The Greatly Augmented Icosidodecahedron, and Its Dual

Posted on 19 April 2015 by RobertLovesPi

If a central polyhedron’s pentagonal and triangular faces are augmented by great dodecahedra and great icosahedra, I refer to it as a “greatly augmented” polyhedron. Here, this has been done with an icosidodecahedron. The same figure appears below, but in “rainbow color” mode.

In the next image, “color by face type,” based on symmetry, was used.

The next image shows the dual of this polyhedral cluster, with face color chosen on the basis of number of sides.

Augmented Icosidodeca colored by whether sides have 5 or 16 sides

Here is another version of the dual, this one in “rainbow color” mode.

Finally, this image of the dual is colored based on face type.

These six images were made with Stella 4d, which may be found here.

The Pyramid-Excavated Icosidodecahedron, and Its Interesting Dual

Posted on 14 April 2015 by RobertLovesPi

To make this, I used the excavation-function of Stella 4d, set to remove pyramids with equal edge length from each face of an icosidodecahedron. You can try this program here.

The dual of this polyhedron is shown below.

Icosidodecahedral Stained Glass

Posted on 27 March 2015 by RobertLovesPi

Polyhedra are one of the areas (there are at least a few others) where the fields of mathematics and art intersect. Stella 4d, the program I used to make this image, is a great tool for the exploration of this region of intersection. This software may be tried for free right here.

Green Icosidodecahedral Cage

Posted on 8 January 2015 by RobertLovesPi

In this icosidodecahedron, the pentagonal faces have been removed, and the triangular faces have been augmented with short pyramids. I used Stella 4d to make it, which you can find here.

Five Variants of the Icosidodecahedron

Posted on 7 December 2014 by RobertLovesPi

I made all of these icosidodecahedron-variants using a program called Stella 4d: Polyhedron Navigator. It may be purchased, or tried for free (as a trial version), at http://www.software3d.com/Stella.php.

A Rhombic Triacontahedron, Made of Icosidodecahedra

Posted on 25 November 2014 by RobertLovesPi

It turns out that it is possible to use multiple icosidodecahedra as building blocks to build that polyhedron’s dual, the rhombic triacontahedron. This construction appears below, in four different coloring-schemes.

These rotating virtual models were constructed using Stella 4d, a program available at http://www.software3d.com/Stella.php.

An Alteration of the Icosahedron/Dodecahedron Compound

Posted on 18 November 2014 by RobertLovesPi

The dual of the icosahedron is the dodecahedron, and a compound can be made of those two solids. If one then takes the convex hull of this solid, the result is a rhombic triacontahedron. One can then made a compound of the rhombic triacontahedron and its dual, the icosidodecahedron — and then take the convex hull of that compound. If one then makes another compound of that convex hull and its dual, and then makes a convex hull of that compound, the dual of this latest convex hull is the polyhedron you see above.

I did try to make the faces of this solid regular, but that attempt did not succeed.

All of these polyhedral manipulations were were performed with Stella 4d: Polyhedron Navigator, available at http://www.software3d.com/Stella.php.

Icosidodecahedra, Icosahedra, and Dodecahedra

Posted on 13 November 2014 by RobertLovesPi

If one starts with a single icosidodecahedron, and then augments its pentagonal faces with dodecahedra, and its trianguar faces with icosahedra, this is the result.

This figure has gaps in it where two pentagons and two triangles meet around a vertex. If one puts icosidodecahedra in those gaps, this is the resulting figure.