A Dozen Dodecahedra, Surrounding an Icosahedron

Posted on 22 January 2020 by RobertLovesPi

I made these virtual models using Stella 4d: Polyhedron Navigator. If you’d like to try this program for yourself — free — the website to visit is http://www.software3d.com/Stella.php.

A Parallelogram-Expanded Snub Dodecahedron

Posted on 21 January 2020 by RobertLovesPi

The snub dodecahedron may be thought of as a dozen regular pentagons, surrounded and separated by a sea of triangles. In this expansion of that Archimedean solid, thirty parallelograms are added to the mix, also surrounded by triangles. In the image above, coloring is by face type — for example, the yellow triangles are those triangles which share an edge with a pentagon. Other triangles have other colors.

The image shown below is of the same polyhedron, but with a different coloring-scheme. In it, all triangles are given the same color, even when their shapes are slightly different.

This polyhedron has an interesting all-pentagon dual, which is shown below. This dual has sixty each of both the small and large pentagons, for a total of 120 faces.

I used Stella 4d: Polyhedron Navigator to create these polyhedra, and to make these rotating images. This program may be tried for free at http://www.software3d.com/Stella.php.

An Embellished Icosahedron

Posted on 20 January 2020 by RobertLovesPi

Created using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.

A Compound of Six Pentadecagonal Prisms

Posted on 19 January 2020 by RobertLovesPi

This chiral polyhedral compound was generated from a partial faceting of the polyhedron shown in the last post here, using Stella 4d‘s faceting function, plus its “try to make faces regular” operation afterwards. Making the six-prism compound in the first place was suggested by Tony Hartley, on Facebook, where I posted a link to that last post in a mathematical group for discussion. Thanks, Tony!

If you’d like to try Stella for yourself, the site to visit for a free trial download is http://www.software3d.com/Stella.php.

An Unusual Polyhedron with 200 Faces

Posted on 18 January 2020 by RobertLovesPi

This polyhedron was made from the faceted rhombicosidodecahedron in the post immediately before this one, by stellating it a few times. Its 200 faces include twenty equilateral triangles (yellow), sixty irregular decagons (green), and two sets of sixty trapezoids each (red and blue). I made it using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.

Two Images of a Toroidal Rhombic Triacontahedron Made of 212 Dodecahedra

Posted on 4 January 2020 by RobertLovesPi

I made these using Stella 4d, a program you can try as a free trial download at http://www.software3d.com/Stella.php.

The Great Icosidodecahedron and Its Dual, the Great Rhombic Triacontahedron

Posted on 1 January 2020 by RobertLovesPi

The great icosidodecahedron is one of the uniform polyhedra, which I do not know well. I stumbled across it while creating facetings of the (lesser) icosidodecahedron. This solid has two face-types, there are red pentagrams and yellow equilateral triangles, as seen in the next two pictures.

The dual of this solid is the great rhombic triacontahedron. Its faces are thirty interpenetrating rhombi.

I used Stella 4d to create these images. You may try this program for free at http://www.software3d.com/Stella.php.

Four Rhombicosidodecahedra Surrounding a Tetrahedron

Posted on 31 December 2019 by RobertLovesPi

I made this using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.

The Icositetrachoron and the Truncated Icositetrachoron, Rotating in Hyperspace

Posted on 29 December 2019 by RobertLovesPi

There are six regular, convex four-dimensional polytopes. Five of them correspond on a 1:1 basis with the Platonic solids (as the tesseract corresponds to the cube), leaving one four-dimensional polytope without a three-dimensional analogue among the Platonics. That polychoron is the icositetrachoron, also named the 24-cell and made of 24 octahedral cells. It also happens to be self-dual.

If, in hyperspace, the corners are cut off just right, new cells are created with 24 cubic cells created at the corners, with 24 truncated octahedral cells remaining from the original polychoron. This is the truncated icositetrachoron:

4-dimensional polytopes have 3-dimensional nets. These nets are shown below — first for the icositetrachoron, and then for the truncated icositetrachoron.