Layer Groups

Scalable tiling patterns let us prove that certain bricks will have multiple solutions on all pyramids larger than a certain size.

group size N = 5, 6
odd   N = 5
even   N = 6

Collisions

A brick can collide with itself in a pyramid in 34 ways. Fixing the lower tile east, there are nine overlaps for a vertical upper tile, and eight for a horizontal one.

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North

South

East

West

Collision Fingerprints

A collision fingerprint is a sequence of 34 booleans (one per collision, in the order above). We can display this as a row of 34 squares or a 9-digit hex code.

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From a pyramid

Each pattern of tiling a pyramid causes certain collisions which can be summarized by a fingerprint.

group
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From a brick

This is a map of bricks, colored by how many collisions it has with itself.

0 conflicts34 conflicts

 

This is the same grid, sorted by post/hole count.

 

Here rows and columns are sorted by the total conflicts in that row/column.

 

code
fingerprint
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Fingerprint Directory

To regenerate: node scripts/pyramid-group-catalog.js --write.

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