I made this using Stella 4d: Polyhedron Navigator. You may try this software, for free, at http://www.software3d.com/Stella.php.
A Transparent Rhombic Triacontahedron
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Tag Archives: polyhedron
The Compound of the Dodecahedron and the First Stellation of the Icosahedron
I made this using Stella 4d, available here.
A Transparent Rhombic Dodecahedron
Created using Stella 4d, a program available at this website: http://www.software3d.com/Stella.php.
A Peek Backwards, as Far Back as Possible
Never before have I deliberately tried to recall my earliest memories. This morning, however, simply to see what might happen, I tried it. In my imagination, I returned to further back than I ever had gone before, to a period before I learned to communicate. In this early period, I could visualize things, with the imagery which appeared being geometric in nature. Later, I had to learn English, as a second language, to express the mathematical ideas in my head. My first word, according to my parents, described one of the two shining round things in the sky: “Moon.” I have always preferred moonlight to sunlight, for the intensity of direct sunlight is painful to me.
At least, that’s how I remember these things; I could be wrong about the earliest parts. All I know is that the image above popped into my head, when I tried to recall my oldest accessible memory. I then made the image above, in a short period of time, using Stella 4d, Polyhedron Navigator, available to try for free at this website. (I’ve used the program for over a decade, and find it an indispensable tool for geometrical investigations, such as this recreation of what I found in this morning’s early-memory-search.)
A Deck of Polyhedra
There are 52 rotating polyhedra below. Starting right now, 52 of anything can be called a “deck” of that thing. The derivation of this term is the the number of cards in a standard playing-card deck. The deck is the smaller “cousin” of the mole, or 6.02 x 10²³ of anything, from chemistry, as well as the “big brother” of the dozen, or twelve of anything. All three units are measures of specific quantities, and can be applied to any objects, at least in principle. One never encounters a mole of people because there aren’t that many of us, but a mole of people can still be imagined. The same thing applies to a deck of Earths.
This deck of virtual, rotating polyhedra was created using Stella 4d, a program you may purchase here. If you would like to see a larger image of any one of these polyhedra, simply click on it.
494 Circles, Each, Adorning Two Great Rhombcuboctahedra, with Different (Apparent) Levels of Anxiety
The design on each face of these great rhombcuboctahedra is made from 19 circles, and was created using both Geometer’s Sketchpad and MS-Paint. I then used a third program, Stella 4d (available here), to project this image on each of a great rhombcuboctahedron’s 26 faces, creating the image above.
If you watch carefully, you should notice an odd “jumping” effect on the red, octagonal faces in the polyhedron above, almost as if this polyhedron is suffering from an anxiety disorder, but trying to conceal it. Since I like that effect, I’m leaving it in the picture above, and then creating a new image, below, with no “jumpiness.” Bragging rights go to the first person who, in a comment to this post, figures out how I eliminated this anxiety-mimicking effect, and what caused it in the first place.
Your first hint is that no anti-anxiety medications were used. After all, these polyhedra do not have prescriptions for anything. How does one “calm down” an “anxious” great rhombcuboctahedron, then?
On a related note, it is amazing, to me, that simply writing about anxiety serves the purpose of reducing my own anxiety-levels. It is an effect I’ve noticed before, so I call it “therapeutic writing.” That helped me, as it has helped me before. (It is, of course, no substitute for getting therapy from a licensed therapist, and following that therapist.) However, therapeutic writing can’t explain how this “anxious polyhedron” was helped, for polyhedra can’t write.
For a second hint, see below.
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Second hint: the second image uses approximately twice as much memory-storage space as the first image used.
A Music Video for “Ride Into the Sun,” by the Velvet Underground
This music video, for a Velvet Underground classic written by Lou Reed, John Cale, Sterling Morrison and Maureen Tucker, was made today — for a song originally recorded in 1969, the year after I was born. I used Windows Movie Maker to assemble it, and “painted” the preview-pic for the video, using MS-Paint. Other programs I used, for other images in the video, include Geometer’s Sketchpad, MS-Paint (again) and Stella 4d: Polyhedron Navigator. Of all these computer programs, my favorite is Stella 4d, which you may try for free at http://www.software3d.com/Stella.php.
Zonohedra, Zonish Polyhedra, and Another Puzzle
In a recent post, I showed many images of zonohedra, then challenged readers to figure out, from the images, what zonohedra are: polyhedra with only zonogons as faces. Zonogons, I then explained, are polygons with (A) even numbers of edges, and with opposite edges always (B) congruent and (C) parallel. Here is another collection of zonohedra. (Individual images may be enlarged with a click.)
The next set of polyhedra shown, below, are not true zonohedra (as all the ones above are), but merely “zonish polyhedra.” From examination of the pictures above and below, can you figure out the difference between zonohedra and zonish polyhedra?
When you are ready to see the solution to the puzzle, simply scroll down.
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While zonohedra have only zonogons as faces, this restriction is “loosened” for zonish polyhedra. Such solids are formed by zonohedrifying non-zonohedral polyhedra, and letting at least some of the faces of the resulting polyhedra remain non-zonogonal. Zonish polyhedra are called “zonish” because many (usually most) of their faces are zonogons, but not all of them — in each case, some non-zonogonal polygons (such as triangles and/or pentagons, with their odd numbers of edges) do appear. Non-zonogonal polygons are not required to have odd numbers of edges, of course: simply having opposite edges be parallel, but of different lengths, is enough to prevent a polygon (such as a hexagon, octagon, or decagon) from being a zonogon.
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Software credit: I used Stella 4d to make these images. This program may be tried for free at this website.
Some Zonohedra, and a Puzzle
Every zonohedron is a polyhedron, but not all polyhedra are zonohedra. Examples of zonohedra appear below. If you don’t already know what zonohedra are, can you figure out the definition from the examples shown, before reading the definition below the pictures?
Answer below (scroll down a bit):
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Zonohedra are polyhedra with only zonogons as faces. A zonogon is a polygon with an even number of sides, and also with opposite sides congruent and parallel.
Software credit: I used Stella 4d to make these virtual, rotating zonohedra. This program may be tried for free at http://www.software3d.com/Stella.php.
Five Polyhedra Featuring Kites, as Well as Other Polygons, as Faces
Software credit: I used Stella 4d (available at http://www.software3d.com/Stella.php) to create these polyhedral images.
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