Saturday, August 22, 2020

[tplpklac] Graphs from lattices

Square lattice

connect nodes separated by distance d=1: square lattice

d=sqrt 2: 2 overlapping square lattices (ferz move graph)

d=2: 4 overlapping square lattices (dabbaba move graph)

d=sqrt 5: knight move graph

d=sqrt 8=2*sqrt 2: 8 overlapping square lattices (alfil move graph)

Triangular lattice

d=1: triangular lattice

d=sqrt 3: 3 overlapping triangular lattices

Hexagonal lattice (centers of a triangular tessellation)

d=1: hexagonal lattice

d=sqrt 3: 2 overlapping triangular lattices

d=2: 4 overlapping hexagonal lattices

Motivation: families of graphs to play games like go 囲碁 or snort.  We can construct unions of graphs within a family to make them connected.  For example, the king move graph is the union of the two ferz move graphs and the square lattice.

Previously, much higher connectivity on the square lattice.

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