Two Symmetrohedra Featuring Regular Pentadecagons

92 faces including 20 reg hexagons and 12 regular pentadecagons

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.

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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:

15_2

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!

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