A Polyhedron Featuring 180 Kites as Faces, Plus Related Polyhedra

If one starts with the great rhombicosidodecahedron, then makes a compound of it, and its dual, and then forms the convex hull of that compound, this is the result:

180 kites 60&60&60

This polyhedron has 180 faces, all of them kites. What’s more, there are equal numbers — sixty each — of the three different types of kites in this polyhedron.

It also has an interesting dual:

180 kites 60&60&60 the dual

These virtual polyhedral models were created using Stella 4d: Polyhedron Navigator, which you can buy, or try for free, right here. Stella contains a “try to make faces regular” function, and here is what appears if that operation is applied to the dual shown above:

180 kites dual with TTMFR

The dual of this figure is similar to the original polyhedron at the top of this post, featuring 180 kites, again: sixty each, of three different types:

180 kites with TTMFR

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The Reason Why My Current Profile Picture on Facebook Is of Matt Murdock / Daredevil

In September, I noticed that friends of mine started suddenly having superheroes appear as their Facebook profile-pictures. After learning that this was being done to support the efforts to find cures for pediatric cancer, I decided to join them. However, I also have adult friends and relatives who are battling cancer, so I cannot limit this to only pediatric cancer. I’m also not changing my profile picture back, just because it is no longer September. Of course, I chose my favorite comic book character. This is what I now “look” like, on Facebook:


Other animated characters I have seen, as profile pictures of friends of mine who are also participating in this effort, include (in alphabetical order) Batman, Cyclops, She-Hulk, Snoopy, Stan Marsh (of South Park), Susan Storm (the Invisible Woman), and the new, female Thor, as well as others.

To those also participating: thank you. To those not yet participating, I invite you to join us.

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My Third Solution to the Zome Cryptocube Puzzle

The President of the Zometool Corporation, Carlos Neumann, gave me a challenge, not long ago: find a solution to the Zome Cryptocube puzzle which uses only B0s, which I call “tiny blue struts.” For the Cryptocube puzzle, though, these “blue” struts actually appear white. Carlos knows me well, and knows I cannot resist a challenge involving Zome. Here is what I came up with, before the removal of the black cube, which is what the Zome Cryptocube puzzle starts with.


In a “pure” Crypocube solution, the red Zomeballs would also be white — not just the “blue” struts. However, when Carlos issued this challenge, I was at home, with all the white Zomeballs I own located at the school where I teach — so I used red Zomeballs, instead, since I had them at home, and did not wish to wait.

Here’s what this Cryptocube solution looks like, without the black cube’s black struts. You can still “see” the black cube, though, for the black Zomeballs which are the eight corners of the black cube are still present. As is happens, this particular Cryptocube solution has pyritohedral symmetry — better known as the symmetry of a standard volleyball.


While the Cryptocube puzzle is not currently available on the Zome website, http://www.zometool.com, it should be there soon — hopefully, in time for this excellent Zome kit to be bought as a Christmas present. Once a child is old enough so that small parts present no choking hazard, that child is old enough to start playing with Zome — and it is my firm belief that such play stimulates the intellectual growth of both children and adults. As far as a maximum age where Zome is an appropriate Christmas gift, the answer to that is simple: there isn’t one.

Also: while I do openly advertise Zome, I do not get paid to do so. I do this unpaid advertising for one reason: I firmly believe that Zome is a fantastic product, especially for those interested in mathematics, or for those who wish to develop an interest in mathematics — especially geometry. Also, Zome is fun!

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Mark Twain, on Fools and Lightning


Source for image, before I added the words: http://saanichfusionfc.ca/wp-content/uploads/2015/06/Lightning_over_city.jpg

Source for the quote, itself: More Maxims of Mark, By Mark Twain, ed. Merle Johnson (New York: Privately printed, November 1927). See, also, this website.

I am grateful to my friend Tom, for helping me correct this Mark Twain quote, an earlier (and apparently incorrect) version of which I posted here.

Regarding the incorrect grammar in the quote: it’s Mark Twain. One does not correct the grammar of a great writer!

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Why I Support Gay Rights

gay rights

It really is exactly that simple.

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Five Polar Polyhedra

Most polyhedra I post have cuboctahedral, tetrahedral, or icosidodecahedral symmetry, or some pyritohedral or chiral variation of one of these symmetry-types. These, however, are exceptions. I call them “polar polyhedra” because they each have an identifiable “North Pole” and “South Pole,” which are, in three of these five images, at the ends of their axes of rotation.

cub isomorph polar and chiral Compound of enantiomorphic pair

polar and chiral cubic isomorpth

Dual Morph 50.0%

polar polyhedrarhombus-elongated trapezohedron with n = 4

These rotating images were created using Stella 4d, software you may try for yourself, right here.

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Richard Feynman Explains When Rules Are Wrong

richard-feynman quote

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