The Seven Zonish Dodecahedra with Zones Added Based on Faces, Edges, and/or Vertices

Posted on 20 December 2014 by RobertLovesPi

If a zonish dodecahedron is created with zones based on the dodecahedron’s vertices, here is the result.

If the same thing is done with edges, this is the result — an edge-distorted version of the great rhombicosidodecahedron.

Another option is faces-only. Although I haven’t checked the bond-lengths, this one does have the general shape of the most-symmetrical 80-carbon-atom fullerene molecule. Also, this shape is sometimes called the “pseudo-truncated-icosahedron.”

The next zonish dodecahedron has had zones added based on the dodecahedron’s faces and edges, both.

Here’s the one for vertices and edges.

Here’s the one for faces and vertices.

Finally, the last of this set of seven has had zones added based on all three: faces, vertices, and edges.

All seven of these were made with Stella 4d, which is available at http://www.software3d.com/Stella.php.

Thirty-Three Polyhedra with Icosidodecahedral Symmetry

Posted on 20 December 2014 by RobertLovesPi

Note: icosidodecahedral symmetry, a term coined (as far as I know) by George Hart, means exactly the same thing as icosahedral symmetry. I simply use the term I like better. Also, a few of these, but not many, are chiral.

The images directly above and below show the shape of the most symmetrical 240-carbon-atom fullerene.

The image above is of the compound of five tetrahedra. This compound is chiral, and the next image is the compound of the compound above, and its mirror-image.

In the next two, I was experimenting with placing really big spheres at the vertices of polyhedra. The first one is the great dodecahedron, rendered in this unusual style, with the faces rendered invisible.

I made these using Stella 4d: Polyhedron Navigator. You may try this program for free at http://www.software3d.com/Stella.php.

The Seven Zonish Icosahedra with Zones Added Based on Faces, Edges, and/or Vertices

Posted on 19 December 2014 by RobertLovesPi

If a zonish icosahedron is created with zones based on the icosahedron’s vertices, here is the result.

If the same thing is done with edges, this is the result.

Another option is faces-only.

The next zonish icosahedron has had zones added based on the icosahedron’s faces and edges, both.

Here’s the one for vertices and edges.

Here’s the one for faces and vertices.

Finally, the last of this set of seven has had zones added based on all three: faces, vertices, and edges.

All seven of these were made with Stella 4d, which is available at http://www.software3d.com/Stella.php.

Compounds of Enantiamorphic Archimedean Solid Duals

Posted on 19 December 2014 by RobertLovesPi

An enantiomorphic-pair compound requires a chiral polyhedron, for it is a compound of a polyhedron and its mirror image. Among the Archimedeans, only the snub cube and snub dodecahedron are chiral. For this reason, only threir duals are chiral, among the Archimedean duals, also known as the Catalan solids.

That’s a compound of two mirror-image snub cube duals (pentagonal icositetrahedra) above; the similar compound for the snub dodecahedron duals (pentagonal hexacontahedra) is below.

Both these compounds were made with Stella 4d, which is available at http://www.software3d.com/Stella.php.

Blue-on-Blue Dodecahedron

Posted on 17 December 2014 by RobertLovesPi

This uses enlarged spheres centered on the dodecahedron’s vertices, overlapping so that they obscure the edges. Also, the faces are rendered invisible. I created it using Stella 4d, available at http://www.software3d.com/Stella.php.

Buckminsterfullerene Molecular Models: Three Different Versions

Posted on 17 December 2014 by RobertLovesPi

Buckminsterfullerene, a molecule made of 60 carbon atoms, and having the shape of a truncated icosahedron, is easily modeled with Stella 4d: Polyhedron Navigator (see http://www.software3d.com/Stella.php to try or buy this program). The first image shows the”ball and stick” version used by chemists who want the bonds between atoms to be visible.

The second model is intermediate between the ball-and-stick version, and the space-filling version, which follows it.

Here’s the “closely packed” space-filling version, taken to an extreme.

Which version better reflects reality depends on the certainty level you want for molecular orbitals. A sphere representing 99% certainty would be larger than one for 95% certainty.

Uniform Polyhedra: A Study, Beginning with the Small Ditrigonal Icosidodecahedron

Posted on 16 December 2014 by RobertLovesPi

A set of polyhedra which I have not (yet) studied much are the uniform polyhedra. The uniform polyhedra do, however, include some sets of polyhedra which I have studied extensively:

The Five Platonic Solids
The Four Kepler-Poinsot Solids
The Thirteen Archimedean Solids

Subtracting these 22 polyhedra (and the infinite sets of prisms and antiprisms), from the uniform polyhedra, leaves 53 uniform star polyhedra, of which 5 are quasiregular and 48 are semiregular. There’s also one other star polyhedron, only counted sometimes, which is different from the others in that it has pairs of edges that coincide. Discovered by John Skilling, it is often simply called Skilling’s figure. There are also 40 “degenerate” uniform polyhedra; these are generally not counted toward the total. I’ve been aware that these 54 polyhedra existed for years, but was preoccupied with the others. Now, it’s time to fix that.

There is a listing of all 75 (or 76) uniform polyhedra at https://en.wikipedia.org/wiki/List_of_uniform_polyhedra, for those who’d like to examine them as a group. My approach will be different: I’m going to study the ones I don’t already know one at a time, starting with one I picked on the basis of aesthetics alone: the small ditrigonal icosidodecahedron. To be a uniform polyhedron, all vertices must be the same (in other words, it is vertex-transitive), and all faces must be regular, with regular star polygons allowed. In this figure, each vertex has three equilateral triangles meet, as well as three star pentagons, with these figures alternating as one moves around the vertex, examining them.

Here are just the twelve star pentagons, with only parallel faces having the same color.

Here are only the twenty equilateral triangles, with only parallel triangles having the same color. As you can see, the triangles interpenetrate.

At least for me, the reason I had trouble understanding this figure, for so long, was that I mistook the small triangular “facelets” (the visible parts of the faces) for the triangular faces, themselves. In reality, the edges of the triangles are just as long as the star polygon edges. Because it has exactly two face-types which alternate around a vertex, it is edge-transitive (not all uniform polyhedra are), and so this polyhedron is part of smaller subset of uniform polyhedron called the quasiregular polyhedra.

Stella 4d, a program I use to study polyhedra, and make these images, will be the primary tool I use to investigate these uniform polyhedra with which I am not already familiar. It is available at http://www.software3d.com/Stella.php.

Four Non-Convex Polyhedra with Icosidodecahedral Symmetry

Posted on 15 December 2014 by RobertLovesPi

All of these were made with Stella 4d, a program you can find at http://www.software3d.com/Stella.php.

Two Polyhedra Featuring Twenty Regular Octadecagons Each

Posted on 14 December 2014 by RobertLovesPi

The first of these two polyhedra also includes isosceles triangles, two types of isosceles trapezoids, and twelve regular pentagons.

It is also possible to make a similar polyhedron where the twelve pentagons are replaced by regular decagons, but only by allowing the twenty octadecagons to overlap.