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Progressing geometricallyfrom ancient thought tofractalsMark McCartneyPublished online: 11 Nov 2010.

To cite this article: Mark McCartney (2001) Progressing geometrically fromancient thought to fractals, International Journal of Mathematical Education inScience and Technology, 32:6, 937-944, DOI: 10.1080/002073901317147924

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3. An application to statistics

Consider a frequency distribution having median class ‰y; y ‡ hŠ with relativefrequency Fy‡h ¡ Fy where Fy‡h ˆ cumulative relative frequency up to the medianclass and Fy ˆ cumulative relative frequency up to the class preceding the medianclass. Also let y0:50 be the median. Then we have the following representation

Fy y

0:50 y0:50

Fy‡h y ‡ h

Then by (4) we have

y0:50 ˆ y ‡ 0:50 ¡ Fy

Fy‡h ¡ Fh

h

which is the well-known formula for the median of grouped data in a frequencydistribution. See [4, p. 72].

AcknowledgmentsThe author acknowledges the excellent research facilities available at King

Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia.

References[1] Lapin, L. L., 1997, Modern Engineering Statistics (New York: Wadsworth Publishing

Co).[2] Newbold, P., 1995, Statistics for Business and Economics (New Jersey: Prentice Hall).[3] Briddle, D. F., 1982, Analytic Geometry (California: Wadsworth Publishing Co).[4] Kvanli, A. H., Guynes, C. H., and Pavur, R. J., 1992, Introduction to Business

Statistics: A Computer Integrated Approach (New York: West Publishing Co).

Progressing geometrically from ancient thought to fractals

MARK MCCARTNEY1

School of Computing & Mathematical Sciences, University of Ulster at Jordanstown,Northern Ireland, BT37 0QB

(Received 14 November 2000)

A range of applications of geometric progressions and their summation areintroduced. Potential classroom applications are emphasized in relation to theuse of geometric progressions to introduce students to the topics of in®niteseries and fractals and as a solution tool in complex numbers, mechanics andmathematical modelling.

Classroom notes 937

1 e-mail: m.mccartney@ulst.ac.uk

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1. Introduction

Geometric progressions, like their sister arithmetic progressions, are simple tode®ne and, with a beautiful piece of minor mathematical insight, elegantlysummed. They are typically used to develop students skills in formulating andsolving equations (Example. In a geometric progression the sum of the third andfourth terms is 18 and the sixth term is eight times the third. Find the ®rst termand the common ratio.) and as an introduction to summation of in®nite series. It isthis second aspect which the current paper focuses on, with examples from ancientphilosophy, complex algebra, mechanics, theology and fractals.

2. De®nition, proof and beauty

A geometric progression with ®rst term a and common ratio r has the ®rst nterms,

a; ar; ar2; ar3; . . . ; arn¡1

To ®nd the sum of the ®rst n terms,

Sn ˆ a ‡ ar ‡ ar2 ‡ ar3 ‡ ¢ ¢ ¢ ‡ arn¡1 …1†

we take equation (1) and multiplying by r obtain

rSn ˆ ar ‡ ar2 ‡ ar3 ‡ ar4 ‡ ¢ ¢ ¢ ‡ arn …2†

Subtracting (2) from (1) and rearranging gives

Sn ˆ a…1 ¡ rn†1 ¡ r

…3†

G. H. Hardy [1] said of good mathematics `Beauty is the ®rst test: there is nopermanent place in the world for ugly mathematics’ and along side beauty heplaced the criteria of economy and a measure of unexpectedness. Taking thesethree as our measure the demonstration of (3) is undoubtedly good mathematics.The derivation is well known, but it is so elegant and concise that it is worthrepeating.

Of course equation (3) can also easily be proved by induction, but this isn’tnearly as satisfying and perhaps serves better as a foil for the above proof.

Interestingly the formula for the sum of a geometric progression goes back to300BC/E with proposition 35 of Book IX of Euclid’s Elements reading:

If as many numbers as we please be in continued proportion, and there besubtracted from the second and the last numbers equal to the ®rst, then as theexcess of the second is to the ®rst, so will the excess of the last be to all thosebefore it.

Writing the nth term of the geometric progression as un this statement isequivalent to [2]

un‡1 ¡ u1

u1 ‡ u2 ‡ ¢ ¢ ¢ ‡ unˆ

u2 ¡ u1

u1…4†

which is equivalent to equation (3).

938 Classroom notes

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3. Zeno’s Dichotomy

The link with ancient thought can be continued with Zeno of Elea (c. 490±435BC/E). One of Zeno’s paradoxes, called the Dichotomy, can be used as a motivatorin teaching the summation of geometric progressions. Zeno argued thus; to travelbetween two points, A and B, an object must ®rst travel half the distance, then halfthe remaining distance (i.e. a further quarter of the total distance), then half of thethen remaining distance (i.e. a further eighth of the total distance) and so onthrough an in®nite number of steps. But this means that for the object to move itwould have to pass through an in®nite number of intermediate steps in a ®nite timeand since (according to Zeno) this is impossible, then motion is impossible.

The fact that this conclusion contradicted observed reality was presumablyconsidered an irritating irrelevance and in that regard it can hardly be thought of asa good `advertisement’ for ancient philosophy. However the Dichotomy doesprovide a useful classroom illustration of a geometric progression. Drawing a lineof unit length on the board it can be divided into two halves, then the second halfinto two quarters, then the second quarter into two eighths and so on, and itbecomes clear that

1 ˆ1

2‡

1

4‡

1

8‡

1

16‡

1

32‡ ¢ ¢ ¢

This exercise has three uses. First, it illustrates that an in®nite set of positivenumbers can be added together to give a ®nite result (an idea which many students®nd surprising), secondly it gives the proof in a simple pictorial fashion, and ®nallyit can be used to show that the sum of the ®rst n terms of the geometric progressionwith ®rst term and common ratio equal to 1

2is given by

Xn

iˆ1

1

2

³ ´i

ˆ 1 ¡ 1

2

³ ´n

4. Putting Zeno into a spin on the complex plane

With Zeno’s Dichotomy we were thinking of motion (or not motion as Zenowould have us believe) along a straight line, but the idea is easily generalized to thecomplex plane [3]. Consider an ant at the origin if the complex plane. It movesalong the positive real axis a distance of a units, then rotates anti clockwise throughan angle ³ and moves forward a further ar units, rotates anti clockwise throughanother angle ³ and moves forward a further ar2 and so on.

The position of the ant after n steps is given by

Pn ˆ a ‡ arei³ ‡ ar2e2i³ ‡ ¢ ¢ ¢ ‡ arn¡1e…n¡1†i³

This of course is simply the sum of the ®rst n terms of a geometric progressionwith ®rst term a and common ratio rei³ and using equation (3) we can write

Pn ˆ a…1 ¡ rnein³†1 ¡ rei³

…5†

Further assuming that r < 1 we can write the position of the ant after an in®nitenumber of steps as

P1 ˆ a

1 ¡ rei³ˆ a…1 ¡ r cos ³†

1 ‡ r2 ¡ 2r cos ³‡ i

ar sin ³

1 ‡ r2 ¡ 2r cos ³…6†

Classroom notes 939

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To arrive at this point on the complex plane the ant has traversed an arc of ®nitelength, but has had to turn through an in®nite number of full revolutions. Theelegance of this result comes from the use of the Euler rei³ form of complexnumbers, without which the solution would be very di� cult!

5. The bouncing ball: Zeno’s paradox in mechanics

One of Isaac Newton’s less well known discoveries is his experimental law forcollisions which states that in a direct collision between two bodies the ratio of therelative velocity of the two bodies after the collision to the relative velocity beforethe collision is a constant . For two bodies 1 and 2 with pre and post collisionalvelocities of u1; u2 and v1; v2 we can write this as

e ˆ ¡v2 ¡ v1

u2 ¡ u1…7†

where e is called the coe� cient of restitution. The name coe� cient of restitutioncomes from the fact that during the collision the two bodies deform undercompression and this is followed by a recovery (or restitution) from the deforma-tion as they separate. If e ˆ 1 we say that the collision is perfectly elastic and ife ˆ 0 (i.e. the bodies coalesce) we say that the collision is perfectly inelastic orperfectly plastic. Values of e are determined experimentally and it should also benoted that the law itself is only approximate [5].

We can use Newton’s experimental law to investigate the behaviour of abouncing ball. Consider a ball dropped from height h0 onto a smooth ¯oor. Usingthe well known equation of motion

v2 ˆ u2 ‡ 2as …8†

(where u and v are the initial and ®nal velocities of a body acted on by constantacceleration a over a distance s) we can calculate that the ball hits the ¯oor withspeed

v0 ˆ����������2gh0

p…9†

where g is acceleration due to gravity.Using equation (7) the ball rebounds with speed v1 ˆ ev0 and similarly the

second time it hits the ¯oor it will rebound with speed v2 ˆ ev1 ˆ e2v0, and thethird time with speed v3 ˆ e3v0 and so on. Thus the nth rebound speed is,

vn ˆ env0 ˆ en����������2gh0

p…10†

And further, using (8), the distance travelled between the nth and …n ‡ 1†threbound is

v2n

gˆ 2e2nh0

Hence the total distance travelled by the ball by the nth impact is

Dn ˆ h0 ‡ 2h0

Xn¡1

iˆ1

e2i …11†

This can be rewritten as

940 Classroom notes

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Dn ˆ 2h0

Xn¡1

iˆ0

e2i ¡ h0 …12†

The ®rst term on the right-hand side of (12) is the sum of the ®rst n terms of ageometric progression with ®rst term h0 and common ratio e2, and using (3) gives

Dn ˆ 2h0

1 ¡ e2n

1 ¡ e2

³ ´¡ h0 …13†

Taking the limit as n tends to in®nity gives the total distance travelled by the ballas

D1 ˆ h01 ‡ e2

1 ¡ e2

³ ´…14†

Note that (13) and (14) do not depend on g. Thus if the experiment was carried outon earth, or, for example, Saturn’s moon Titan, the distance travelled by the ballby its nth bounce would be the same.

At this point it is very tempting to let the following idea enter your mind: Sincethe ball bounces an in®nite number of times, it must continue bouncing forever.This, however, is wrong. Using the same procedure as was used to ®nd thedistance travelled we ®nd that the time taken until the nth bounce is

Tn ˆ 2v0

g

1 ¡ en

1 ¡ e

³ ´¡ v0

g…15†

giving the total time until the ball stops bouncing as

T1 ˆv0

g

1 ‡ e

1 ¡ e

³ ´…16†

This is, perhaps, a more telling and subtle example of Zeno’s paradox, becausehere we have a physical system which undergoes an in®nite number of changes in a®nite period of time. Unlike Zeno’s Dichotomy, which is unlikely to ¯uster anystudent (and instead will probably merely leave him or her with the idea that Zenoand his listeners weren’t very clever), the bouncing ball is likely to cause someperturbed and surprised looks in class. This is because hidden unchallenged inmany students’ minds is Zeno’s erroneous premise that an in®nite number ofchanges cannot occur in a ®nite period.

6. Theological re¯ections: a mathematical model

A wise person once said `The Lord gave us two ears but only one mouth, so weshould listen twice as much as we speak.’

Two friends, Alice and Bob, decide to take this saying to heart in theirconversation. Alice starts by speaking for x minutes. How long will the conversa-tion last?

We assume that Alice and Bob are going to take turns in speaking and sincethey are each going to listen twice as much as they speak, the initial reply Bob willgive to Alice will last for only half the time which Alice spoke i.e. 1

2x, and Alice’s

subsequent reply will last only half as long as Bob’s previous comment i.e. …12†2x,

and so on. Thus the nth statement in the conversation will last for

Classroom notes 941

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Sn ˆ 1

2

³ ´n¡1

x …17†

and this statement will be made by Alice if n is even, and Bob if n is odd. Clearlythe duration of the statements form a geometric progression with ®rst term x andcommon ratio 1

2. Thus using equation (3) the conversation up to the end of the nth

statement has lasted

Cn ˆ 2x 1 ¡1

2

³ ´n³ ´…18†

and letting n tend to in®nity the total conversation time is 2x minutes.Of course a more realistic model of the situation would include the fact that

there is usually a gap in between a statement and response, and that there is also ashortest possible reply (e.g. `No!’). It is left as an exercise for the reader toformulate and solve such a model.

7. The Koch curveNiels Fabian Helge von Koch (1870±1924) was a Swedish mathematician who

in 1904 [4] introduced to mathematics the fractal curve which bears his name. Theconstruction of the Koch curve is illustrated in ®gure 1. We begin (step k ˆ 0 in®gure 1) with a straight line (in fractal terminology this is called the initiator).Dividing the line into three equal parts, use the middle segment as the base for anequilateral triangle and then remove the base (step k ˆ 1), the resulting ®gure iscalled the generator. In the next step …k ˆ 2† this process is repeated for each of thefour lines making up the generator. Repeating this process an in®nite number oftimes on each segment produces the Koch curve. This last point is worthemphasizing; all the curves generated along the way at steps k ˆ 1; 2; 3 . . . arenot Koch curves but are merely approximations to it, and they are not fractalseither, but prefractals. A regular fractal is an object which contains exact copies ofitself across all magni®cations, hence an in®nite number of steps are required in itsconstruction¤ [6].

Koch’s motivation was to produce a curve which was continuous everywherebut di� erentiable nowhere, and indeed the Koch curve satis®es these pathologicalcriteria, as it is e� ectively constructed entirely from corners and hence at no pointon the curve can a tangent be drawn. Thus the Koch curve stands unmoved by themighty bulwark Newtonian±Leibnizian calculus. Geometric progressions howevercome to the rescue, and with them we can evaluate both the length of, and areabound by, the Koch curve and its prefractals.

First consider the length. If we take the length of the initiator in ®gure 1 to be lthen the length of the initiator, generator and subsequent prefractals at stepsk ˆ 0; 1; 2; 3; 4 are given by the geometric progression

l;4

3l;

4

3

³ ´2

l;4

3

³ ´3

l;4

3

³ ´2

l

942 Classroom notes

* Note that there are two types of fractal, regular and random. Regular fractals areexactly self-similar across all scales, but random fractals are only statistically self similar,with each part of the fractal having the same statistical properties as the whole.

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Thus by the kth step the length of the Koch prefractal is given by

Lk ˆ 4

3

³ ´k

l …19†

Clearly as k approaches in®nity the length of the prefractal diverges and thus theKoch curve has in®nite length.

We de®ne the area under the Koch curve as the area bound by the curve and itsinitiator. The area bounded by the initiator and the prefractal for the ®rst threesteps are given by

step k ˆ 1 A1 ˆ l2

12���3

p

step k ˆ 2 A2l2

12���3

p ‡ 4

9

l2

12���3

p

step k ˆ 3 A3

l2

12���3

p ‡ 4

9

l2

12���3

p ‡ 4

9

³ ´2 l2

12��s

p

Classroom notes 943

Figure 1. The construction of the Koch Curve.

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Clearly for the kth step the area Ak of the prefractal is given by the sum of the ®rstk terms of a geometric progression with ®rst term …l2=12

���3

p† and common ratio

…4=9†. Thus using equation (3) gives

Ak ˆ���3

p

20l2 1 ¡

4

9

³ ´kÁ !

…20†

and letting k tend to in®nity we ®nd the area of the Koch curve to be …���3

p=20†l2:

8. Concluding remarks

As was noted in the introduction, geometric progressions are often seen astools to develop students skills in the formulation and solution of equations. Thispaper has attempted to show other aspects of geometric progressions which can beused to (hopefully!) enthuse students and introduce them to other areas ofmathematics.

AcknowledgmentsFigure 1 is reproduced from Figure 2.10 of [6] by kind permission of the author

Dr Paul Addison of Napier University, Edinburgh, and Institute of PhysicsPublishing.

References[1] Hardy, G. H., 1992, A Mathematicians Apology, p. 85 Canto Edition (Cambridge

University Press).[2] Boyer, C. B., and Merzbach, U. C., 1989, A History of Mathematics, 2nd edition

(Wiley).[3] Nahin, P. J., 1998, An Imaginary Tale: The Story of

�������¡1

p(Princeton University Press).

[4] Von Koch, H., 1904, Arkiv foÈ r Mathematik, 1, 681±704[5] Lambe, C. G. 1970, Advanced Level Applied Mathematics, 3rd edition (Hodder and

Stoughton).[6] Addison, P., 1997, Fractals and Chaos an Illustrated Course (Institute of Physics

Publishing).

944 Classroom notes

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