About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things, a large portion of which are geometrical. Welcome to my own little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with those of my employer, nor any other organization, nor institution, of any kind.

Golden Triangles and Golden Gnomons

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A Compound of Three Elongated Octahedra

Posted on 14 March 2020 by RobertLovesPi

This compound is the 16th stellation of the tetrakis hexahedron, the Catalan solid which is the dual of the Archimedean truncated octahedron. I made it using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.

An Icosahedron, Augmented with Twenty Triangular Cupolae

Posted on 13 March 2020 by RobertLovesPi

To make the polyhedron shown above, I started with an icosahedron . . .

. . . and then I augmented each face with a triangular cupola, with this polyhedron’s hexagonal faces pointed outward.

I made these images using Stella 4d, which you can try for free at this website.

A Rhombic Triacontahedron, Vertices Surrounded By Smaller Rhombic Triacontahedra, and Its Interesting Dual

Posted on 8 March 2020 by RobertLovesPi

The first image shows a central yellow rhombic triacontahedron, with smaller, blue rhombic triacontahedra attached to each of its thirty-two vertices. The second polyhedron shown is the dual of the first one, with colors chosen by the number of sides per face in the second image — pentagons red, and triangles yellow. The convex hull of this second polyhedral complex shown would be an icosidodecahedron, itself the dual of the rhombic triacontahedron.

I use software called Stella 4d: Polyhedron Navigator to make the rotating polyhedral images on this blog. You can try Stella for yourself, for free, at http://www.software3d.com/Stella.php.

My Primary Vote, 2020

Posted on 7 March 2020 by RobertLovesPi

I wonder how many of the other 6,202 people who voted for Bill Weld in the Arkansas Republican primary were doing so simply because they’re allergic to Trump and Trumpism? That reason was good enough for me to proudly say “Republican” when asked which ballot I wanted, even though I more often vote for Democrats in recent years. I don’t pass up a chance to vote against Trump, as directly as possible.

If anyone ever tries to tell me that one vote makes no difference, I’ll refer them to this statistic, along with my previous such vote in 2016. I know exactly how much difference my vote makes.

[Image source: here.]

Tessellation Featuring Rhombi and Concave, Equilateral Dodecagons

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This particular tessellation is full of angles measuring 20 degrees, 40 degrees, and other angles which are not constructable using the traditional rules of Euclidean constructions. This is because this tessellation is based on a matrix which includes regular enneagons.

Twelve Pentagrammic Prisms

Posted on 23 February 2020 by RobertLovesPi

I made this by augmentation (with prisms) of the faces of a small stellated dodecahedron, using Stella 4d, which you can try for free at this website.

A Symmetrohedron Featuring Twelve Regular Pentagons, Thirty Regular Octagons, and Twenty Equilateral Triangles

Posted on 18 February 2020 by RobertLovesPi

In addition to the sixty-two regular faces, this polyhedron also has two sets of sixty isosceles trapezoids each, shown in different shades of blue. That’s 182 faces in all. I made it using Stella 4d, which you can try for free at www.software3d.com/Stella.php.

A Partial Faceting of the Rhombic Octagonoid

Posted on 16 February 2020 by RobertLovesPi

The rhombic octagonoid appears here. These polyhedra were made using Stella 4d, which you can try for free at this website.

The Rhombic Octagonoid, a Zonohedron With Ninety Faces

Posted on 15 February 2020 by RobertLovesPi

To make this zonohedron with Stella 4d (available as a free trial download here), start with a dodecahedron, and then perform a zonohedrification based on both faces and vertices. It is similar to the rhombic enneacontahedron, with thirty equilateral octagons replacing the thirty narrow rhombic faces of that polyhedron.

I’ve run into this polyhedron from time to time, and have also had students make it. It is the largest zonohedron which can be built using only red and yellow Zome (available here) of a single strut-length (short, medium, or long). I thought it needed a name, so I made one up.