2i/n, e4i/n, . . . , 1]T and Xn1 ¼ X1 .
3.10. Lecture 12: explaining the Buffon puzzle
Ask students to express an affine transformation in a complex form:
z0 ¼ z þ z þ 6¼ The vector X1 represents n equally spaced points on the unit circle z ¼ eit i.e. the vertices of a
regular nth order polygon. We see that the vector ð1 lk1 X1 þ n1 lk
1 X1 Þ is the affine image of the
regular nth order polygon with vertices on the ellipse z0 ¼ 1 eit þ n1 eit . Thus the prediction is
that the iterated image tends to the affine regular version of Pn1 . This is precisely what is observed.
In Part 2 to this article, we show how this basic result may be extended in two ways:
. dividing polygonal edges by ratios other than 11 and by complex ratios and
. considering the effect of the Buffon transformation on polyhedra
These extensions are delivered over a further 12 lectures.
TEACHING MATHEMATICS AND ITS APPLICATIONS, 2006
11 of11
4. Assessment
By its very nature this not a normal mathematics module driven by the lemma/theorem approach.
It is intended to be less rigid and more free-flowing. There certainly will be areas in which normal
assessment through the vehicle of examination questions might be appropriate. The work on
linear algebra, eigenvalue analysis, complex variables and affine transformations, etc. might all be
assessed in the usual way, particularly as the topics are covered during the module. However,
examining in this module should reflect the open-ended approach of experimental mathematics.
Project work is more appropriate. Students could be asked to carry out further research and
present a written report of their work. The project could be on the historical aspects of the subject
area. Students could work individually or in groups. This module is more a taking part than a
winning module. My own preference is that it should be a pass or fail module.
References
1.
2.
3.
4.
MOSAIC 11. (1984) Journal of Asprom.
Wells, D. (1995) You are a Mathematician, London: Penguin Books.
Veselov, A. P. and Ward, J. P. (to be submitted for publication).
Berlekamp, E. R., Gilbert, E. N. and Sinden, F. W. (1965) American Mathematical Monthly, March,
233–241.
5. Davis, P. I. (1979) Circulant Matrices, New York: John Wiley.
6. Coxeter, H. S. M., Longuet-Higgins, M. S. and Miller, J. C. P. (1954) Uniform polyhedra. Phil. Trans.
Roy. Soc. London Ser A, 246, 401–450.
Address for correspondence: Department of Mathematical Sciences, Loughborough University,
Loughborough, Leicestershire, LE 11 3TU, UK. E-mail: [email protected]