3D cubic grid symmetries and naming

muzik · Post by **muzik** » June 29th, 2025, 2:50 pm

Asked about this on Polytope Discord around a year ago. The response was this, I'm not sure how comprehensive it is:

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symmetries with singular central/fixed points: 
cubic symmetry centered on a cube or vertex
chiral cubic symmetry centered on a cube or vertex
pyritic symmetry centered on a cube or vertex
tetrahedral symmetry centered on a cube or vertex
chiral tetrahedral symmetry centered on a cube or vertex
triangular antiprismatic symmetry centered on a cube or vertex
triangular proantiprismatic symmetry centered on a cube or vertex
chiral triangular prismatic symmetry centered on a cube or vertex
square prismatic symmetry centered on a cube, face, edge, or vertex
square proprismatic symmetry centered on a cube, face, edge, or vertex
chiral square prismatic symmetry centered on a cube, face, edge, or vertex
brick symmetry centered on a cube, face, edge, or vertex (there are two different orientations centered on each)
digonal antiprismatic symmetry centered on a cube, face, edge, or vertex
espic symmetry centered on a cube, face, edge, or vertex (there are two different orientations centered on each)
chiral brick symmetry centered on a cube, face, edge, or vertex (there are two different orientations centered on each)
prodappic symmetry centered on a cube, face, edge, or vertex (there are two different orientations centered on each)
central inversion symmetry centered on a cube, face, edge, or vertex

symmetries with central/fixed lines:
triangular pyramidal symmetry (line goes through vertices and cubes)
chiral triangular pyramidal symmetry (line goes through vertices and cubes)
square pyramidal symmetry (line goes through cubes and faces)
square pyramidal symmetry (line goes through edges and vertices)
chiral square pyramidal symmetry (line goes through cubes and faces)
chiral square pyramidal symmetry (line goes through edges and vertices)
digonal pyramidal symmetry (line goes through cubes and faces)
digonal pyramidal symmetry (line goes through edges and vertices)
digonal pyramidal symmetry (line goes through cubes and edges)
digonal pyramidal symmetry (line goes through faces and vertices)
chiral digonal pyramidal symmetry (line goes through cubes and faces)
chiral digonal pyramidal symmetry (line goes through edges and vertices)
chiral digonal pyramidal symmetry (line goes through cubes and edges)
chiral digonal pyramidal symmetry (line goes through faces and vertices)

symmetries with central/fixed planes (these are just reflections):
plane goes through cubes, edges, and faces
plane goes through faces, edges, and vertices
plane goes through cubes, faces, edges, and vertices

symmetries with central/fixed everything:
no symmetry

which is like around 90

Assuming someone hasn't went ahead and done so already, do we have a naming system for symmetries on the cubic grid like we do on the square grid (specifically the Catagolue style system)? If not, how would we go about devising one? The highest level of symmetry would be D48_1 or D48_8 (cubic symmetry centered on a cube or vertex), and the lowest C1. I don't know how we'd sort all of the rest of these, however.

This list is currently present on the cubic honeycomb page on the wiki as a commented out block of text and it'd be great if we can finally get it set up in a readable way (alongside a bunch of other symmetry-related things I've been putting off for longer than I should have).

muzik · Post by **muzik** » June 30th, 2025, 2:51 pm

Simple example cases:

Much like the tub is a simple example of something with D8_1 symmetry in 2D: