## Three Polyhedra Which Feature Regular Pentadecagons

### Image

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.

## 482-Faced Polyhedron Featuring 32 Regular Pentadecagons

### Image

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.

## 422-Faced Polyhedron Featuring 32 Regular Pentadecagons

### Image

To create this, I augmented the red pentadecagons in the polyhedron found on the last post with antiprisms, augmented the slightly larger green pentadecagons with prisms, and then took the convex hull of the result. I like this one better than the last one, on purely aesthetic grounds.

Software credit: I used Stella 4d to make this, which is available at http://www.software3d.com/Stella.php.

## 542-Faced Polyhedron Featuring 32 Regular Pentadecagons

### Image

I’ve been obsessing about symmetrical polyhedra which include regular pentadecagons all day, and this is the latest result.

Software credit: I used Stella 4d to make this, which is available at http://www.software3d.com/Stella.php.

## A 102-Faced Polyhedron Featuring Regular Pentadecagons

### Image

This is a stellation of a slightly-modified version of the second polyhedron shown in the last post here. It includes twelve regular pentadecagons, thirty rectangles, and sixty irregular pentagons, grouped in twenty sets of three pentagons each.

Software credit: please visit http://www.software3d.com/Stella.php if you’d like to try a free trial download of Stella 4d, the program I use to make these virtual models.

## Two Symmetrohedra Featuring Regular Pentadecagons

### Image

I’ve posted “bowtie” symmetrohedra on this blog, before, which I thought I had discovered before anyone else — only to find, later, that other researchers had found the exact same polyhedra first. Those posts have now been edited to include credit to the original discoverers. With polyhedra, finding something interesting, for the first time ever, is extremely difficult. This time, though, I think I have succeeded — by starting with the idea of using regular pentadecagons as faces.

Software credit: Stella 4d was the tool I used to create this virtual model. You can try a free trial download of this program here: http://www.software3d.com/Stella.php.

*** *** ***

Update:  once again, I have been beaten to the punch! A bit of googling revealed that Craig Kaplan and George Hart found this particular symmetrohedron before I did, and you can see it among the many diagrams in this paper: http://archive.bridgesmathart.org/2001/bridges2001-21.pdf.

You’ll also find, in that same paper, a version of this second pentadecagon-based symmetrohedron:

There is a minor difference, though, between the Kaplan-Hart version of this second symmetrohedron, and mine, and it involves the thirty blue faces. I adjusted the distance between the pentadecagons and the polyhedron’s center, repeatedly, until I got these blue faces very close to being perfect squares. They’re actually rectangles, but just barely; the difference in length between the longer and shorter edges of these near-squares is less than 1%. I have verified that, with more work, it would be possible to make these blue faces into true squares, while also keeping the pentadecagons and triangles regular. I may actually do this, someday, but not today. Simply constructing the two symmetrohedra shown in this post took at least two hours, and, right now, I’m simply too tired to continue!