A Polyhedron with 36 Pentagonal Faces

Posted on 16 December 2015 by RobertLovesPi

36 pentagons

This polyhedron was created using Stella 4d, software which can be found here. It has pyritohedral symmetry.

Tessellation Featuring Regular Pentagons and Regular Pentadecagons

Posted on 12 December 2015 by RobertLovesPi

tess new

Other polygons included in this tessellation include several types of rhombi, as well as triconcave octadecagons. The pattern is chiral, but the chirality is subtle. (Hint: look near the pentagons.)

A Rhombic Enneacontahedron, Adorned with Jumpy Tessellations Which Resemble Rhombic Enneacontahedra

Posted on 8 December 2015 by RobertLovesPi

Software credit: I made this using Stella 4d, available here.

A Compound of Ten Thin Parallelopipeds, Together with Its Final Stellation

Posted on 4 December 2015 by RobertLovesPi

The polyhedron above also appeared in the post immediately before this one, as the second of three images. However, here it is presented in “rainbow color mode.”

This is its final stellation:

Both virtual models were created with Stella 4d, software you may try for yourself at this website.

A Compound of Ten Elongated Octahedra Which Is Also a Particular Faceting of the Rhombicosidodecahedron, Together with Its Dual

Posted on 3 December 2015 by RobertLovesPi

Thinking about the post immediately before this one led me to see if I could connect opposite triangular faces of a rhombicosidodecahedron to form a ten-part compound — and it worked with Stella 4d just as it had when I “previewed” it in my head.

The interesting dual of the above polyhedral compound, also a ten-part compound, I was not able to preview in my head (although that would be a nice ability to have), but creating it was easy with Stella:

It is difficult, in the dual, to tell what the ten components are. To help with this, in the next image, all but one component has been removed. This reveals the components of the dual to be rhombus-faced parallelopipeds which are quite flattened, compared to most parallelopipeds I have seen before. This polyhedron is isomorphic to the cube, just as the elongated octahedra in the first compound were each isomorphic to the Platonic octahedron. Given that the cube and octahedron are duals, this is no surprise.

Stella 4d may be tried for free, as a trial download, at this website: http://www.software3d.com/Stella.php.

A Compound of Fifteen Cuboids — Which Is Also a Particular Faceting of the Icosidodecahedron

Posted on 1 December 2015 by RobertLovesPi

The creator of Stella 4d, the program I used to make these rotating polyhedral images, is Robert Webb (and the software itself may be tried for free here). Recently, on Facebook, he displayed a paper model of this compound of fifteen cuboids, pointed out that it is a faceting of the icosidodecahedron, and I (being me) took that as a challenge to make it myself. Here is my first result, in which all fifteen cuboids have different colors.

I then realized that RW had rendered his in only five colors, so I studied his post more carefully, and made the appropriate adjustments to do the same:

If you’d like to find the Stella page on Facebook, here is a link to it.

Focus

Posted on 29 November 2015 by RobertLovesPi

On your nth birthday, you turn n – 1 years old.

Posted on 27 November 2015 by RobertLovesPi

birthday cake

As a teacher, I have had variants of this conversation many times. The specific details, however, are fictional, for this changes, somewhat, each time it happens.

Student: Guess what? It’s my birthday!
Me: Congratulations! How old are you?
Student: I’m seventeen!
Me: Well, happy 18th birthday, then!
Student: Huh?
Me: Look, on that one day, 17 years ago, when you were born, that was your birthday. That day has a better claim on being your birthday than any other day, because it’s the day you were born. That was your first birthday. But you weren’t one year old yet. You turned one year old a year later, on your next birthday . . . your second birthday. A year later, on your third birthday, you turned two years old. Do I need to continue?
Student: So I’m 18? I can buy cigarettes without a fake ID, and vote, and stuff?
Me: No, not for another year, because you’re only 17 years old — but you have had 18 birthdays. Say, here come some of your friends. Use this bit yourself, if you want to, and have fun with it.
Student, to other students: Hey, guys, it’s my birthday! I’m 18 today!