## A symmetry question

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

### A symmetry question

I wrote on the wiki about D4.7 having "a 4D variant of" pyritohedral symmetry.

This got me thinking though - just because pyritohedral is 3D, what is actually any different about the symmetry of D4.7 - "pyritohedral axial" symmetry?

Similarly, is there even any difference between the axial symmetries in 3D, and the n-gonal symmetries in 2D?

They work in exactly the same way, and have the same orders, and so on - it is just that it seems strange to say a 4D object has a 3D symmetry group, etc.

But I can't see how it would be ambiguous. Using a 3D symmetry group to describe a 4D object cannot mean anything other than it has those symmetries along an axis, and no others; one may as well just say it belongs to that symmetry group.

Is this something I should be doing?

Keiji

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### Re: A symmetry question

Well, you should think of what kind of ...-like symmetry you are dealing with. D4.7 indeed clearly has just pyritohedral symmetry, but e.g. D4.9.1 has pyritohedral symmetry, combined with the property that "north"is the same as "south", thereby doubling the order of pyritohedral symmetry. you can think of that as follows: a thawro clearly has trigonal symmetry: it can be mirrored along planes that correspond to the mirrors of a 2-dimensional triangular symmetry. but a triangular prism has the additional property that it can be flipped, and thus it has a symmetry of double order. According to wikipedia, dihedral symmetry is just normal symmetry with a additional flip, and cyclic symmetry is normal symmetry without such a flip. therefore I think D4.7 has symmetry that could be called cyclic pyritohedral symmetry, and D4.9.1 dihedral pyritohedral symmetry. the names cyclic and dihedral are clearly used wrongly here, so I hope anyone has a better sugestion

student91
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student91
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### Re: A symmetry question

Well, the thawro / triangular prism distinction is simply axial vs prismatic symmetry (I tabulated all the 3D ones on the Symmetry group page).

I was going to call D4.7's symmetry "pyritohedral axial" symmetry, but my question was really about whether the word "axial" is redundant, and therefore it's just "pyritohedral" symmetry straight up.

Keiji

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### Re: A symmetry question

A lower-dimensional symmetry can certainly also occur in higher-dimensional objects. A hexagonal prism, for example, has hexagonal symmetry, even though hexagonal symmetry only needs 2 dimensions to be faithfully represented. Some lower-dimensional symmetries are also subsymmetries of a higher-dimensional symmetry; for example, a tetrahedron clearly has triangular symmetry, but it also has much higher order of symmetry, which we call tetrahedral symmetry. The triangular symmetry is just a subsymmetry of tetrahedral symmetry.

The fact that a higher-dimensional object "only" has a lower-dimensional symmetry just means that it doesn't have a full-fledged n-dimensional symmetry, but only exhibits a lower order of symmetry. A trigonal bipyramid, for example, does not have tetrahedral symmetry, though both it and the tetrahedron have trigonal symmetry as a subsymmetry.

As student91 pointed out, there's also a difference between, say, a trigonal bipyramid, vs. a triangular cupola. Both have trigonal symmetry, but the trigonal bipyramid is more symmetrical than the triangular cupola because it also has reflective symmetry about the plane that bisects its trigonal-symmetry axis, whereas the triangular cupola doesn't have such a symmetry (reflecting it about the hexagonal face makes it no longer the same as before).

There are other kinds of variations of a basic symmetry group; the 24-cell, for example, has the basic 24-cell symmetry, which includes various reflections across the hyperplanes of symmetry. You can think of this as 24-cell symmetry being composed of various subsymmetries: a 3-fold rotational symmetry around its triangular faces, a 2-fold reflective symmetry about the plane of its triangular faces, a 4-fold rotational symmetry about its edges, a 24-fold rotational symmetry around its octahedral cells, etc.. The snub 24-cell (*ahem* snub demitesseract ), however, loses some of the reflective symmetry of the 24-cell, so its symmetry group is only a subset of the full 24-cell symmetry, hence we call it the diminished 24-cell symmetry.

On the other hand, the bitruncted 24-cell has more symmetries than the 24-cell itself: in addition to all the reflections and rotations in 24-cell symmetry, it also has an additional set of symmetries that map the octahedra of a 24-cell to its vertices. These additional symmetries are not present in the 24-cell itself, because this mapping causes a visible change in the orientation of the 24-cell. The bitruncated 24-cell, however, has truncated cubes in both positions, so it is invariant under these additional symmetries. Hence, we call its symmetry group the augmented 24-cell symmetry.

Generally, self-dual polytopes will exhibit an augmented symmetry group where its facets are interchanged with its vertices. Such a symmetry is not an isometry of the polytope itself, but generally some derivation of the polytope will exhibit this augmented symmetry. So, all the n-simplices have an associated augmented n-simplex symmetry; in 2D, the augmented triangular symmetry is hexagonal symmetry (seen in the hexagon as the truncated triangle); in 3D, the augmented tetrahedral symmetry is the same as octahedral/cubic symmetry (seen in the octahedron being the rectified tetrahedron). In 4D, the augmented 5-cell symmetry no longer coincides with one of the other families, but diverges into its own "augmented 5-cell symmetry", which is exhibited by the 5-cell family members whose CD diagrams are palindromic -- e.g., the bitruncated 5-cell o3x3x3o, the runcinated 5-cell x3o3o3x, and the omnitruncated 5-cell x3x3x3x.

Similarly, if a base symmetry group can be alternated, then the alternated group forms a subsymmetry in which some reflections are no longer in the symmetry group. In 2D, the alternated 2n-gon produces the n-gon, with a reduction in symmetry from 2n-gonal symmetry to n-gonal symmetry. In 3D, the alternated cube produces the tetrahedron, so here we see the alternation of the cubic group being the inverse of the augmentation of the tetrahedral group. This, however, is peculiar to 3D; in 4D, the alternated tesseractic group forms a distinct symmetry group from the augmented 5-cell group, so here things diverge again. Of course, it just so happens that the alternated tesseract is the 16-cell, which actually has a higher order of symmetry than the alternated tesseractic group (in fact, it has the same symmetry as the tesseract!); a better example of the alternated tesseractic group (or demitesseractic group) would be the snub 24-cell (snub demitesseract). D4.11 is also an example. In 5D, the alternated tesseractic group coincides with the Gosset 1_21 group, but in 6D, the two diverge into the demi-hexaractic group and the Gosset 2_21 group. The Gosset groups go up to 9_21, where they flatten out into an 8D tessellation, and thereafter they become hyperbolic. The demicube symmetry groups, however, exist in all dimensions as their own series.
quickfur
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### Re: A symmetry question

Keiji wrote:Well, the thawro / triangular prism distinction is simply axial vs prismatic symmetry (I tabulated all the 3D ones on the Symmetry group page).

I was going to call D4.7's symmetry "pyritohedral axial" symmetry, but my question was really about whether the word "axial" is redundant, and therefore it's just "pyritohedral" symmetry straight up.

Probably redundant, I think, since pyritohedral symmetry is specific to 3D. I've tried to look for the 4D equivalent of pyritohedral symmetry before, but I don't think it exists, because you can't assign a bidirectional orientation to the tesseract's 8 cells such that no adjacent cube has a parallel orientation.
quickfur
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### Re: A symmetry question

For the polytopes, the symmetry groups mainly correspond to using stott-nodes over odd groups.

You get in general

simplex ES (even coordinates = rotary, AS = all coordinates). A third group is possible using a wander ASW. You might get ESW as well, in even dimensions.

for the cross-polytope and cubic, there are 5, EPECS (rotary semicubic), APECS (full semicubic), EPACS (pyritotopic), EP&CS (rotary-cubic), and APACS (cubic).

The 343 gives five symmetries 3+43+, 343+, (343)+, 343 and ((343)). There might be a (3+43+) etc.

The pentagonal groups have two EI and AI, being the even icosahedral and all-icosahedral
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### Re: A symmetry question

E.g. in 3D you have several different embeddings of 2D groups.
Those correspond to the not necessary polyhedra based on a regular n-gon.
• n-gonal pyramid - that one just has all 2D symmetries, but no additional along the axis. - I usually refer to that symmetry simply by "n-gonal pyramidal".
• n-gonal prism - that one adds a further reflection plane, which is orthogonal to the axis. - I usually refer to that one simply by "n-gonal prismatic".
• n-gonal antiprism - that one adds to the 2D symmetries a different possible action, a combinated mirror with a simultanuous drill. - I usually refer to that one simply by "n-gonal antiprismatic".

Thus you might have correspondingly within 4D the corresponding embeddings of 3D pyritohedral symmetry:
• pyritohedral pyramidal - no additional sym.
• pyritohedral prismatic - additional top bottom reflection
• pyritohedral antiprismatic - additional top bottom symmetry, where the bottom is a 90 degrees rotated copy of the top

--- rk
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