Truncated octahedra are among the special polyhedra which can fill space without leaving any gaps. There are others, as well. This image was created using Stella 4d, software you may try, for yourself, right here. There is a free “try it before you buy it” download available.
This tessellation appears, at first, to be regular (or “Archimedean,” by analogy with the Archimedean solids), for all of the polygons included are regular. However, it is not vertex-transitive, which keeps it from qualifying as a regular or Archimedean tessellation.
This is the truncated cube, one of the thirteen Archimedean solids.
If the truncation-planes are shifted, and increased in number, in just the right way, this variation is produced. Its purple faces are regular dodecagons, and the orange faces are kites — two dozen, in eight sets of three.
Applying yet another truncation, of a specific type, produces the next polyhedron. Here, the regular dodecagons are blue, and the red triangles are equilateral. The yellow triangles are isosceles, with a vertex angle of ~41.4 degrees.
All three of these images were produced using Stella 4d, available at this website.
The yellow faces of this polyhedron are parallelograms, while the red ones are trapezoids. To demonstrate its chirality, here is the compound of it, and its own mirror-image.
Both of these “virtual polyhedra” were made using Stella 4d: Polyhedron Navigator, a program available at this website. It has a free trial download available.
Created using Stella 4d: Polyhedron Navigator; see this website to try it for yourself!
Math jokes are almost universally awful — or, at least, it seems that way to me, since I spend a lot of time around ninth graders. Hearing “Gee, I’m a tree” or “Pi are square? No, pies are round, and cake are square!” will generally elicit a groan from me, and each new cadre of students seems to think they invented these fossilized puns. An even worse “joke” is the giggling one should expect from, say, 7th graders, if one squares the number thirteen in their presence.
I do know exactly one good math joke, though. I didn’t hear it from a student. If you’re curious, read on. Only the embellishments are original; I didn’t make up the joke, itself, though, nor do I know who did.
My source for the image below is this fellow WordPress blogger’s photography blog.
So a physicist, chemist, and a lawyer enter a balloon race together. Theirs is the last balloon to leave, because the lawyer had been in court, arrived late, and caused a short delay in departure. The consequences of this were serious, though, for a sudden cross-wind blew them off course, right after takeoff. Soon, they couldn’t even see any of the other balloons in the race, and none of them recognized any landmarks in the landscape below.
Soon, they had no idea where they were, and started getting worried about making it to their next classes on time — or back to court, in the case of the lawyer. The chemist was particularly worried. “What are we going to do?” asked the chemist.
The physicist replies, “I have an idea!” He cups his hands, leans out, and yells, as loud as he can, “Hello! Where are we?”
The balloon flies on for at least two long, anxious minutes as the trio waits, silently, for an answer. Eventually, they hear, from a great distance, a voice. “Hello! You’re lost!”
The physicist looks at the other two, and says, “That, my friend, was a mathematician.”
“How,” asked the lawyer, “could you possibly know that?”
“Three things,” replied the physicist. He held up one finger. “First, it took him a long time to answer.”
“Second,” he continued, holding up two fingers, “the answer, when it finally came, was absolutely correct.”
A third finger joined the first two. “Third, the answer, when it finally came, was completely useless!”
This polyhedron has been described here as a “bowtie cube.” It is possible to augment its six dodecagonal faces with additional bowtie cubes. Also, the bowtie cube’s hexagonal faces may be augmented by truncated octahedra.
These two polyhedra “tessellate” space, together which square pyramidal bifrustrums, meeting in pairs, which fill the blue-and-green “holes” seen above. This last image shows more of the “honeycomb” produced after yet more of these same polyhedra have been added.
This pattern may be expanded into space without limit. I discovered it while playing with Stella 4d, software you may try for free at this website.
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