This chiral polyhedral compound was generated from a partial faceting of the polyhedron shown in the last post here, using Stella 4d‘s faceting function, plus its “try to make faces regular” operation afterwards. Making the six-prism compound in the first place was suggested by Tony Hartley, on Facebook, where I posted a link to that last post in a mathematical group for discussion. Thanks, Tony!
If you’d like to try Stella for yourself, the site to visit for a free trial download is http://www.software3d.com/Stella.php.
The blue polygons are regular pentadecagons, and the yellow polygons are irregular dodecagons. There are also equilateral hexagons (orange), red squares, and black concave pentagons.
This polyhedron has 92 faces: twelve regular pentadecagons, twenty equilateral triangles, and sixty isosceles triangles, each with a vertex angle of ~42.5°. Its first stellation appears below.
Both models were created using Stella 4d, software you can try for yourself at http://www.software3d.com/Stella.php.
Because I did not start this blog until mid-2012, I sometimes encounter things I made before then, but have not yet posted here. I made this image in 2011, after reading that the ancient Greeks discovered how to combine the Euclidean constructions of the regular pentagon and the equilateral triangle, in order to create a construction for the regular pentadecagon. Having read this, I felt compelled to try this for myself, without researching further how the Greeks did it — and, as evidenced by the image above, I successfully figured it out, using the Euclidean tools embedded in a computer program I often use, Geometer’s Sketchpad.
What I did not do at that time was show the pentagon’s sides (so it is rather hard to find in the image above, but its vertices are there), nor record step-by-step instructions for the construction. For those who wish to try this themselves, I do have some advice: construct the pentagon before you construct the triangle, and not the other way around, and you are likely to find this puzzle easier to solve than it would be, if this polygon-order I recommend were reversed.
I also have two more hints to offer: 108º – 60º = 48º, and half of 48º is 24º. Noticing this was, as I recall, the key to cracking the puzzle.
Other polygons included in this tessellation include several types of rhombi, as well as triconcave octadecagons. The pattern is chiral, but the chirality is subtle. (Hint: look near the pentagons.)
In the last post here, there were two polyhedra shown, and the second one included faces with nine sides (enneagons, also known as nonagons), as well as fifteen sides (pentadecagons), but those faces were not regular.
The program I use to manipulate polyhedra, Stella 4d (available at http://www.software3d.com/Stella.php), has a “try to make faces regular” function included. When I applied it to that last polyhedron, in the post before this one, Stella was able to make the twenty enneagons and twelve pentadecagons regular. The quadrilaterals are still irregular, but only because squares simply won’t work to close the gaps of a polyhedron containing twenty regular enneagons and twelve regular pentadecagons. These quadrilaterals are grouped into thirty panels of four each, so there are (4)(30) = 120 of them. Added to the twelve pentadecagons and twenty enneagons, this gives a total of 152 faces for this polyhedron.
I haven’t encountered many polyhedra which feature regular pentadecagons, and geometry textbooks generally don’t even use that word, calling them “15-gons,” instead. The pentadecagon happens to be one of my favorite polygons, though, and has been ever since I independently figured out, a few years back, how to duplicate the ancient Greeks’ accomplishment of combining the Euclidean constructions for the regular pentagon and equilateral triangle, in order to construct a regular pentadecagon.
The one above also includes regular decagons as faces — but I had to let the pentadecagons intersect each other to get that to work.
This third polyhedron resembles a truncated icosahedron, but with pentadecagons replacing that solid’s twenty hexagons. The pentagons are still in place, with two types of trapezoid and some very thin rectangles needed to fill the gaps.
These images were all created using Stella 4d, software you may try or buy at http://www.software3d.com/Stella.php.
I’ve been obsessing about symmetrical polyhedra which include regular pentadecagons all day, and this is the latest result. I feel like I’m getting closer to what I’m looking for . . . but whatever it is I’m looking for seems to be receding from me, at the same time. At least for me, this is a familiar feeling when researching polyhedra, trying to find interesting shapes never seen before by anyone. It’s a high bar to try to reach, but I enjoy a challenge, if it involves something I find of interest. Clearly, this qualifies.
Software credit: I used Stella 4d to make this, which is available at http://www.software3d.com/Stella.php.