The Runcinated Snub 24-cell


The runcinated snub 24-cell, also variously known as the prismato­rhombato snub 24-cell, or the runcinated snub demi­tesseract, or by the Bowers Acronym prissi, is a CRF polychoron derived from the snub 24-cell. Its surface consists of 288 vertices, 1008 edges, 960 polygons, and 240 cells (24 icosa­hedra, 24 truncated tetra­hedra, 96 triangular cupolae (J3), and 96 triangular prisms).

The runcinated
snub 24-cell

It can be constructed by radially displacing the icosahedral cells of the snub 24-cell outwards until they are an edge length apart, and filling in the resulting gaps with truncated tetrahedra, triangular cupolae, and triangular prisms. It is vertex-transitive but not uniform, and thus belongs to the category of scaliform polytopes.

It was discovered by Richard Klitzing in 2005.

Structure

We shall explore the runcinated snub 24-cell by means of its parallel projections into 3D, centered on an icosahedral cell.

The Near Side

Parallel
projection of the runcinated snub 24-cell, showing nearest icosahedron

This image shows the nearest cell to the 4D viewpoint, an icosahedron. Of its 20 triangular faces, 8 are joined to triangular prisms, shown next:

Parallel
projection of the runcinated snub 24-cell, showing 8 triangular prisms

The remaining 12 faces of the icosahedron are joined to 12 triangular cupolae:

Parallel
projection of the runcinated snub 24-cell, showing 3 of 12 triangular
cupolae

Parallel
projection of the runcinated snub 24-cell, showing 6 of 12 triangular
cupolae

Parallel
projection of the runcinated snub 24-cell, showing 9 of 12 triangular
cupolae

Parallel
projection of the runcinated snub 24-cell, showing 12 of 12 triangular
cupolae

These cupolae come in 6 pairs, with their hexagonal faces sharing an edge. These hexagonal faces are joined to 6 truncated tetrahedra:

Parallel
projection of the runcinated snub 24-cell, showing 6 truncated
tetrahedra

These truncated tetrahedra touch 6 of the starting icosahedron's edges.

The remaining exposed square faces of the triangular cupolae are joined to 12 more triangular prisms:

Parallel
projection of the runcinated snub 24-cell, showing 12 more triangular
prism

These triangular prisms look distorted, but that's only because they are being seen from an oblique angle with respect to the 4D viewpoint. In 4D, they are perfectly uniform.

Between these triangular prisms and the hexagonal faces of the truncated tetrahedra are 12 more triangular cupolae:

Parallel
projection of the runcinated snub 24-cell, showing 12 more triangular
cupolae

It should be obvious by now that the depressions forming around the triangular faces of the triangular prisms we've seen so far are where another 8 icosahedra fit into:

Parallel
projection of the runcinated snub 24-cell, showing 8 more icosahedra

The exposed square faces of the triangular cupolae are joined to 24 more triangular prisms:

Parallel
projection of the runcinated snub 24-cell, showing 24 more triangular
prisms

Nestled between these triangular prisms, the icosahedra, and the previous set of triangular prisms are another 12 triangular cupolae:

Parallel
projection of the runcinated snub 24-cell, showing 12 more triangular
cupolae

These are all the cells that lie on the near side of the polychoron. Next, we come to the cells that lie on the limb of the projection (the equator of the polytope).

The Equator

On the equator of the runcinated snub 24-cell are 6 icosahedra:

Parallel
projection of the runcinated snub 24-cell, showing 6 equatorial
icosahedra

These icosahedra have been foreshortened into hexagons, because they lie perpendicular to the 4D viewpoint. In reality, they are perfectly regular icosahedra.

Between these icosahedra are 12 truncated tetrahedra:

Parallel
projection of the runcinated snub 24-cell, showing 12 equatorial truncated
tetrahedra

These truncated tetrahedra lie at a 90° angle to the 4D viewpoint, and so they have been foreshortened into irregular pentagons. In 4D, however, they are perfectly uniform truncated tetrahedra. Their tips that touch the icosahedra are actually edges that have been foreshortened into points; thus they form bridges between some of the edges of the 6 icosahedra.

Between these truncated tetrahedra are 8 triangular prisms:

Parallel
projection of the runcinated snub 24-cell, showing 8 equatorial triangular
prisms

These triangular prisms lie at a 90° angle from the 4D viewpoint, and thus have been foreshortened into triangles. The edges that they share with the truncated tetrahedra have been foreshortened into points.

The final gaps in the above image are filled in by 24 triangular cupolae:

Parallel
projection of the runcinated snub 24-cell, showing 24 equatorial triangular
cupolae

These cupolae appear as irregular trapeziums because they lie at a 90° angle with the 4D viewpoint, and are seen laterally.

These are all the cells that lie on the limb of the runcinated snub 24-cell. Past this point, the arrangement of cells on the far side mirror the arrangement on the near side.

Summary

The following table summarizes the cell counts of the runcinated snub 24-cell:

Region Layer Icosahedra Triangular prisms Triangular cupolae Truncated tetrahedra
Near side 1 1 8 12 0
2 8 12 12 6
3 0 24 12 0
Subtotal 9 44 36 6
Equator 6 8 24 12
Far side 3 0 24 12 0
2 8 12 12 6
1 1 8 12 0
Subtotal 9 44 36 6
Grand total 24 96 96 24
240 cells

Coordinates

The Cartesian coordinates of the runcinated snub 24-cell, centered on the origin with edge length 2 are all even permutations of coordinate and all changes of sign of:

where φ is the Golden Ratio, (1+√5)/2.


Last updated 30 Apr 2018.

Powered by Apache Runs on Debian GNU/Linux Viewable on any browser Valid CSS Valid HTML 5! Proud to be Microsoft-free