Spiraling Stars

spiraling stars

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Polyhedral Butterfly

polyhedral butterfly

Created using Stella 4d, available at www.software3d.com/Stella.php.

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Tessellation Featuring Squares, Regular Hexagons and Dodecagons, and Thirty Degree Rhombi

tess

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My Lost Discovery from ~2006: The Triamond Pentagonal Bifrustrum

I just had the strange experience of encountering a polyhedral discovery of mine on the Internet from about eight years ago — one that I had completely forgotten, but had shared with others, who posted it online, and were kind enough to give me credit for the discovery. It’s the fifth polyhedron shown on this page: http://www.interocitors.com/polyhedra/Triamonds/ — and is shown with a .stel file, so I was able to use polyhedral-manipulation-and-imaging software, Stella 4d (available at www.software3d.com/Stella.php) to make a rotating image of it:

Triamond Pentagonal Bifrustum

It’s a member of a class of polyhedra which have, as faces, only regular polygons with unit size, as well as “triamonds.” Triamonds are 1:1:1:2 trapezoids composed of three coplanar, equilateral triangles.

Back in 2006 or earlier, my guess is that I simply made a physical model out of card-stock paper and tape, and then took a photograph of it — something I haven’t done in a very long time, now that making moving pictures of virtual models has become so easy. Another possibility is that I used Zome, an excellent ball-and-stick modelling system available at www.zometool.com. Zome, like Stella, I still use — and I will be using Zome often with students, in class, when school starts next month. Fortunately, I have a lot of Zome!

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Infinite Families of Polyhedra

Many polyhedra appear in finite sets. The most well-known example of such a set is the five Platonic solids. Many know them from role-playing games.

BluePlatonicDice

Other finite sets include the thirteen Archimedean, four Kepler-Poinsot, thirteen Catalan, and 92 Johnson solids, and the eight convex deltahedra, among others. There is even a finite set, the near-misses, with an unknown number of members, due to the “fuzziness” of its definition. The symmetrohedra is another set with “fuzzy” criteria, but there are still only so many symmetrohedra to be found. We simply haven’t found them all yet, or, if we have, we don’t know that we have, but it would not be reasonable to claim that infinitely many await discovery.

However, not all groups of polyhedra are finite. Some polyhedra appear, instead, in infinite families. What is needed to generate such infinite families (at the cost of some forms of symmetry, compared to, say, Platonic or Archimedean solids) is the use of bases — special polyhedral faces which play a stronger role in the determination of that polyhedron’s shape than do the other faces. For the familar prism, there are two bases.

5 Penta Prism

In a pentagonal prism, the bases are pentagons, as seen above. For a pentagonal prism, n = 5, for n is simply the number of sides of the base. The smallest value of n which is possible, 3, yields a triangular prism. There’s no upper limit for n, either. Here’s a regular hexacontagonal prism, where each base has sixty sides.

60- Prism

Obviously, n can be increased without limit, although for very large values of n, the prism will be hard to distinguish from a cylinder.

Another infinite family may be found by taking the dual of each prism. This is the dual of the pentagonal prism:

5 Penta Prism dual

Taking a dual of a prism produces a dipyramid, with its n-gonal base hidden between the puramids, but with no guarantee that the triangular faces will be regular — and in this case, they are not. It is possible for a pentagonal dipyramid to have only regular faces . . .

5-dipyramid J13

. . . but regular faces will not work if n = 6 (because the dipyramid collapses to zero height), or indeed any number other than 3, 4, or 5. Therefore, dipyramids will not be an infinite family unless non-equilateral triangles are permitted as faces.

Half of a dipyramid, of course, is simply a pyramid, with the single base now visible:

5- Pyramid

Like the dipyramids, and for the same reason, there are only three pyramids (n = 3, 4, or 5) which can have all faces regular. An infinite family of right, regular pyramids do exist, though, if isosceles triangles are permitted as lateral faces.

While pyramids and dipyramids have only one base each, the already-described prisms have two. Prisms also can maintain regularity of all faces, no matter how large n becomes, unlike pyramids and dipyramids. Moreover, prisms can be transformed into another infinite family of regular-faced polyhedra by rotating one base by one half of one-nth of a rotation, relative to the other base, and replacing the n square lateral faces with 2n equilateral triangles. These polyhedra are called antiprisms. The pentagonal antiprism looks like this:

5 Penta Antiprism

Antiprisms with all faces regular do not have the “only 3, 4, and 5″ limitation that affected the pyramids and dipyramids. For example, here is one with all faces regular, and dodecagonal bases:

12- Antiprism

One more infinite family of polyhedra may be found, using the antiprisms: their duals. The dual of the pentagonal antiprism looks like this:

5 Penta Antiprism dual

There is no regularity of faces for any members of this family with n > 3, for their faces are kites. (When n = 3, the kites become squares, and the polyhedron formed is simply a cube.) Unfortunately, “kiteohedron” looks ridiculous, and sounds worse, so efforts have been made to find better names for these polyhedra. “Deltohedra” has been used, but is too easily confused with the deltahedra, which are quite different. The best names yet invented for this infinite family of kite-faced polyhedra are, in my opinion, the antidipyramids, and the trapezohedra.

(Note:  the rotating polyhedral images above were generated using Stella 4d:  Polyhedron Navigator, available at http://www.software3d.com/Stella.php.)

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Splat!

splat

This is based on the well-known tessellation of the plane with hexagons. Each side of each hexagon has been replaced by a set of three semicircles.

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The Rhombic Dodecahedron and Its First Stellation, Decorated in Three Colors

Rhombic Dodeca

The image on the faces of these polyhedra appeared, in black and white form, in the last post here. I used Geometer’s Sketchpad and MS-Paint to make it. It took a third program, Stella 4d, to project the image onto each face of a rhombic dodecahedron (as seen above), as well as the rhombic dodecahedron’s first stellation (below), and then generate these rotating .gif images. You can try Stella for free, as a trial download, at www.software3d.com/Stella.php.

First Stellation Rhombic Dodeca

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