Three Different Views of the 600-Cell

Posted on 1 July 2024 by RobertLovesPi

The 600-cell is a four-dimensional, convex, regular polychoron. Its dual is the 120-cell, also known as the hyperdodecahedron. Since the dual of the dodecahedron is the icosahedron, the 600-cell is sometimes called the “hypericosahedron,” That might lead one to think that the cells of a 600-cell are icosahedra, but they are actually tetrahedra. These tetrahedra meet twenty at a vertex, which is another way the icosahedron is involved in this figure.

I just finished a Zome model of a three-dimensional “shadow” of a 600-cell. It’s a challenging model to make. Built with B1, Y1, R1, and R0 Zomestruts, it’s about the size of a basketball. Instructions for building it may be found at https://www.pmedig.com/Zome_600cell.html.

If you’d like to try Zometools for yourself, or want some for your kids, the website to visit is http://www.zometool.com. I’ve been a happy customer of theirs for over twenty years.

Here are two more models of the 600-cell. Rather than being physically built, these are virtual models. They show the polychoron rotating in hyperspace, thus changing the appearance of its three-dimensional “shadow” on a continual basis. This rotation in hyperspace differs in the two images shown, along with the coloring of the models.

These virtual models were created using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.

The Hyperspace Analog of the Dodecahedron/Icosahedron Compound

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The Hyperspace Analog of the Dodecahedron/Icosahedron Compound

The dodecahedron and the icosahedron are dual to each other, and can be combined to make this well-known compound.

In hyperspace, the analog to the dodecahedron is the hyperdodecahedron, also known as the 120-cell, as well as the hecatonicosachoron. Its dual is the 600-cell, or hexacosichoron, made of 600 tetrahedral cells. The image at the top is the compound of these two polychora, rotating in hyperspace.

These images were made using Stella 4d, available at http://www.software3d.com/Stella.php.

600 Undulating Tetrahedra

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600 Undulating Tetrahedra

This is a 600-cell, one of the regular polychora (four-dimensional polytopes), with its edges and vertices rendered invisible, and its cells shrunk so that they do not touch. It’s rotating in hyperspace, and what you are seeing at any given moment is a particular three-dimensional “shadow,” or projection, of the entire figure.

It’s easy to make this sort of thing with software called Stella 4d, written by an Australian friend of mine. Here’s a link to a site where you can try it, as a free trial download, before deciding whether or not to purchase the fully-functioning version: http://www.software3d.com/Stella.php.

Another View of the 600-Cell

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Not long ago, I posted one rotating projection of the regular polytope known as the 600-cell. Here is another.

Software used: see http://www.software3d.com/stella.php.

The 600-Cell

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This regular 4-d polytope is made of 600 tetrahedral unit cells. It is rotating in hyperspace, continually changing the three-dimensional “slice” of it which you can see. The software used to make this image may be purchased, or tried for free, at http://www.software3d.com/Stella.php.