Kaleidoscope of 3D Life

Animated 3D generalization of the John Conway's Game of Life. 3D Life was investigated by Carter Bays. In a M-dimentional Life a point has 3M - 1 neighbors, i.e. 8 for 2D (M = 2) and 26 for 3D (M = 3). Therefore in 3D we can get richer rules and structures.

You can play with rules and grid size to find interesting "life forms".

Will the life die whenever? :)
9x9x9 Game goes to period 8 cycle after about 217 iterations.

Press & Drag mouse to rotate a life form.
Press Enter to set new parameters values from the text fields.
You can choose any reasonable N - size of the grid (e.g. N < 25 for PII-400). The applet will restore the initial "ooo" structure when you change N.
Selector Periodic turns on/off periodic boundary conditions.
Press the Set button to set your own initial "life form". A blue ball will turn into a red one if you click it by mouse and vice versa (you need to hit into the center of a red ball).

Rules:
a new ball will appear if the number of neighbors (Sum) is equal or more than r1 and equal or less than r2.
a ball will die if the Sum is more than r3 or less than r4.

   if ( (L[p]==0)&&(Sum[i][j][k]>=r1)&&(Sum[i][j][k]<=r2) ) L[p]=1;
   else if ( (L[p]!=0)&&((Sum[i][j][k]>r3)||(Sum[i][j][k]<r4)) ) L[p]=0;

You can set your own "initial structure" (e.g. "OOO") in a HTML file by the applet tag
    <param name=points value="3,-1 0 0,0 0 0,1 0 0">
where the first 3 is the number of points, then (i j k) coordinates of points go. The center of coordinate is in the center of the grid.

3D Life "Zoo"

It is evident that:
a small 2x2x2 cube is stable for R = (5,5,7,7) and a 2x2x1 square is stable for R = (5,5,3,3)
a 3x3x3 octahedron for R = (6,6,5,3) has an "oscillating" ball in the center and a 3x3x1 rhomb for N = 5, 9 and R = (3,3,4,4) makes moving, oscillating structures too (see simple oscillators). You can try Random Game to find your own oscillators and gliders.

Michael Toftdal sent me several oscillators too.

Carter Bays found amazing 3D Gliders. See also Which "Life"-Like Systems Have Gliders? by David Eppstein.

Variety of "Life"

Since I've got your messages with new suggestions, therefore I make a few examples too. See also Eric Weisstein's Treasure Trove of the Life CA, Stephen Silver's Life Page (but they don't like Java) and alife.co.uk. I was amazed when I found in Internet all these evident 2D grids and even 3,4D gliders. David Eppstein kindly sent me a reference about A CA run on Penrose Tiles by David Griffeath.

2D lattices

In 2D space there are 5 different Bravais lattices (lattices with translation symmetry which consist of equivalent nodes). A node in the square lattice has 8 neighbors (see "a" picture below) but one can use only 4 nearest neighbors too ("b" case). A node in the hexagonal lattice has 6 neighbors ("c" picture). Neighbors in the three other Bravais lattices are similar to ones in the square or hexagonal lattices.
a *--*--*  b .--*--.   c *---*      d     *---.
  |  |  |    |  |  |    / \ / \    \     /     \
  *--o--*    *--o--*   *---o---*    *---o       .--
  |  |  |    |  |  |    \ / \ /    /     \     /
  *--*--*    .--*--.     *---*            *---.
Honeycomb like lattice ("d" case) has non-equivalent nodes with 3 neighbors.

3D lattices

There are 14 different Bravais lattices in 3D. A node in the simple cubic lattice has 26 neighbors (or only 6 nearest nodes). There are 8 and 12 neighbors in the body centered and face centered cubic lattices correspondingly. A node in the close packed hexagonal lattice has 12 neighbors too.
Diamond has a non-Bravais lattice with non-equivalent nodes and 4 neighbors.
And one can use quasi-crystall lattices (see Introduction to Quasicrystals). I think that every N-dimentional cube has a N-order rotation symmetry axis. Therefore we can make quasicrystals with any rotation symmetry axis by an appropriate projection of N-dimentional cubic lattice to 3D space. Where can I find these lattices for my students?

Don't think we need all these lattices. Really cellular automata seem me a little "artificial" yet. The Mandelbrot set is more "simple" and rich IMHO :)

6 neighbors Game


E-notes,   Game 700x700 pixels,   "Random" Game,   free sources.
updated 9 Nov - 8 Dec 2000