Medium: electrons and photons, arranged carefully using Stella 4d: Polyhedron Navigator, software I bought at http://www.software3d.com/Stella.php. There is a free trial download available at the same website.
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Twenty Rotating Triskelions, Made of Kites
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Created using Stella 4d, software available at http://www.software3d.com/Stella.php.
Stellated Polyhedron Featuring Self-Intersecting Regular Decagons
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I created this using Stella 4d: Polyhedron Navigator, a program you can find athttp://www.software3d.com/Stella.php.
Two Chiral Polyhedra
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To make this, I started with the dual of the great rhombicosidodecahedron, a polyhedron known as the dysdyakis triacontahedron. I then augmented half of its faces with tall prisms (thereby creating the chirality in this polyhedron), and took the convex hull of the result. The sixty red triangles are the tops of the augmentation-prisms.
A stellation of the above polyhedron, and a color-change, produced this result, also chiral. It may be enlarged with a click.
These polyhedra were created using Stella 4d, a program which you may buy — or try for free, as a trial download — at http://www.software3d.com/Stella.php.
A Polyhedron Featuring Sixty Irregular, Convex Hexagons and Thirty Rhombi
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I created this using Stella 4d: Polyhedron Navigator, a program you can find at http://www.software3d.com/Stella.php.
A Polyhedron Featuring Sixty Octagons and Sixty Triangles
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If someone had asked me if it were possible to form a symmetric polyhedra out of irregular triangles and octagons, using exactly sixty of one type each, I would have guessed that it were not possible. Why does it work here? Part of the reason is that each triangle borders three octagons, and each octagon borders three triangles — a necessary, but not sufficient, condition. This is a partial truncation of an isomorph of the pentagonal hexacontahedron, the dual of the snub dodecahedron. As such, no surprise — it’s chiral.
This was made while stumbling about in the wilderness of the infinite number of possible polyhedra using Stella 4d: Polyhedron Navigator. You can get it here: http://www.software3d.com/Stella.php.
The Convex Hull of a Prism-Augmented Icosidodecahedron As a (Possibly) New Near-Miss Candidate
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To make this polyhedron using Stella 4d: Polyhedron Navigator (a program which is available at this website), I started with an icosidodecahedron, augmented all faces with prisms of height 1.6 times greater than their bases’ edge length, and then took the convex hull of the result. I’m proposing it as a candidate for the loosely defined group of polyhedra called near-misses to the 92 Johnson solids: convex polyhedra which are almost, but not quite, Johnson solids, due to slight irregularity in some of their faces.
In this case, the pentagons and green triangles are regular, and have the same edge length. The blue triangles, however, are isosceles, with vertex angles of ~67.6687 degrees. The yellow almost-squares are actually rectangles, with edges next to blue triangles which are ~2.536% longer than the edges next to pentagons or green triangles.
I stumbled upon this design earlier today, while simply exploring polyhedra more-or-less randomly, using Stella. Below is the prototype I found at that time, which I merely made a .gif of, but did not perform measurements on.
In this prototype, the most significant difference I can detect is in the yellow faces, which are trapezoids, rather than rectangles, since the pentagon edge-length is slightly longer than that of the green triangles.
Stella has a “try to make faces regular” function built-in to try to help improve upon polyhedra such as these, but here’s what happens when that function is used on the first polyhedron shown above:
Behold! It worked — all of the faces are perfectly regular. However, that caused another problem to appear, and you can see it most easily by looking at the blue triangle-pairs: this polyhedron is slightly non-convex. It’s also easily described as a truncated dodecahedron, with each of the twelve decagonal faces augmented by a pentagonal rotunda.
I’ll show this to some other people who are polyhedron-experts, and will update this post with what I find after I’ve talked to them. My questions for them, as usual in such situations, are two in number:
1. Has this polyhedron been found before?
2. Is it close enough to regularity to qualify for “near-miss” status?
If it hasn’t been found before, but is judged unworthy of “near-miss” status, it will at least join the newly-described group I call “near near-misses” — polyhedra which don’t quite qualify for near-miss status, by visual inspection. Obviously, this new group’s definition is even more “fuzzy” than that of the near-misses, but there is a need for such a category, nonetheless.
[Update: Robert Webb, who wrote Stella 4d (and is not the blogger here, despite our sharing a first name), has seen this before, so it isn’t an original discovery of mine. He doesn’t accept it as a near-miss on the grounds that it naturally “wants” to be non-convex, as seen in the last of the three images in this post, and I agree with his reasoning. I’m therefore considering this to be a “near-near-miss.”]
Twelve Rotating Images of Barred Spiral Galaxy NGC1300
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After using Google to find the image of this galaxy, I used software called Stella 4d (available at http://www.software3d.com/Stella.php) to project it onto the twelve pentagonal faces of an icosidodecahedron, and then hid the triangular faces, as well as the vertices and edges — and then set the galaxies to rotate on the faces, as well as around the axis of the polyhedron.
Another View of the Regular Henkaipentacontagon
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This is simply the inverted-color version of the second image in the last post on this blog, where I explain how to construct this 51-sided polygon using compass and straightedge, starting with the heptadecagon, which has 17 sides.
Constructing the Heptadecagon
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I have just completed my first construction of the regular heptadecagon — a construction that even the ancient Greeks were never able to figure out. They did figure out how to construct a regular pentadecagon (by combining the constructions for the regular pentagon and triangle), and I once replicated that discovery, meaning that I figured it out independently.
The regular heptadecagon construction, however, I did not figure out independently. I used instructions found here (http://www.mathpages.com/home/kmath487.htm), which built on the work of Carl Friedrich Gauss, who, in 1796, at the age of 19, became the first person in history to determine that such a construction is possible with the traditional Euclidean tools.
A word of warning, if you attempt to replicate this construction yourself: points M and G are merely close together, but are not in the same place. Point M is the center of the circle which passes through points D and V17, while point G is one of the two points of intersection of (1) the line passing through points O and V17, and (2) the circle centered at C, and passing through E.
Gauss (and other mathematicians, building on his work) also showed, later, that constructions are possible for regular polygons with 257 sides, as well as 65,537 sides. I might, someday, replicate the construction of the regular polygon with 257 sides.
A man named Johann Gustav Hermes once spent ten years completing a 200-page manuscript showing how to construct the regular polygon with 65,537 sides, and I believe he actually performed the construction, as well. I will not be constructing this polygon — ever. I will, however, figure out a proper name for it. Let’s see . . . it’s the heptakaitriacontakaipentacosioikaipentachilikaihexamyriagon. Try saying that five times in a row, quickly!
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