494 Circles, Each, Adorning Two Great Rhombcuboctahedra, with Different (Apparent) Levels of Anxiety


Trunc Cubocta

The design on each face of these great rhombcuboctahedra is made from 19 circles, and was created using both Geometer’s Sketchpad and MS-Paint. I then used a third program, Stella 4d (available here), to project this image on each of a great rhombcuboctahedron’s 26 faces, creating the image above.

If you watch carefully, you should notice an odd “jumping” effect on the red, octagonal faces in the polyhedron above, almost as if this polyhedron is suffering from an anxiety disorder, but trying to conceal it. Since I like that effect, I’m leaving it in the picture above, and then creating a new image, below, with no “jumpiness.” Bragging rights go to the first person who, in a comment to this post, figures out how I eliminated this anxiety-mimicking effect, and what caused it in the first place. 

Trunc Cubocta

Your first hint is that no anti-anxiety medications were used. After all, these polyhedra do not have prescriptions for anything. How does one “calm down” an “anxious” great rhombcuboctahedron, then?

On a related note, it is amazing, to me, that simply writing about anxiety serves the purpose of reducing my own anxiety-levels. It is an effect I’ve noticed before, so I call it “therapeutic writing.” That helped me, as it has helped me before. (It is, of course, no substitute for getting therapy from a licensed therapist, and following that therapist.) However, therapeutic writing can’t explain how this “anxious polyhedron” was helped, for polyhedra can’t write.

For a second hint, see below.



[Scroll down….]



Second hint: the second image uses approximately twice as much memory-storage space as the first image used.

Some Zonohedra, and a Puzzle

Every zonohedron is a polyhedron, but not all polyhedra are zonohedra. Examples of zonohedra appear below. If you don’t already know what zonohedra are, can you figure out the definition from the examples shown, before reading the definition below the pictures?

Answer below (scroll down a bit):








Zonohedra are polyhedra with only zonogons as faces. A zonogon is a polygon with an even number of sides, and also with opposite sides congruent and parallel.

Software credit: I used Stella 4d to make these virtual, rotating zonohedra. This program may be tried for free at http://www.software3d.com/Stella.php.

Explaining China, Part I: The Scope of This Series, Which Includes the PRC, the ROC, the Han, and Greater China

China and environs

I’m bringing a new topic to my blog. I’m going to attempt to explain things about China, the largest nation in which the Han (that’s the way to write, in English, the Chinese name for the Chinese people, as an ethnic group) form the majority, as well as the largest nation on Earth, by population. The map above comes from this website. If you’re wondering why, in the map above, Taiwan is the same color as the People’s Republic of China, this series of blog-posts is definitely for you. In a future post, I will deal with the historical reasons for the China/Taiwan puzzle, and the current state of that interesting situation. (“May you live in interesting times” is not a nice thing to say directly to any of the Han, by the way, no matter where they live. It is considered by many people to be part of an ancient Chinese curse, although the veracity of this claim is disputed — a topic for another post, later in this series.)

If you find China, Taiwan, puzzles in general, mysteries which are not fictional, history, current events, and/or the Han to be interesting topics, then this irregularly-published series of blog-posts is for you. If you aren’t interested in any of those topics, my assumption is that you wouldn’t have read this far, anyway. To those who miss the other topics about which I blog, don’t worry: posts in this series will not be the only topic I blog about, by any means, for the fact that I am interested in many things, and blog on many topics, is not going to change.

The People’s Republic of China is also known as mainland China, Red China, the PRC, Communist China, or simply “China.” The government of the PRC is often referred to simply as “Beijing,” the city which is the capital of the PRC. Taiwan, by contrast, is officially known as the Republic of China, or the ROC, or even, by some people, “Taiwan, China” (a term I tend not to use). The ROC’s government can be referred to as “Taipei,” the ROC’s capital, to distinguish it from the government in Beijing. My preferred way to refer to the nation-state which is actually under the control of the Beijing government is to call it the PRC, and I use ROC, often, to refer to the nation-state actually under the control of the Taipei government, which most people call Taiwan, a term I also use. When I only write “China,” I mean the PRC. I also use the term “Greater China,” which is explained below.

The Han are in the majority in both the PRC and the ROC, and these two regions are collectively known as “Greater China,” which sounds like, and in some ways actually is, one nation with two governments, since both governments claim to be the only legitimate government of the nation which is all of Greater China (and, yes, that is confusing, along with “China Proper” on the map above). All of these topics: the nations, governments, regions, and people, are mysteries for most people on Earth — and topics for future posts in this series.

I am not of the Han. I do not speak, read, nor write any variety of the Chinese language. Also, I have yet to visit any part of Greater China. By contrast, I am known as a teacher of both science and mathematics, as someone who does “math problems for fun” (as my blog’s heading-cartoon, which I did not write, puts it), as well as a blogger on many topics that have previously had little to do with China, until this post, from yesterday, which analyzed current events worldwide, starting with recent developments in China. I do not want anyone to think I just started studying China yesterday, for that would not be correct. I do feel that I owe anyone who has read this far an explanation for exactly one thing: why should anyone care what I have to say on these subjects? I will explain that in Part II of this ongoing series . . . and tackling the PRC/ROC puzzle will be coming later, as will other topics.

A Hollow Faceting of the Rhombicosidodecahedron, and Its Hollow Dual

The images above all show a particular faceting of the rhombicosidodecahedron which, to my surprise, is hollow. It has the vertices of a rhombicosidodecahedron, but two different face-types, as seen in the smaller pictures: yellow hexagons, and red isosceles trapezoids. (To enlarge any image in this post, simply click on it.)

The dual of this polyhedron is even more obviously hollow, as seen below. Its faces, as seen in the still picture, are crossed hexagons — with edges which cross three times per hexagon, no less.

The software I used to make these polyhedra, Stella 4d, will return an error message if the user attempts to make a polyhedron which is not mathematically valid. When I’ve made things that look (superficially) like this before, I used “hide selected faces” to produce hollow geometrical figures which were not valid polyhedra, but that isn’t what happened here (I hid nothing), so this has me confused. Stella 4d (software you can buy, or try for free, here) apparently considers these valid polyhedra, but I am at a loss to explain such familiar concepts as volume for such unusual polyhedra, or how such things could even exist — yet here they are. Clarifying comments would be most appreciated.

A Logic Problem Involving Marvel Super-Heroes


Iron Man, Daredevil, Spider-Man, Captain America, and Wolverine each have a favorite food, a favorite beverage, own one pet, and have a single hobby. Based on the clues which follow, find out these things:

  • Which hero’s favorite food is (A) pizza, (B) green eggs and ham, (C) apple pie, (D) Chinese take-out, and (E) caviar?
  • Which hero’s favorite beverage is (A) beer, (B) vodka, (C) Coca-Cola, (D) water, and (E) chocolate milkshakes?
  • Which hero owns (A) a black cat, (B) a porcupine, (C) a robot dog, (D) an iguana, and (E) a real dog?
  • Which hero’s hobby is (A) coin collecting, (B) stamp collecting, (C) collecting comic books, (D) collecting seashells, and (E) collecting rocks?

Here are the clues. Answers will be revealed in the comments, but only after someone solves the puzzle (to avoid spoiling anyone’s fun).

  1. Wolverine drinks beer.
  2. Daredevil is blind. The other four heroes can all see.
  3. Spider-Man eats pizza.
  4. Wolverine has a mutant healing factor that allows him to rapidly heal from injuries.
  5. Iron Man is the only one of these five heroes who wears a suit of armor.
  6. The hero whose favorite food is apple pie always eats it with his favorite drink, Coca-Cola.
  7. Iron Man drinks vodka.
  8. All of the heroes who can see refuse to eat green eggs and ham.
  9. Of these five heroes, no one without either a mutant healing factor or a suit of armor would be dumb enough to keep a porcupine as a pet.
  10. Iron Man, an accomplished inventor, refuses to own a pet which he did not build himself.
  11. The hero who eats apple pie doesn’t like chocolate, nor chocolate-flavored anything.
  12. Iron Man has more money than all the other heroes combined.
  13. The hero whose favorite food is pizza does not own a dog.
  14. The seashell-collector is blind.
  15. The owner of a real dog also collects stamps. 
  16. The porcupine-owner doesn’t like apple pie.
  17. Spider-Man likes the black cat, but has to visit the cat’s owner in order to see her.
  18. The richest hero eats caviar.
  19. The coin collector doesn’t like pizza, nor porcupines.
  20. The comic-book collector hates drinking water. He also doesn’t like milkshakes of any kind.
  21. The owner of the black cat is lactose-intolerant, and, for this reason, doesn’t drink milkshakes.

The first person to leave the solutions in the comments wins bragging rights.

[Source of image: http://www.hdwallpaperpc.com/show-wallpaper/Spider_man_DareDevil_Iron_Man_Captain_America_Wolverine_Black_43340.html].

87 Rotating Non-Convex, Non-Chiral Polyhedral Images Featuring Icosidodecahedral Symmetry, Plus Four Which Snuck In with Cuboctahedral Symmetry — Can You Find All the Intruders?

To see larger versions of any of these, simply click on the images.

24 to this point….

That’s 40 so far…

Now the count is at four dozen.

That was 26 more, so there are 48 + 26 = 74 so far.

Now the count is up to 83.

So there were 91 of these stored on my hard drive, from all my “hard work” playing with polyhedra using Stella 4d: Polyhedron Navigator. (It will be good for my computer to get all that hard drive space back!) If you’d like to try playing with the same program — for free — just try the free download at http://www.software3d.com/Stella.php.

A Euclidian Construction of the Regular Pentadecagon (from 2011)

pentadecagon construction 2011

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.

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!

Geometry Problem Involving Two Circles (See Comments for Solution)

This is a puzzle I made up not long ago. After trying to solve it for a bit (no success yet, but I haven’t given up), I decided to share the fun.

A small circle of radius r is centered on a large circle of radius R. It is a given that 0 < r < R. In terms of r and R, what fraction of the smaller circle’s circumference lies outside the larger circle?

two circles

I am 90% certain there is an extremely simple way to do this, using only things I already know. It’s frustrating that the answer isn’t simply leaping out of the computer screen, at me. For simple math problems, that’s what usually happens . . . so either this is merely deceptively simple, or I am missing something.

Silly U.S. Map Puzzle #5

What do the colors on this map mean?


If you wish to check your answer, or just what to know what the solution is, just scroll down.








And keep on scrollin’. . . .






Of the other 49 states in the USA, how many are adjacent to this one? The answer to this question determines the color of each state.

One point of clarification: if it takes a lengthy trip by boat or ship to get there, I didn’t count it as an adjacent state . . . so, for example, Minnesota and Michigan didn’t make each other’s lists. Simply going over a bridge isn’t enough for this sort of separation, though, which is why Arkansas and Tennessee did make each others’ lists of adjacent states. Had I interpreted water borders differently, this map would have some differences.

Another way this map could be altered would be to count states that meet others only at a single point, rather than a border with non-zero length. This would change the colors of the “four corners” states of Arizona, Utah, New Mexico, and Colorado, but would have no effect on the other 46 states.