page_title: Ulam–Warburton automaton
section_title: Toothpick sequence
hierarchical_section_title: Ulam–Warburton automaton / Relationship to / Toothpick sequence
context_page_description: The Ulam–Warburton cellular automaton 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 and the Scottish engineer, inventor and amateur mathematician Mike Warburton.
context_section_description: 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.
caption: Toothpick sequence - generation 13