The Octahedral Ursachoron


The octahedral ursachoron is a CRF polychoron belonging to a family of polytopes that generalize the tridiminished icosahedron (J63) to higher dimensions. Its surface consists of an octahedron, 8 tridiminished icosahedra (J63), 6 square pyramids, and a cuboctahedron.

The octahedral
ursachoron

It is the bigger brother of the tetrahedral ursachoron, having octahedral symmetry rather than tetrahedral.

Structure

We shall explore the structure of the octahedral ursachoron by means of its parallel projections into 3D.

Side-view

First, we shall look at the octahedral ursachoron from a side-view, centered on one of the edges encircled by 4 tridiminished icosahedra.

Near Side

Parallel projection of
the octahedral ursachoron, showing a square pyramid

The above image shows one of the square pyramid cells that sit directly under the edge closest to the 4D viewpoint. It is somewhat foreshortened, because its base is actually slanting away into the 4th direction.

The next 4 images show each of the 4 tridiminished icosahedral cells that surround this edge, sitting above the square pyramid:

Parallel projection of
the octahedral ursachoron, showing 1 of 4 J63's

Parallel projection of
the octahedral ursachoron, showing 2 of 4 J63's

Parallel projection of
the octahedral ursachoron, showing 3 of 4 J63's

Parallel projection of
the octahedral ursachoron, showing 4 of 4 J63's

These cells appear squashed, due to perspective foreshortening, because they lie at an angle to the 4D viewpoint. In 4D, they are perfectly normal tridiminished icosahedra.

These are all the cells that lie on the near side of the polytope.

Equator

On the limb, or equator of the polytope, there are 4 square pyramids, shown below:

Parallel projection of
the octahedral ursachoron, showing 4 of equatorial square pyramids

These square pyramids have been foreshortened into isosceles triangles, because they lie at a 90° angle to the 4D viewpoint. In 4D, they are perfectly normal square pyramids.

The top cell, an octahedron, also appears on the equator from the current 4D viewpoint:

Parallel projection of
the octahedral ursachoron, showing octahedron

Like the square pyramids, this octahedron has been foreshortened into a square, due to its 90° angle to the 4D viewpoint.

The bottom cuboctahedron likewise:

Parallel projection of
the octahedral ursachoron, showing cuboctahedron

The remaining pentagonal faces are not the projection images of any cell; they are where the 4 tridiminished icosahedra on the near side of the polychoron touch their 4 counterparts on the far side.

Summary

The arrangement of cells on the far side of the polytope mirror the arrangement of cells on the near side. Therefore, there are another 4 tridiminished icosahedra and a square pyramid on the far side. The following table summarizes the cell counts:

Region Octahedra Tridiminished
		icosahedra Square
		pyramids Cuboctahedra
Near side 0 4 1 0
Equator 1 0 4 1
Far side 0 4 1 0
Grand total 1 8 6 1
16 cells

Top-view

The side-view projections of the octahedral ursachoron show its similarity to its tetrahedral analogue and the tridiminished icosahedron. However, its octahedral symmetry may not be immediately obvious. So now, we turn to its top-view projections: those centered on its octahedral cell.

Near side

Octahedron-centered
parallel projection of the octahedral ursachoron, showing octahedral
cell

This image shows the top octahedral cell. The octahedral symmetry of the entire polychoron is obvious from this viewpoint.

The faces of this octahedron are joined to 8 tridiminished icosahedra, shown below in 4 pairs:

Octahedron-centered
parallel projection of the octahedral ursachoron, showing first pair of 8
J63's

Octahedron-centered
parallel projection of the octahedral ursachoron, showing second pair of 8
J63's

Octahedron-centered
parallel projection of the octahedral ursachoron, showing third pair of 8
J63's

Octahedron-centered
parallel projection of the octahedral ursachoron, showing fourth pair of 8
J63's

The next image shows them all together:

Octahedron-centered
parallel projection of the octahedral ursachoron, showing all 8 J63's

These tridiminished icosahedra appear quite squished along the direction of their 3-fold axis of symmetry, because of the angle they lie with respect to the 4D viewpoint.

Together with the octahedron, these are all the cells that lie on the near side of the polytope from this 4D viewpoint.

Far side

Now we come to the far side of the polychoron.

Octahedron-centered
parallel projection of the octahedral ursachoron, showing antipodal
cuboctahedron

This image shows the cuboctahedral cell, which lies antipodal to the octahedron. Its square faces are joined to 6 square pyramids, shown below:

Octahedron-centered
parallel projection of the octahedral ursachoron, showing 6 square
pyramids

These cells are joined directly to the J63 cells of the near side; there are no equatorial cells from this 4D viewpoint.

Summary

The following table summarizes the cell counts from the top-down 4D viewpoint:

Region Octahedra Tridiminished
		icosahedra Square
		pyramids Cuboctahedra
Near side 1 8 0 0
Far side 0 0 6 1
Grand total 1 8 6 1
16 cells

Coordinates

The Cartesian coordinates of the octahedral ursachoron with edge length 2 are:

where φ=(1+√5)/2 is the Golden Ratio.

Like the tetrahedral ursachoron, the octahedral ursachoron's vertices come in 3 sets, each lying on a parallel hyperplane: the vertices of an octahedron, a φ-scaled octahedron, and a cuboctahedron (rectified octahedron). The hyperplanes are suitably positioned so that regular pentagons are formed between the vertex layers. This follows the same pattern as the construction of a tridiminished icosahedron (J63): a triangle, a φ-scaled triangle, and a dual triangle (rectified triangle).

This construction is general, and applies to all dimensions, consisting of a base shape B, a φ-scaled copy of B, and a rectified version of B in parallel hyperplanes. In 2D, it yields the regular pentagon. In 3D, using an equilateral triangle as a base shape, it yields J63. It is possible to use a square as a base shape in 3D instead; the result is a direct lower-dimensional equivalent of the octahedral ursachoron, the square ursahedron. However, the square ursahedron is not CRF (it contains isosceles triangles with a base of length √2), and thus does not belong to the set of Johnson solids.

In 4D, the octahedral ursachoron is CRF. It is also possible to construct a cubical ursachoron, but it contains square ursahedra as cells and is thus non-CRF. In general, if the base shape of the construction is a regular (n-1)-dimensional polytope that whose 2D faces are all triangles, then the corresponding ursatope will contain CRF cells. The ursatope itself will be CRF provided it is convex and full-dimensioned. Specifically, the n-simplex ursatope exists and is CRF in all dimensions.

In 4D, there are 3 CRF ursachora:

Other ursachora are possible, but they are non-CRF.


Last updated 03 Sep 2018.

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