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The Ulam–Warburton cellular automaton (UWCA) is a 2-dimensional fractal pattern that grows on a regular grid of cells consisting of squares. Starting with one square initially ON and all others OFF, successive iterations are generated by turning ON all squares that share precisely one edge with an ON square. This is the von Neumann neighborhood. The automaton is named after the Polish-American mathematician and scientist Stanislaw Ulam[1] and the Scottish engineer, inventor and amateur mathematician Mike Warburton.[2][3]

The first twenty iterations of the Ulam-Warburton cellular automaton

Properties and relationsEdit

The UWCA is a 2D 5-neighbor outer totalistic cellular automaton using rule 686.[4]

The number of cells turned ON in each iteration is denoted   with an explicit formula:

  and for  


where   is the Hamming weight function which counts the number of 1's in the binary expansion of   [5]


The minimum upper bound of summation for   is such that  

The total number of cells turned ON is denoted  


Table of  ,   and  Edit

The table shows that different inputs to   can lead to the same output.

This surjective property emerges from the simple rule of growth – a new cell is born if it shares only one-edge with an existing ON cell - the process appears disorderly and is modeled by functions involving   but within the chaos there is regularity.

0 0 0 0 10 2 12 101
1 1 1 1 11 3 12 113
2 1 4 5 12 2 36 149
3 2 4 9 13 3 12 161
4 1 12 21 14 3 36 197
5 2 4 25 15 4 36 233
6 2 12 37 16 1 108 341
7 3 12 49 17 2 4 345
8 1 36 85 18 2 12 357
9 2 4 89 19 3 12 369

  is OEIS sequence A147562 and   is OEIS sequence A147582

Counting cells with quadraticsEdit

Total number of ON cells   in the Ulam–Warburton CA and quadratics   and  

For all integer sequences of the form   where   and  



(  is OEIS sequence A130665)

Then the total number of ON cells in the integer sequence   is given by [6]


Or in terms of   we have


Table of integer sequences   and  Edit

0 1 1 3 9 5 25 7 49
1 2 5 6 37 10 101 14 197
2 4 21 12 149 20 405 28 789
3 8 85 24 597 40 1,621 56 3,157
4 16 341 48 2,389 80 6,485 112 12,629
5 32 1,365 96 9,557 160 25,941 224 50,517

Upper and lower boundsEdit

Total number of ON cells in the Ulam–Warburton cellular automaton

  has fractal-like behavior with a sharp upper bound for   given by


The upper bound only contacts   at 'high-water' points when  .

These are also the generations at which the UWCA based on squares, the Hex–UWCA based on hexagons and the Sierpinski triangle return to their base shape.[7]

Upper and lower bounds of  

Limit superior and limit inferiorEdit

We have


The lower limit was obtained by Robert Price (OEIS sequence A261313 ) and took several weeks to compute and is believed to be twice the lower limit of   where   is the total number of toothpicks in the toothpick sequence up to generation  [8]

Relationship toEdit

Hex-Ulam-Warburton cellular automaton - generation 11

Hexagonal UWCAEdit

The Hexagonal-Ulam–Warburton cellular automaton (Hex-UWCA) is a 2-dimensional fractal pattern that grows on a regular grid of cells consisting of hexagons. The same growth rule for the UWCA applies and the pattern returns to a hexagon in generations  , when the first hexagon is considered as generation  . The UWCA has two reflection lines that pass through the corners of the initial cell dividing the square into four quadrants, similarly the Hex-UWCA has three reflection lines dividing the hexagon into six sections and the growth rule follows the symmetries. Cells whose centers lie on a line of reflection symmetry are never born.

The Hex-UWCA pattern can be explored here.

Sierpinski triangleEdit

Sierpinski triangle - generation 16

The Sierpinski triangle appears in 13th century Italian floor mosaics. Wacław Sierpiński described the triangle in 1915.

If we consider the growth of the triangle, with each row corresponding to a generation and the top row generation   is a single triangle, then like the UWCA and the Hex-UWCA it returns to its starting shape, in generations  

Toothpick sequenceEdit

Toothpick sequence - generation 13

The toothpick pattern is constructed by placing a single toothpick of unit length on a square grid, aligned with the vertical axis. At each subsequent stage, for every exposed toothpick end, place a perpendicular toothpick centred at that end. The resulting structure has a fractal-like appearance.

The toothpick and UWCA structures are examples of cellular automata defined on a graph and when considered as a subgraph of the infinite square grid the structure is a tree.

The toothpick sequence returns to its base rotated ‘H’ shape in generations   where  

The toothpick sequence and various toothpick-like sequences can be explored here.

Nim and LIMEdit

The classic game of Nim and a related impartial combinatorial ruleset called LIM are also connected with the Ulam–Warburton cellular automaton in interesting and surprising ways.[9][10]


The beginnings of automata go back to a conversation Ulam had with Stanislaw Mazur in a coffee house in Lwów Poland when Ulam was twenty in 1929.[11] Ulam worked with John von Neumann during the war years when they became good friends and discussed cellular automaton. Von Neumann’s used these ideas in his concept of a universal constructor and the digital computer. Ulam focussed on biological and ‘crystal like’ patterns publishing a sketch of the growth of a square based cell structure using a simple rule in 1962. Mike Warburton is an amateur mathematician working in probabilistic number theory who was educated at George Heriot's School in Edinburgh. His son's mathematics GCSE coursework involved investigating the growth of equilateral triangles or squares in the Euclidean plane with the rule – a new generation is born if and only if connected to the last by only one-edge. That coursework concluded with a recursive formula for the number of ON cells born in each generation. Later, Warburton found the sharp upper bound formula which he wrote up as a note in the Open University’s M500 magazine in 2002. David Singmaster read the article, analysed the structure and named the object the Ulam-Warburton cellular automaton in his 2003 article. Since then it has given rise to numerous integer sequences.


  1. ^ S. M. Ulam, On some mathematical problems connected with patterns of growth of figures, Mathematical Problems in BiologicalSciences, 14 (1962), 215–224.
  2. ^ M. Warburton, One-edge connections, M500 Magazine of The Open University, 188 (2002), 11
  3. ^ D. Singmaster, On the cellular automaton of Ulam and Warburton, M500 Magazine of The Open University, 195 (2003), 2–7
  4. ^ OEIS - Index to 2D 5-Neighbor Cellular Automata,[1],
  5. ^ Applegate, David; Pol, Omar E.; Sloane, N. J. A. (2010). "The Toothpick Sequence and Other Sequences from Cellular Automata". Congressus Numerant. 206: 157–191. arXiv:1004.3036.
  6. ^ Mike Warburton, "Ulam-Warburton Automaton - Counting Cells with Quadratics " ,arXiv:1901.10565
  7. ^ Tanya Khovanova, Eric Nie, Alok Puranik, "The Sierpinski Triangle and the Ulam-Warburton Automaton" ,arXiv:1408.5937
  8. ^ Steven R. Finch, Mathematical Constants II, 364-365
  9. ^ Alex Fink, Avierzri S. Fraenkel, Carlos Santos, LIM is not slim. International Journal of Game Theory, 2014, vol. 43, issue 2, 269–281
  10. ^ Tanya Khovanova, Joshua Xiong, Nim Fractals,arXiv:1405.5942
  11. ^ S. M. Ulam, Adventures of a Mathematician, p32

External linksEdit