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archimedean.md

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Archimedean tilings

There are only 11 combinations of regular p-gons at the full vertex being able to cover an infinite flat surface.

Called “Archimedean” tilings, they are tilings with only 1 type of vertex, that is they are monogonal.

Monogonal: every vertex, together with its incident edges, forms a figure congruent to that of any other vertex and its incident edges. It simply means that vertices with its incident edges form one congruent class.

They can be divided into:

Regular

There are only 3 regular tilings, they are monohedral.

Monohedral: every tile Ti i in the tiling T is congruent to one fixed set T, meaning all the tiles are of the same shape and size.

  1. (▲⁶)

    Mono.expandPattern(Full.s("(3*6)"), 30)

  2. (■⁴)

    Mono.expandPattern(Full.s("(4*4)"), 30)

  3. (⬣³)

    Mono.expandPattern(Full.s("(6*3)"), 30)

Semiregular

The other 8 tilings are 2-hedral or 3-hedral.

  1. (▲⁴.⬣)

    Mono.expandPattern(Full.s("(3*4.6)"), 30)

  2. (▲³.■²)

    Mono.expandPattern(Full.s("(3*3.4*2)"), 30)

  3. (▲².■.▲.■)

    Mono.expandPattern(Full.s("(3*2.4.3.4)"), 30)

  4. (▲.■.⬣.■)

    Mono.expandPattern(Full.s("(3.4.6.4)"), 30)

  5. (▲.⬣.▲.⬣)

    Mono.expandPattern(Full.s("(3.6.3.6)"), 30)

  6. (▲.12²)

    Mono.expandPattern(Full.s("(3.12*2)"), 30)

  7. (■.⬣.12)

    Mono.expandPattern(Full.s("(4.6.12)"), 30)

  8. (■.⯃²)

    Mono.expandPattern(Full.s("(4.8*2)"), 30)