242 SCIENCE AND CULTURE, JULY-AUGUST, 2018

MICE PROBLEM AND FRACTALS

ACHINTYA PAL*

The celebrated Mice Problem – that of n objects chasing each other from corners of an n-sidedpolygon – in mathematics is a special case of more general Pursuit Problem. A new approach tosolution of this problem, not found in standard texts, is given in this article using the conceptinvolved in fractals, an unconventional geometric shape that can be divided into parts, each ofwhich is a smaller duplicate of the whole. Similar parts of such objects as cloud or branches of atree can be obtained by rotation and scaling of another part. Precisely, this mathematics of successiverotation and scaling has been utilized in finding the solution to the Mice Problem.

ARTICLE

* Retired Exploration Geophysicist of ONGC.e-mail : achintya.pal1952@gmail.com, babulan@gmail.com

Introduction

In mathematics, the mice problem is a celebrated onein which a number of mice (or beetles, dogs, missiles,etc.) are placed at the corners of a regular polygon. In

this problem, also called the beetle problem, n mice startat the corners of a regular n sided polygon of side lengtha, each always heading towards its closest neighboringmouse in a counter-clockwise direction at constant speedv. The questions posed are:

(1) What is the distance covered by each mouse?

(2) When and where the mice meet?

(3) What is the trajectory traced out by each?

As a matter of fact, the mice problem belongs to themore general class of ‘pursuit problems’ – on a plane oron a curved surface such as a sphere as discussed below.

A Bit of History of General Pursuit Problems1

Modern mathematical pursuit analysis is generallyassumed to begin with a problem posed and solved by theFrench mathematician Pierre Bouguer (1698– 1758) in1732. Bouguer treated the case of a pirate ship pursuing afleeing merchant vessel, as illustrated in the followingfigure:

The pirate ship and the merchant vessel are taken tobe at (0, 0) and (x0, 0) at time t = 0, respectively, theinstant the pursuit begins, with the merchant vesseltravelling at constant speed Vm along the vertical line x =x0. The pirate ship travels at constant speed Vp along acurved path such that it is always moving directly towardthe merchant, that is, the velocity vector of the pirate shippoints directly at the merchant vessel at every instant oftime. Bouguer’s problem was to determine the trajectorywhich he called the curve or line of pursuit. It was alsonoticed from the earliest days of falconry that in its attack,the falcon always flies directly at the instantaneous location

VOL. 84, NOS. 7–8 243

of its prey – this is the very definition of what is nowcalled pure or direct pursuit. In comparison, the MiceProblem may be termed as cyclic pursuit.

In May 1877, the French mathematician EdouardLucas (1842-1891) formally posed the cyclic pursuitproblem. Interestingly, the problem had actually appearedyears earlier on the famous Cambridge UniversityMathematical Tripos Examination of January 1871. Theproblem of cyclic pursuit remained one known only tomathematicians through the first half of the 20th century,until the Polish mathematician Hugo Steinhaus (1887–1972) included it in his 1950 puzzle book MathematicalSnapshots. But what really brought it widespread fame wasits use by Martin Gardner in his famous ‘MathematicalGames’ column in Scientific American (July 1965).

The Cyclic Pursuit of n Mice on a Regular n-sided Polygon

While this problem may be known to some of thereaders, one other objective of this article is to demonstrateits relation or correspondence to (mathematics of) fractals,discussed in Section V. As a result, the method of solutionpresented here is different from that found elsewhere onthe web or in printed texts, say for example, in Ref1.

For simplicity and ease of visualization, we considera square M1 M2 M3 M4 of side a, with the mice at therespective vertices at time t = 0 (Fig.1). After some small(infinitesimal) time interval ΔΔΔΔΔt, the mice will be at positionsM1´ M2´ M3´ M4´ which is again a square because theymove at equal speed heading directly towards the nearestneighbor. It may be easily seen that this square is rotatedby an angle β with respect to the previous one and is alsoscaled down in size with side a´ where

cos sinaa

β β′ =

+

1 2 1 4 1 1, ,M M a M M a M M tν⎡ ⎤′ ′ ′ ′= = = Δ⎢ ⎥⎣ ⎦ (1)

and sin t f or small ta tνβ βνΔ

≈ = Δ− Δ

It is shown below that the general formula for anypolygon of side n ≥ 3 (Fig.2) is2:

sinsin( ) sin

a Aaa β β

′ =+ + (2)

where 21An

π⎛ ⎞= −⎜ ⎟⎝ ⎠

is each of the internal angles of the

polygon.

Eqn. (2) reduces to Eqn. (1) for n = 4 (square) whenA = π/2.

From 1 1 3 ,sin sin( )

a a v tM M MA A β′ − Δ′ ′Δ =

+

sin sin( ) sinv t a

Aβ β βΔ

= =+ + [by addendo]

sinsin( ) sin

a Aa afA β β

′ = ≡+ + verifying (2)

To find the length of the trajectory up to the meetingpoint at the centre of the polygon, we have to add up line

elements 1 1 1 1,M M M M′ ′ ′′ ...

1 1sinsin

M M v t aAβ′ ′= Δ =

1 1sinsin

M M aAβ′ ′′ ′′= ... etc.

Total length ( ) sin...sin

L a a aAβ′ ′′ ′′′= + + +Fig. 1 : Mice on vertices of a square

Fig. 2 : Mice on corners of an equilateral triangle.

244 SCIENCE AND CULTURE, JULY-AUGUST, 2018

( )2 3 sin....sin

af af afAβ

= + + +

sin1 sin

aff A

β=

− where

sinsin( ) sin

AfA β β

=+ +

(3)

sin1 sin( ) sin sin

f Af A Aβ β=

− + + −

sin(sin cos cos sin ) sin sin

AA A Aβ β β

=+ + −

sin(1 cos )

AA β

=+ [in the limit β → 0, cosβ → 1,

sinβ → β]

Hence, sin sin(1 cos ) sin

AL aA A

ββ

=+

2 21 cos 1 cos 1 cos

a a aA

n nπ ππ

= = =+ ⎛ ⎞ ⎛ ⎞+ − −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

[in the limit β → 0]

2 ( ), ( ),3

L a triangle a square⎛ ⎞= ⎜ ⎟⎝ ⎠

115

a⎛ ⎞+⎜ ⎟

⎝ ⎠ (pentagon), 2a (hexagon)

This answers Question (1) above. From the symmetryof the problem and the successive rotated and scaled downpolygons, it is obvious that the mice will meet at the centrereferred coordinate geometrically at (0,0) and after a timeL/v. To answer Question (3), the nature of the trajectory,we switch to the notation n dθ for the infinitesimal angleβ Referring to Fig. 3,

1 1 1 1, ,In M M O let OM r OM r dr′ ′′ ′ ′′Δ = = −

1 1 1, cot cot2 2A AM P rd M P M P rdθ θ⎛ ⎞ ⎛ ⎞′′ ′ ′′= = =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

Also, 1M P dr′ =

Hence, cot2

dr A dr

θ⎛ ⎞= ⎜ ⎟⎝ ⎠

By simple integration, log loge er b Cθ= + ,

where cot2Ab ⎛ ⎞= ⎜ ⎟

⎝ ⎠

This is the equation of a logarithmic or a log spiral :r = Cebθ as shown in Fig. 3.

The Direct Pursuit of Merchant Vessel byPirate Ship on Spherical Earth

Now we revisit a pirate ship-merchant vessel directpursuit problem – this time on a sphere instead of a plane.If the pirate is sailing in the direction of the merchant andthe merchant is trying to avoid the pirate holding a fixedcourse, the resulting trajectory is a spherical spiral calleda loxodrome (Fig. 5(a)). Holding a fixed course on theearth’s surface implies crossing the meridians at a constantangle, not equal to a right angle. When projected on to amap on a flat surface in Mercator projection (Fig. 4), themeridians or longitudes appear as uniformly spaced verticallines, the latitudes as non-uniformly spaced horizontal linesand the fixed course appears as a straight line across themeridians – the so called ‘rhumb line’ (Fig. 5(b)). A rhumbis a course on the earth of constant bearing – the course isa rhumb, the curve is a loxodrome. While Fig. 6(a) offers

Fig. 3: The chasing mice spiral around to meet at the centre of thetriangle.

Fig. 4: Spherical earth projected on a flat surface by opening up theimage on a surrounding cylinder – the Mercator projection.

VOL. 84, NOS. 7–8 245

Fig. 7: Different examples of fractals seen in nature. Notice that the section of a nautilus is almost exactly a logarithmic spiral!

Fig. 5(a): Loxodrome on a sphere Fig. 5(b): ‘Rhumb Line’ in Mercator projection on a map

Fig. 6(a): View of Loxodrome from 45 degree Fig. 6(b): Looking down over the North Pole

246 SCIENCE AND CULTURE, JULY-AUGUST, 2018

a nice view of the loxodrome from aperspective view at 45 degree to theequator, the view from over the pole(Fig. 6(b)) resembles the spiral on aplane.

Relation of the Solution ofMice Problem to Mathematicsof Fractals

The essential feature of theprocedure employed here, namely,successive rotation and scaling is thecharacteristic of an interesting class ofobjects called fractals. A fractal is ageometrical shape that can be dividedinto parts, each of which is a smallerduplicate of the whole. This property ofduplication is called self-similarity. Inmathematics, a fractal is based upon anequation that undergoes an iteration orrecursion – that is, a repetition of itself.A few pictorial examples will make itmore comprehensible (Fig. 7).

Conversely, if one zooms on any of the above figures– be it a broccoli, cloud, section of a nautilus or branchingof a tree – one may see the replica of the whole or anyother part. The famous Koch Curve is most frequently usedto illustrate the general characteristics of fractal objects. Astraight line is divided into three equal parts and the middlesegment is replaced by sides of an equilateral trianglewithout the base. This procedure is repeated on the linesegments so obtained and continued ad infinitum. Theresulting figure shown in Fig. 8 has two features: (1) Eachsmaller part, when magnified, looks like the whole and (2)The length of the curve goes to infinity after many iterationsbut the area enclosed by the curve and imaginary baseremains finite!

While there are several definitions and mathematicalmethods of generating fractals, self-similarity is oneindisputable feature. This is obtained from a given initialgeometrical shape by affine transformation. An affinetransformation is an important class of linear 2-D geometrictransformations which maps a coordinate (x1, y1) of an inputimage into a new one (x2, y2) of the output image byapplying a linear combination of translation, rotation,scaling and/or shearing (i.e. non-uniform scaling in somedirections) operations. Affine transformation is a linearmapping method that preserves points, straight lines, andplanes. Sets of parallel lines remain parallel after an affinetransformation.

Fig. 8: Common characteristics of fractals exemplified by Koch Curve

The general affine transformation is commonly writtenas a matrix equation:

2 1

2 1[ ] [ ]

x xA B

y y⎡ ⎤ ⎡ ⎤

= +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

where [A] is a 2X2 transformation matrix and [B] containsthe translation companents.

If 1 0

[ ]0 1

A⎡ ⎤

= ⎢ ⎥⎣ ⎦

(unit matrix) and 1

2[ ]

bB

b⎡ ⎤

= ⎢ ⎥⎣ ⎦

(pure translation)

If cos sin

[ ]sin cos

Aβ ββ β

−⎡ ⎤= ⎢ ⎥⎣ ⎦

and 0

[ ]0

B⎡ ⎤

= ⎢ ⎥⎣ ⎦

(pure rotation)

If 11

22

0[ ]

0a

Aa

⎡ ⎤= ⎢ ⎥⎣ ⎦

and 0

[ ]0

B⎡ ⎤

= ⎢ ⎥⎣ ⎦

(pure scaling)

If 12

11

1[ ]

1a

Aa⎡ ⎤

= ⎢ ⎥⎣ ⎦

and 0

[ ]0

B⎡ ⎤

= ⎢ ⎥⎣ ⎦

(pure shear)

A nice example of general affine transformation hasbeen given by Barnsley3 (Fig.9) :

VOL. 84, NOS. 7–8 247

Fig. 9: An illustrative example of affine transformation by MichaelBarnsley3.

employing the above algorithm such that corners of thesuccessive polygons are rotated by some pre-assigned smallangle β and scaled down each time by a factor f given byEqn. (3) above through the recursive equations

( )cos sin *x x y fβ β′ = −

( )sin cos *y x y fβ β′ = −

which may be written in matrix notation :

[ ] 00

x x fA

y y f′⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥′⎣ ⎦ ⎣ ⎦ ⎣ ⎦

where sin

sin( ) sinAf

A β β=

+ + and cos sin

[ ]sin cos

Aβ ββ β

−⎡ ⎤= ⎢ ⎥⎣ ⎦

is the rotation matrix.

References1. Chases and Escapes: The Mathematics of Pursuit and Evasion

by Paul J Nahin (Princeton University Press, Princeton, NewJersey)

2. Fractals: Endlessly Repeated Geometrical Figures by HansLauwerier (Princeton University Press, Princeton, New Jersey)

3. Fractals Everywhere by Michael Barnsley (Academic Press, NewYork)

(Figs. 3 to 9 are taken from different sources on the internet. Figs. 1and 2 drawn by the author)

2 1 1 2 2 1

2 1 1 2 2 1

cos sinsin cos

x r r xy r r y

θ θθ θ

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

The above discussion connects mathematics of thefractals to the mice problem. The original square (or anyother polygon) is repeatedly subjected to the same similaritytransformation. It has been rotated and reduced each timein such a way that the vertices of the new polygon lie onthe sides of the previous one. Actually Figs. 1 and 2 hasbeen drawn (by the author) through a computer program