An 1800-Faced Polyhedron, with Its 960-Faced Dual

Posted on 7 January 2015 by RobertLovesPi

The polyhedra shown above and below are duals. The one above has 1800 faces and 960 vertices. The one below has 960 faces, and 1800 vertices. This “flipping” of the face and vertex numbers always happens with dual polyhedra.

Also, two dual polyhedra always have the same number of edges, which can be found by subtracting two from the sum of the numbers of faces and vertices (this is based on Euler’s Formula, F + V = E + 2). In this case, each of these polyhedra have 1800 + 960 – 2 = 2758 edges.

These virtual models were created using Stella 4d: Polyhedron Navigator, software you can try for yourself right here.

[Later note: I have noticed a lot of referrals to this post through stumbleupon.com, and wish to thank the unknown person who posted the link there for all this increased traffic to my blog. To those who are finding me via StumbleUpon, welcome! I invite you to check out other posts here, as well. The “topic cloud” on the right side of the page should help you find stuff of interest to you, of the 1000+ posts here, many of which are also about polyhedra.]

A Polyhedron with Eight Regular Hexagonal Faces and Twenty-Four Irregular Pentagonal Faces

Posted on 6 January 2015 by RobertLovesPi

There are many polyhedra that include only hexagons and pentagons as faces — infinitely many, in fact. Most of the well-studied ones include twelve regular pentagonal faces, though, but this polyhedron has twenty-four pentagons, none of which are regular, in six panels of four pentagons each. Its regular faces are the eight hexagons, in the face-planes of the faces of an octahedron.

I made this with Stella 4d, software you can try here.

A Rhombic Enneacontahedron, Augmented with Sixty Rhombic Dodecahedra, Along with Its Dual

Posted on 3 January 2015 by RobertLovesPi

The dual to this cluster-polyhedron appears below. Both virtual models were created using Stella 4d: Polyhedron Navigator, software available here.

A Kite-Faced Polyhedron Based on the Cube, Octahedron, and Rhombic Dodecahedron

Posted on 27 December 2014 by RobertLovesPi

Above is the entire figure, showing all three set of kites. The yellow set below, though, lie along the edges of a rhombic dodecahedron.

The next set, the blue kites, lie along the edges of an octahedron.

Finally, the red set of kites lies along the edges of a cube — the dual to the octahedron delineated by the blue kites.

These images were made using Stella 4d, which is available here.

“Fractured” Octahedra

Posted on 27 December 2014 by RobertLovesPi

These are variations of the octahedron. I made them all with Stella 4d, which is available here. “Fractured” isn’t an official term, as “truncated or stellated” are, but I can’t come up with a better description, at least not yet. Other suggestions are welcome.

$fractured octahedron 2$

$fractured octahedron 3 rb$

$fractured octahedron$

A Variant of the Octahedron Which Features Regular Dodecagons and Quadrilaterals of Three Types

Posted on 27 December 2014 by RobertLovesPi

(See here for more information on Stella 4d, the software used to create this image.)

Six Hexagons and Four Triangles, As Faces of a Small Polyhedron

Posted on 27 December 2014 by RobertLovesPi

One could call this a half-truncated cube. A fully truncated cube has eight triangular faces, created by truncation, and this has half as many.

(See here for more information on Stella 4d, the software used to create this image.)

A Survey of Polyhedra with Pyritohedral Symmetry

Posted on 24 December 2014 by RobertLovesPi

The simplest way for many to understand pyritohedral symmetry is simply to realize that it is the symmetry of the seams in a volleyball. The first time I encountered this unusual symmetry-type was in the golden icosahedron I blogged about here, a figure which much resembles this pyritohedral icosahedron, except the dozen isosceles triangles in this one have a leg-to-base ratio which is not the golden ratio.

Earlier today, I went on a search for polyhedra with pyritohedral symmetry. I found several, but the worthwhile findings from the search are far from exhausted. Here are some others I found, exploring and manipulating polyhedra using Stella 4d, which you can try at this website.

In the version of the pyritohedral icosahedron above, the twelve green triangles have become heptagons which use very short sides to approximate triangles. The one below is of a similar figure, but one in which truncations has happened, so I call it a truncated pyritohedral icosahedron.

There also exist many pyritohedral polyhedra based, more or less, on the cube. These are a few I have found:

Now, is this next one a pyritohedral cube, or a pyritohedral dodecahedron? A case could be made for either, so it inhabits a “gray zone” between varying categories.

Here is a pyritohedral icosidodecahedron:

This one could probably be described in multiple ways, also, but it looks, to me, like a rhombic dodecahedron with its six four-valent vertices being double-truncated in a pyritohedral manner, with pairs of isosceles trapezoids appearing where the truncations took place.

One thing that this one, and the last, have in common is that the largest faces are heptagons. It appears to be a pyritohedral dodecahedron which has been only partially truncated.

This survey could not have been performed without a program called Stella 4d, which I rely on heavily for polyhedral investigations. It may be purchased, or tried for free, at http://www.software3d.com/Stella.php.

If You Have Enough Platonic Dodecahedra Around, and Glue Them Together Just Right, You Can Make a Rhombic Triacontahedron.

Posted on 22 December 2014 by RobertLovesPi

Aren’t you glad to know that? As soon as I found out icosahedra can form a rhombic dodecahedron (see last post), I knew this would be true as well. Why? Zome explains why, actually. It’s at http://www.zometool.com. Anything buildable with yellow Zome can be built out of icosahedra. Dodecahedra con form anything buildable with red Zome. Finally, if you can make it with blue Zome, it can be built out of rhombic triacontahedra. It follows that rhombicosidodecahedra can build anything Zome-constructible — but one look at a Zomeball makes that easy to believe, since Zomeballs are modified rhombicosidodecahedra.

Anyway, here’s the rhombic triacontahedron, made of dodecahedra: