The Square Antitegmatic Hecaton­tetraconta­tetrachoron


The square antitegmatic hecatontetracontatetrachoron is a Catalan polychoron bounded by 144 square trapezohedra, 576 kites, 672 edges (288 short, 384 long), and 240 vertices. It is the dual of the runcinated 24-cell.

The square
antitegmatic hecatontetracontatetrachoron

The cells are transitive, and are square trapezohedra with an edge length ratio of 2√(5−3√2) : √7, or approximately 0.658 : 1. The following image shows an individual cell:

A single
facet of the square antitegmatic hecatontetracontatetrachoron

The dual of this cell is a non-uniform square antiprism whose lacing edges (the edges linking its two square faces) are longer than the edges of its square faces by a factor of √2.

Structure

We will explore the structure of the square antitegmatic hecatontetracontatetrachoron using its parallel projections into 3D.

The First Layer

The following images show three pairs of square trapezohedral cells situated around the vertex that the 4D viewpoint is looking at.

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing a
pair of square trapezohedra

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing 2
pairs of square trapezohedra

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing 3
pairs of square trapezohedra

The Second Layer

Surrounding these 6 cells are a layer of another 24 cells, each sharing a face with one of these cells. The following images incrementally show these cells, in groups of 8:

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing
8/24 cells in the 2nd layer

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing
16/24 cells in the 2nd layer

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing
24/24 cells in the 2nd layer

The Third Layer

Six more square trapezohedra fit into the bowl-shaped depressions, touching the vertices of the cells from the first layer:

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing 6
more cells in a 3rd layer

The Fourth Layer

Straddling the shallow valleys between the 2nd layer cells are another 24 cells, shown next:

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing 8/24
lateral cells in a 4th layer

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing 16/24
lateral cells in a 4th layer

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing 24/24
lateral cells in a 4th layer

These are all the cells that lie on the near side of the polytope, its northern hemisphere. Following this, we come to the limb, or equator, of the polytope.

The Equator

There are 24 cells that lie on the limb of the polychoron, in 3 rings of 8 cells each, as shown below:

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing
8/24 equatorial cells

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing
16/24 equatorial cells

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing
24/24 equatorial cells

These cells lie at a 90° angle to the 4D viewpoint, and so appear to be flat. In 4D, of course, they are full-bodied square trapezohedra. For the sake of clarity, we have omitted the cells we saw earlier.

Past this point, the layers we have seen so far repeat in reverse order in the southern hemisphere of the polytope.

The remaining gaps in the equator do not correspond to any cells; they are where the 4th layer cells of the northern hemisphere touch the corresponding 4th layer cells in the southern hemiphere.

Summary

In summary, there are 6 cells in the first layer, 24 cells in the 2nd layer, 6 cells in the 3rd layer, and 24 cells in the 4th layer, for a total of 60 cells in the northern hemisphere. There are 24 cells in the equator, and another 60 cells in the southern hemisphere, for a grand total of 60 + 24 + 60 = 144 square trapezohedral cells.

Relation to Hopf Fibration

The cells in the square antitegmatic hecatontetracontatetrachoron can be partitioned into 3 sets of 48, each set consisting of six rings, each of which consists of 8 cells joined to each other by their apical vertices. Each set of rings represents a discrete subset of the Hopf fibration of the 3-sphere.

The following images show one such set of rings. The first image shows the first ring:

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing 1st
ring

Apparently only 4 cells are shown here, but actually there are 8 cells: 4 cells on the far side of the polytope project to the same images as the 4 cells on the near side.

The next image shows the second ring:

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing 2nd
ring

Four of the cells are shown here only in outline, because they lie on the far side of the polychoron.

The next image shows the 3rd ring, again with the far side cells only in outline:

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing 3rd
ring

The following image shows the 4th ring:

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing 4th
ring

And the 5th ring:

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing 5th
ring

Finally, the last ring wraps around the equator:

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing 6th
ring

Here are all the rings together:

Parallel
projection of the square antitegmatic hecatontetracontatetrachoron, showing 6
rings together

For the sake of clarity, we show only the cells on the near side of the polytope.

Coordinates

The Cartesian coordinates of the square antitegmatic hecatontetracontatetrachoron are all permutations of coordinate and changes of sign of:

These coordinates correspond with a dual runcinated 24-cell of edge length (1−√2/2).


Last updated 24 Oct 2023.

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