The First of the Smith Polyhedra: An Example of Mathematical Collaboration

Recently, a reader of this blog contacted me about a polyhedron he wished to model. His name is Dave Smith, and he had already done much of the work involved, but needed help finishing off his project. Here’s the picture he e-mailed me.


The visible faces are pentagons — four of them. The invisible faces are isosceles trapezoids. I e-mailed Smith, and told him the truth: I didn’t have a clue how to make this in Stella 4d, the program I use to make the rotating polyhedra on this blog (including the one below). I also told him I wasn’t giving up — merely enlisting help with his puzzle.

And, with that, I went to Facebook, posting the image above, along with an explanation, and request for help finishing it. This may not be what most people think of when they consider Facebook, but I have deliberately sought out experts there in many fields, including geometry, to make the social-networking site useful in unusual ways, such as getting help with geometrical puzzles I can’t solve alone. Three geometricians with skills which exceed mine (Wendy Krieger, Tom Ruen, and Robert Webb, who wrote Stella 4d) began discussing the figure. One of them, Tom Ruen, sent me .stel files (That’s what Stella 4d uses) for multiple figures, getting closer each time. With the last such virtual model Tom sent me, I was able to “tweak” it to get the pentagons regular.

Smith's puzzle

The figure has two edge lengths, the shorter appearing only twice, as the shared, shorter base of isosceles trapezoids — and these two lengths are in the golden ratio. Smith’s octahedron, as it could be called (since it has eight faces), also has an interesting form of symmetry — it reminds me of pyritohedral symmetry, but .

And it only took five people to figure all of this out!

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When the Westboro Baptist Church Protests Leonard Nimoy’s Funeral, What Is the Appropriate Phaser Setting?


I’ve been trying to determine the appropriate phaser setting for dealing with these people, and have decided to go with “heavy stun.”

Heavy stun is kinder than the WBC adults deserve, but some of those WBC people are infants and children, and they have a chance of throwing off their brainwashing as they grow up. I would not deny them that chance.

[Photo credit: This website is where I found this image. It’s a story about the WBC announcing their intent to protest Leonard Nimoy’s funeral.]

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Campaign Flyer, with Corrected Dates, for the PACT Election


Please note the new dates for the PACT election. The change was caused by the weather.

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A Polyhedron with 542 Faces


I used Stella 4d to make this.  This program’s name, in the last sentence, is a link; if you follow it, you’ll be taken to a site where you can give it a try for free.

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5, 10, and 15 (from 2012)

5 10 and 15 from 2012

I recently found a bunch of my “lost” geometrical art which never found its way to this blog before, and here’s the latest piece of it. Created in 2012, it has a central pentadecagon, five orange decagons partially hidden behind other polygons, and many pentagons, all of them regular.

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Nine (2015) / Nine (2013)

First, the newer version I just made:

nine 2015

Next, the 2013 version, which I recently found, along with a bunch of other previously-lost stuff from around then. The two are simply color-inversions of each other, according to the rules for color-inversion used by MS-Paint.

nine 2013

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Maria Dee Austin for President of PACT

Dee flyer

Some who teach in my school district know my wife by her former name, Maria Dee Giddens. She has taught here longer than I have, and is the mother of two of our district’s students. Our professional organization and union for teachers is having an election March 2-6, and I am proud to publicly endorse my wife’s campaign to serve the members of PACT as our president. Members will need to see the PACT Rep. in their school to cast their vote.

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