The game of life

A special square tiling

Magic squares

A collection of bare links to mathematical topics can be found on my link page.

Paul Callahan's Game of Life page.
Actually the Game of Life is not a game, it really is a cellular
automaton - an infinite set of possible states, with rules defining
how one state evolves into another. Paul's pages are excellent,
containing java applets, further pointers, and many interesting Life
patterns thematically presented. One of the main interests in the Game
of Life is the quest for and construction of patterns that exhibit a
particular kind of behaviour. Examples are the well known `traveling'
patterns or *spaceships*, *guns* - oscillating patterns emitting
spaceships at regular intervals, *puffers* - traveling or propagating
patterns whilest leaving clouds of debris behind, *breeders* - (..),
*reflectors, oscillators, fuses, inductors*,

the Heisenburp device, *a Life object which can detect the passage of a glider without affecting the glider's path or timing*,

a lightspeed wire, *a wick that can burn non-destructively at the speed
of light, which might be useful for sending a signal or constructing
oscillators with periods not currently attainable. Unfortunately, no
way has been found to return a signal to its starting point, either by
reflecting it or bending it around a corner*,

and many many other constructs. The flush of
Life terminology
reflects the richness of the game. As shown by the Heisenburp device's
description, Life researchers combine patterns to create new patterns
with more complex behaviour. It seems that the Game of Life can even
be used to simulate the universal computing device known as **Turing
machine.** As an opening door to a small
universe in its own right, Paul's page is highly recommended.

The picture shows a beautiful tiling of a square of dimension 112x112, with tiles that are squares of different dimensions and no two tiles the same. One of my PostScript projects was to project this tiling onto each of its constituting elements, i.e. make it recurse. I copied this tiling from the december 1998 problem page by Erich Friedman.

I could first not believe it when I saw the magic square below in Pythagoras, a mathematical magazine for youth of all ages (especially young youth though) in the Netherlands. The accompanying text has this to say about the square:

The square was constructed by C.J. Taale [clearly someone with
*strong magical powers*].
The square has magic constant 260. The four main 4x4 sub-squares are
magic with constant 130. All main 2x2 sub-squares sum to 130 as well. If each
entry is squared, a new magic square is created with constant 11180 (this
makes it *bimagic* according to the terminology pointed to below).
There seem to exist even *trebly magical* or *trimagic* squares -
after some searching I finally managed to get hold of one (see further below).
Trimagic squares seem to require the dimension to be at least 32.
Bimagic squares exist for smaller dimensions. The one below has dimension 8.

.----.----.----.----.----.----.----.----. | 3 | 50 | 55 | 6 | 48 | 29 | 28 | 41 | .----.----.----.----.----.----.----.----. | 45 | 32 | 25 | 44 | 2 | 51 | 54 | 7 | .----.----.----.----.----.----.----.----. | 22 | 39 | 34 | 19 | 57 | 12 | 13 | 64 | .----.----.----.----.----.----.----.----. | 60 | 9 | 16 | 61 | 23 | 38 | 35 | 18 | .----.----.----.----.----.----.----.----. | 40 | 21 | 20 | 33 | 11 | 58 | 63 | 14 | .----.----.----.----.----.----.----.----. | 10 | 59 | 62 | 15 | 37 | 24 | 17 | 36 | .----.----.----.----.----.----.----.----. | 49 | 4 | 5 | 56 | 30 | 47 | 42 | 27 | .----.----.----.----.----.----.----.----. | 31 | 46 | 43 | 26 | 52 | 1 | 8 | 53 | ^----^----^----^----^----^----^----^----^

For terminology and further pointers, check
the
magical page at wolfram's mathworld.
It appears that magical squares can have still other magical properties,
such as magicalness of *all generalized* diagonals (a generalized
diagonal is obtained by continuing a diagonal at the other side). The
latter kind of square is called *panmagical*, *diabolical*,
or *pandiagonal*.

Another cool site on magical squares is Holger Danielsson's. View for example this picture of a trimagic square of dimension 32, one of the many gems to be found on the site. Other gems include magical squares composed of primes, of consecutive primes, bordered magical squares, a square in which consecutive numbers form a closed knight's tour, and more.

From Holger's pages I understand that John R. Hedricks is another
name to remember if magical squares is the game. He is into things
like finding the first bimagic *cube* (which he did) and more
of such exquisite and exotic exceptionalities.