OCA:H-trees
| H-trees | |
| Rulestring | 012345678/1 B1/S012345678 |
|---|---|
| Rule integer | 261634 |
| Character | Explosive |
| Black/white reversal | B/S01234568 |
|
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H-trees, also known as Christmas Life, is a Life-like cellular automaton with rulestring B1/S012345678.
It is an exploding rule where H-shaped branches grow from pattern borders.
No spaceships can exist in this rule: with B1, any pattern expands in all directions and with S012345678, any cell in any pattern would never die.
It seems that for any finite initial pattern, choosing any cell and casting a ray pointing diagonally from it, the sequence of eventual states of cells along this ray becomes periodic, with a period of a power of 2.
Single-cell evolution
The number of cells on generation n is given by the OEIS sequence A151725(n+1), and the number of births in the nth iteration is given by its first differences, A151726(n+1). Recurrences are known that allow these sequences to be computed efficiently, but explicit forms and generating functions are not.[1]
It "completes" a (2*n+1)2 square on generations n that are one less than powers of 2, and on generations shortly preceding these, the set of cells within the neighbourhood of its living cells approaches the H-trees.
On generation 2n-1, within the (2n+1-1)2 bounding box, it has
- (16*4n+24*n-7)/9 alive cells, and a total edge length of 4*(8*4n+12*n-17)/9. Its asymptotic density is 4/9, and cells have on average 2 exposed edges.
- Apart from an asymptotically infinitesimal proportion of cells along the diagonals, it creates a binary tree structure at which trunk cells have two edges, branching cells have only one and leaf cells have three.
- 4*(4n+6*n-10)/9 branch cells, density 1/9 (for n>=1)
- 4*(2*4n-6*n+7)/9 stem cells, density 2/9 (for n>=1)
- 4*(4n+6*n-1)/9 leaf cells, density 1/9
- When two branches of the tree meet, they exhibit crown shyness, and a 3×(2n-1) rectangular region is left unfilled.
- (5*4n-24*n-32)/6 dead cells are in these rectangles, with density 5/24,
- (4n+32)/6-4*n cells are in the rectangles' central rows or columns (off with no on neighbours), with density 1/24.
It can be seen as a generalisation from the von Neumann neighbourhood to the Moore neighbourhood of the Ulam-Warburton automaton (encodable as the isotropic non-totalistic rule B1e2ak3nqy4ny5e/S012345678, with population sequence A147562). The change to population on the nth iteration is 4*3n.bit_count()-1, where n.bit_count() is the number of bits set to 1 in n's binary representation.
It may also be defined in the hexagonal grid as B1/S0123456H, with population sequence A151723.
| Starting with only the origin occupied, repeatedly add to the crystal any site with exactly 1 of its 8 neighbors previously occupied (click above to open LifeViewer) |
| The Ulam-Warburton automaton (click above to open LifeViewer) |
| The hexagonal equivalent (click above to open LifeViewer) |
See also
- Life without death, a rule with similar survival conditions
References
- ↑ David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), p. 30-34
External links
- Totalistic Growth Rules with Moore Neighborhood
- Christmas Life (discussion thread) at the ConwayLife.com forums
- H-trees at Adam P. Goucher's Catagolue
- H-trees at David Eppstein's Glider Database
- Interesting Rules at Fano Experimental Web Server (archived copy)