the life zones

The Life ZonesAuthor(s): Martin HansenSource: Mathematics in School, Vol. 20, No. 1 (Jan., 1991), pp. 34-37Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214752 .

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THE

THE

ONE

by Martin Hansen

The Schools, Shrewsbury

Many readers will be familiar with the name "John Con- way" for he not only devised a fascinating "Mathematical Universe" but also created within it the thing that makes our own Universe special - "Life"! It's not "Life" as we know it, however ....

His brainchild is beautiful in its simplicity. Essentially, all he did was take a sheet of squared paper, imagine that the squares were "cells" and invent a set of rules by which, as time progressed, cells were born, cells survived and cells died. It's an idea that continues to fire the imagination of mathematicians and later on in this article I shall talk a little about recent "real world" developments.

Unfortunately, all of these well known "Cellular Auto- mata", as they are now called, require fast computing facilities to be investigated, which makes them impractical to use as GCSE coursework material. Much to my delight, however, I have discovered that the ideas can be repack- aged and that by keeping to "Life Forms" which only grow, some marvellously attractive children's work results. What follows is also great fun to teach but to make the most of the situation you must use the "official" terminology of the "Federation"! Don't talk about "colouring pencils" - they are "SpaceCol Sticks". Your "glue" has become "Galactic Glook" and the "scissors" are "QuickCut Tools". The epic space voyage begins with a lesson for all cadets at the Space Academy ...

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Let's start at the beginning in that far off time when the Cosmic Clock read t= 0. As in the "other" Universe, the Big Bang had just occurred (At t = - 0) and the seeds of life were scattered throughout all zones of the Universe. As the Cosmic Clock ticked out time, t= 1, t=2, t=3, t=4 ... seeds in some parts of the Universe started to grow. These parts of the Universe became known as The Life Zones. In other places, the seeds grew very little, if at all. It was as if they were frozen, and so such parts of the Universe were named The Frozen Zones.

The way in which the "A" Life Form expands in the Square Sector of the One Zone is shown in Fig. 1. This Life Form has a very straight forward type of expansion and is thus a good one upon which to start a class together.

Lifeform Type: A A AI0A Cell.X comes alive

Zone: a Zone: ea ofthesurrounding

Zone: One cells is alive,

Fig. 1

Mathematics in School, January 1991



As we are investigating a Life Form of the "Square" sector, squared paper is required and we begin by selecting an appropriate "S" (for Square) seed from "The Seed Cata- logue" of Fig. 5. Notice that seeds can be reflected and rotated without being considered to have changed.

As a "starter example", pick the bent 3S-seed. Having made a choice, the investigation proceeds by copying the chosen seed onto the squared paper using the t=0 start colour. My version is shown to the left in Fig. 2. Using the "Touch Rule" of Fig. 1 each of the cells around the seed are considered in turn. Any that have their "Touch Rule" satisfied are coloured in using the t= 1 colour. There is no need to draw a new diagram - simply add the "births" around the original three squares of the seed. Stress that

Lifeform Type:tA

Setor: Square

Zoneo:One

Fig. 2

this should be in a different colour and that it helps if in future diagrams the same colour code is used. I've redrawn the whole thing to make it clear what is going on and

The 1H-seed

The 2H-seed The 3H-seeds The 4H-seeds

The IS-seed

The 2S-seed

The 3S-seeds The 4S-seeds (Quadominoes) The 5S-seeds (Pentominoes)

The IT-seed

The 2T-seed The 3T-seed The 4T-seeds The 5T-seeds

The 1FS-seed

The 2FS-seeds The 3FS-seeds

The 1FT-seed

The 2FT-seeds The 3FT-seeds IFig. 5

Mathematics in School, January 1991 35



diagrams should at this point look like the central one of Fig. 2. The Cosmic Clock strikes t = 2 and, using a third colour, "cadets" should be starting to see how to continue without further prompting.

Having shown in picture form, how the seed grows one more "push" is needed to set things going in the right direction; How might we describe the growth using mathematics? This is not too difficult and simply involves some careful counting of squares and the recording of the results in some sort of table. My suggestion for how the above example could be set out is shown in Fig. 3. Attention should be drawn to the last line of this mathematical description which gives general formulae that can be used to work out "The Number of New Cells" and "The Total Number of Alive Cells" for any tick of the cosmic clock. I point out that such formulae are worth quite a few marks but only if it is explained how they were obtained. Often, the geometry of squares will help as shown in Fig. 4.

State of the The number The total number Cosmic Clock of new cells of alive cells

t=0 3 3 t = 1 12 15 t= 2 20 35 t = 3 28 63 t = 4 36 99 t = 5 44 143 t = n 8n+4 4n2+8n+3

(for n>0 ) ( for all n)

Fig. 3

The real strength of this investigation lies in what can be attempted after the class has been left to pursue the "A" type Life Form of the Square Sector in the One Zone, as detailed above, for two or three lessons. Firstly, we can change the "Life Form Touch Rule" so that, for example, only a touch through the edge of a cell brings it to life. The "Sector" can be altered so that work is on "Hexagonal" or "Triangular" paper. Or the "One Zone" can be left behind. This is so called because just one (or more) alive cells needs to satisfy the "Life Form Touch Rule" to bring a dead cell to life. So in the two in the "Two Zone", it is required that two (or more) alive cells satisfy the "Life Form Touch Rule". Figures 6 to 14 illustrate a few of the possibilities and at this point "cadets" should be encour- aged to go their separate ways. Some Life Forms are quite awkward to draw and although they all have formulae associated with them, some of these are not easily found. However, finding out what is what is all part of the adventure!

Much of what we teach in school was discovered many, many years ago. For me, the pleasure surrounding this investigation came out of the newness of it. As a topic, "Cellular Automata" could only be developed once fast computers had been built as these are essential for explor- ing sets of promising rules upon a large numbers of cells quickly. As I point out to my classes, quite a few mathematicians have become "famous" upon inventing an interesting set of rules and there are, no doubt, still exciting fresh discoveries just waiting to be made.

The most spectacular "Cellular Automata" of recent years was invented by David Griffeath of the University of Wisconsin. His version of "Life" is called "Demon" and involves a rule by which some cells "eat" their neighbours!

36

When t = 1 No Alive = (4 x 4)- 1 When t = 2 N' Alive = (6 x 6)- 1 When t = 3 No Alive=(8 x 8)- i

So when t = n No Alive = ( [ 2n+2] x [ 2n+2] ) - i.e. N' Alive = 4n2+8n+3

Fig. 4

Lifefrm Type:

A CellX comes alive

if Anyoneormore Seor:Hexagonalof the surrounding Zone: One cells is alive,

Example

Fig. 6

Lifeform Type: E Cell X comes alive E

if0one0or more Sector: Square E

..srrounding E Zone: One Edge cells is alive,

Example

Fig. 7

LifeformType:D D D Cell X comes alive

D if one or1more

Sector:Square D D ofthe surrounding Zone: One Diagonal cells is alive,

Example

Fig. 8




Yet another recent concoction on the "Life" theme was devised by M Gerhardt and H Schuster at the University of Bielefeld in West Germany. Their "gimmick" was to tag cells as being either "active", "dead" or "sick" and their Life Form they called "The Hodgepodge Machine". The resulting colour coded "Life Cycle" is most beautiful to watch.

Lifeform Type: CE

Sector Fractured Fractured

Zone: One

both CEIXICEI

both

CellX comes alive if one or0more of it's Edges is Completely lined with live cells,

Example

Fig. 9

Lifeform Type: A

Sector:Triangular Zone: One

;4 A A~ AA

>~~ IAA - I

CelIXcomesalive if Any one or more ofthe surrounding cells is alive,

Example

Fig. 10

Lifeform Type: E


>E

CellXcomesalive if one or more ofthe surrounding Edge cells is alive.

Example

Fig. 11


Lifeform Typel: D


CellX comes alive if oneor more of the surrounding Diagonal cells is alive.

Example

Fig. 12

Lifeform Type :NED

Sector: Triangular Zone', One

< D

9iD 9 E

CelloX comes alive if one or more of the surrounding Not Edge nor Diagonal cells is alive,

Example

Fig. 13

Lifeform Type: CE Sector': Fractured

Triantgular

Zone :One

both CE

Cell X comes alive if one or more of it's Edges is Completely lined with live cells,

Example

Fig. 14

One day, I suspect that schoolchildren will have access to computers in the number and for the time needed to be able to investigate these dazzling displays of mathematics. Adventurous teachers with a computer at hand will find that "Cellular Automata" feature regularly in the various magazines for computer enthusiasts. They may well be able to give their classes a display of what is to come.

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the life zones

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