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!

…At least I try. Also, sometimes, the educational outcome is better than in this fictionalized example.

[Image source: http://www.decorationnako.tk/birthday-cake/]

Ebony Against Onyx, with Low Albedo, but High Ilumination

Posted on 16 November 2015 by RobertLovesPi

Dodeca icosa low albedo

This is the compound of the icosahedron and its dual, the dodecahedron. I made this rotating image using Stella 4d , which is available here.

A Faceted Truncated Tetrahedron, or Seven Tetrahedra Joined at Their Edges — Your Choice

Posted on 14 November 2015 by RobertLovesPi

I made this by faceting a truncated tetrahedron, giving it faces which are interpenetrating, red, equilateral triangles, as well as yellow crossed-edged hexagons. It can also be viewed as a central tetrahedron, with six more tetrahedra attached to its edges. This was made with Stella 4d, available at this website.

A Cluster of 33 Truncated Icosahedra

Posted on 13 November 2015 by RobertLovesPi

There is one truncated icosahedron at the center of this cluster, and each of its 32 faces is augmented with another truncated icosahedron, for a total of 33. I built this cluster using Stella 4d, software available here.

A Box of Heptagons

Posted on 13 November 2015 by RobertLovesPi

It is hard to find polyhedra which contain regular heptagons, but that simply makes the search more challenging, and therefore more fun. This polyhedron features six regular heptagons, and was created with Stella 4d, software available here.

A Large Collection of Polyhedra with Pyritohedral Symmetry, Some of Them Chiral

Posted on 13 November 2015 by RobertLovesPi

It’s quite an informal way to define it, but pyritohedral symmetry is the symmetry-type of a standard volleyball. These images of pyritohedral polyhedra were made using Stella 4d, software available at http://software3d.com/Stella.php.

The Small Ditrigonal Icosidodecahedron, Together with Its Fifth Stellation

Posted on 11 November 2015 by RobertLovesPi

I made the polyhedron above by performing a faceting of the dodecahedron, and only realized, after the fact, that I had stumbled upon one of the uniform polyhedra, a set of polyhedra I have not yet studied extensively. It is called the small ditrigonal icosidodecahedron, and its faces are twelve star pentagons and twenty equilateral triangles, with the triangles intersecting each other. Below is its fifth stellation, which appears to be a compound of a yellow dodecahedron and a red polyhedron which I do not (yet) recognize, although it does look quite familiar.

Both images were created using Stella 4d, software you can try right here.

Partially Truncated Platonic and Rhombic Dodecahedra

Posted on 11 November 2015 by RobertLovesPi

Each of these dodecahedra were modified by truncations at exactly four of their three-valent vertices. As a result, each has four equilateral triangles as faces. In the one above, the Platonic dodecahedron’s pentagonal faces are modified into a dozen irregular hexagons by these truncations, while, in the one below, the rhombic dodecahedron’s faces are modified into twelve irregular pentagons.

Both of these polyhedra were created using Stella 4d, software you can try for yourself at this website.

A Mandala Made of Hexagons, Enneagons, and Dodecagons

Posted on 11 November 2015 by RobertLovesPi

$recreational math from 2011$

I recently re-discovered this “lost work,” which I made using Geometer’s Sketchpad, in 2011 — before I started this blog, which is why it has not appeared here before.