Edge symmetries in tilings

Higher-dimensional geometry (previously "Polyshapes").

Edge symmetries in tilings

Postby Marek14 » Sun May 19, 2019 12:43 pm

Here's one neat thing I found out a few months ago:

When you have a uniform tiling, there are various centers of symmetries in the centers of the faces, sometimes in vertices, and sometimes in the edges. Or you can have axes of symmetry.

Here's a question: How many kinds of edges exist, symmetry-wise?

You can have edges without symmetry, or edges with 2-fold center of symmetry, or edges with an axis of symmetry perpendicular to them, or highly symmetrical edges that have two axes of symmetry and a 2-fold center of symmetry.

But what's surprisingly rare is an edge that has an axis of symmetry running THROUGH it, but has no other axis or center.

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I only managed to find an example for eight polygons per vertex. In this tiling, there are axes of symmetry that run through edges between green squares, but those edges have no other symmetries.
Marek14
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