More Polyhedra, Including Some Compounds, from the Stellation-Series of the Tetrakis Cube

Posted on 16 September 2014 by RobertLovesPi

The next one is a compound of eight off-center pyramids. By this point, I had gone so far into the stellation-series (a search I began when preparing the post before this one) that I had lost count.

This one is a compound of three short square-based dipyramids:

This one, according to Stella 4d, is a compound of three parts, but I can’t quite figure out what the parts are!

Here is another “mystery compound,” this one with two parts:

Stella 4d, which I used to make these, may be tried here.

Two Compounds of Dipyramids from the Stellation-Series of the Tetrakis Cube

Posted on 16 September 2014 by RobertLovesPi

The 16th stellation of the tetrakis cube, the dual of the truncated octahedron, is a compound of three elongated octahedra, or square dipyramids:

The 65th stellation of this same polyhedron is of another compound of dipyramids, but these are triangular dipyramids with obtuse faces, and there are four of them:

I generated both of these images with Stella 4d: Polyhedron Navigator, available right here.

Two Dodecahedra with Varying Rotation-Types of the Same Design Shown On Their Faces

Posted on 15 September 2014 by RobertLovesPi

The images shown on the faces of these dodecahedra appeared in the last post, and were made using Geometer’s Sketchpad and MS-Paint.

Assembling the polyhedral images and creating these rotating .gif files required another program, Stella 4d, which is available at http://www.software3d.com/Stella.php.

This Space Station for Geometricians Has, as Outer Hulls, Twelve Trapezoids, and Six Parallelograms with One Square Window / Docking Port Each

Posted on 14 September 2014 by RobertLovesPi

I can’t think of any good reasons for geometricans not to have their own space station, and I know what we’d do there: we’d work on geometry (also known informally as “playing with shapes”).

My suggestion for this space station’s design was created with Stella 4d, and you may find that program (to try or guy) here: http://www.software3d.com/Stella.php.

Another Faceting of the Icosidodecahedron

Posted on 13 September 2014 by RobertLovesPi

Check out http://www.software3d.com/stella.php to try the software used to make this image.

Two Wrinkled Polyhedra, and One of Their Duals

Posted on 13 September 2014 by RobertLovesPi

I recently stumbled upon some wrinkled polyhedra. Such polyhedra have an unusually high amount of surface area, compared to their volumes, relative to most more-familiar polyhedra. The next one shown, below, is the dual of the first one. shown, above.

The two above were made before I received a request to slow down the spin-rate of the polyhedra I post, most of which, in the past, have had a rotational period of four seconds. This next one has the new, slower spin-rate (with T = 6 seconds) which I am now using:

I hope all of my readers prefer this change, especially since it takes 50% more memory, per file, to slow these down to 2/3 their previous rotational speed.

These were all made using Stella 4d, software which can be tried or bought here: http://www.software3d.com/Stella.php.

How to Get Fair Dice with Various Numbers of Possible Results, from Two to Thirty, Using Different Polyhedra

Posted on 13 September 2014 by RobertLovesPi

For a d2, number a cube’s faces with three ones and three twos.
For a d3, number a cube’s faces 1,1,2,2,3,3.
The standard d4 is a Platonic solid, the tetrahedron. Double-numbered (two ones, two twos, etc.) octahedra are sometimes used as d4s, also.
For a d5, an icosahedron can be renumbered with four each of the numbers one through 5. Double-numbering a pentagonal dipyramid or pentagonal trapezohedron also works.
The d6 is the familiar cube.
For a d7, one option is to roll an octahedron, but re-roll 8 each time it comes up.
For a d9, one option is to roll a d10, but reroll 10s.
For d10s, pentagonal dipyramids and pentagonal trapezohedra both work. There is also the option of double-numbering an icosahedron.
For a d11, one option is to roll a d12, but reroll 12s.
For a d12, the Platonic and rhombic dodecahedra both work.
For a d13, roll a d14, but reroll 14s.
For a d14, one option is to roll a d7 and a d2, then add 7 to the d7 result iff the d2 shows 2. Another is to roll a d15, but reroll 15s.
For a d15, simply double-number the thirty faces of a rhombic triacontahedron.
For a d16, roll a d2 and a d8 together, using the d8 result, but adding 8 to the d8 result iff the d2 result is 2.
For d17, roll a d18, but reroll 18s.
For d18, roll a d2 and a d9 together, using the d9 result, but adding 9 to the d9 result iff the d2 result is 2.
For d19, roll a d20, but reroll 19s.
For d20, icosahedra are used.
For d21, one option is to roll a d24, rerolling any result from 22-24. Another is roll a d7 and a d3 together, using the d7 result, but adding 7 to this d7 result iff the d3 result is 2, but adding 14 to the d7 result iff the d3 result is 3.
For d22, one option is to roll a d24, rerolling any 22s and 24s. Another is to roll a d2 and a d11 together, using the d11 result, but adding 11 to this d11 result iff the d2 result is 2.
For d23, use a d24, and re-roll 24s.
Options for the d24 include the triakis octahedron, the tetrakis cube, the deltoidal icositetrahedron, and the pentagonal icositetrahedron, all of which are Catalan solids (duals of the Archimedeans). Another d24 can be made by rolling a d2 and a d12 together, and using the d12 result, but adding 12 to this d12 result iff the d2 result is 2.
For a d25, roll two distinguishabale d5s, called d5a and d5b. The 1-25 random number is (d5a)+ (5)(d5b-1).
For a d26, roll a d13 and a d2, then add 13 to the d13 result if the d2 shows 2. Another is to roll a d15, but reroll 15s. The first option may require two different d2s, so they will have to be distinguishable, in that case, or one d2 must be rolled twice, each for a different purpose.
For a d27, roll a d9 and d3 together. The result is (d9) + (9)(d3-1).
For a d28, roll a d14 and d2 together, using the d14 result, but adding 14 to it iff the d2 result is 2. Another option: roll a d30, but reroll results of 28 or 29. The first option may require two different d2s, so they will have to be distinguishable, in that case, or one d2 must be rolled twice, each for a different purpose.
For a d29, roll a d30, and reroll 30s.
The most common d30 is a rhombic triacontahedron. After the Platonic solids and the d10, these d30s are the most commonly available example of polyhedral dice.