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# Seemingly Unnamed Apeirohedron

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What is the specific name of this particular apeirohedron? I've had no luck with reverse image searches, all the results simply list it as "apeirohedron" or "infinite skew polygon" (another name for apeirohedron). Does it not have a specific name?

Edited by Anchovyforestbane
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Try this:

"nonuniform honeycomb with truncated octahedra and hexagonal prisms (as ditrigonal trapezoprisms). "

https://en.m.wikipedia.org/wiki/Bitruncated_cubic_honeycomb

I've done a bit more research with this information, and it seems another fitting name would be "hexaprismatic truncated octahedrille".

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I've done a bit more research with this information, and it seems another fitting name would be "hexaprismatic truncated octahedrille".

That more describes a unit cell while missing the fact that what you picture is a honeycomb.

Why do you ask and what do you mean by "graphing"

the structure?

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What is the specific name of this particular apeirohedron?

SpongeBob SquarePants seems right.

I agree with Turtle It’s definitely a cubic honeycomb of some sort. I think Turtle even made a kite once that looked like this.

The vertex configuration of 4.4.4.6 should be a clue for those who know about these things.

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That more describes a unit cell while missing the fact that what you picture is a honeycomb.

Why do you ask and what do you mean by "graphing"

the structure?

Actually, in the information you sent me, it is detailed that the apeirohedral "honeycombs" can be denoted with suffix "drille" (for example, truncated octahedrille, oblate tetrahedrille, etc.) Given this information and more that I've found, I do believe my description to be accurate.

As for the latter question, I mean is there any function I can use to graph the object with an x, y, z axis? I'm not as experienced with graphing, so any pointer in the right direction would be appreciated.

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SpongeBob SquarePants seems right.

I agree with Turtle It’s definitely a cubic honeycomb of some sort. I think Turtle even made a kite once that looked like this.

The vertex configuration of 4.4.4.6 should be a clue for those who know about these things.

Sponge Bob=pineapple=Fibonacci for those who don't know. Lol

Actually, in the information you sent me, it is detailed that the apeirohedral "honeycombs" can be denoted with suffix "drille" (for example, truncated octahedrille, oblate tetrahedrille, etc.) Given this information and more that I've found, I do believe my description to be accurate.

As for the latter question, I mean is there any function I can use to graph the object with an x, y, z axis? I'm not as experienced with graphing, so any pointer in the right direction would be appreciated.

Acknowledge various acceptable names. A hexaprismatic truncated octahedrille by any other name will fly as well. So where did you get the image you posted? Who drew it? What software did they use? Is it public domain? What's your endgame? Just a mental exercise?

I found graphing/drafting different views so challenging for all but the simplest polyhedra, I instead built models and photographed them.

Question: If a hexaprismatic truncated octahedrille space frame has flexible vertices, will it fold?

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Acknowledge various acceptable names. A hexaprismatic truncated octahedrille by any other name will fly as well. So where did you get the image you posted? Who drew it? What software did they use? Is it public domain? What's your endgame? Just a mental exercise?

The image is from here and includes the full name

Apeirohedron truncated octahedra and hexagonal prism 4446

The 4.4.4.6 is the vertev configuration that I mentioned earlier

Question: If a hexaprismatic truncated octahedrille space frame has flexible vertices, will it fold?

Sure, if you sit on it.

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It could be an E8 honeycomb symmetry

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So where did you get the image you posted? Who drew it? What software did they use? Is it public domain? What's your endgame? Just a mental exercise?

I found graphing/drafting different views so challenging for all but the simplest polyhedra, I instead built models and photographed them.

Question: If a hexaprismatic truncated octahedrille space frame has flexible vertices, will it fold?

I found the image in a wikipedia article about apeirohedrons, and immediately found it to be the most fascinating one on the page. I'd like to get into graphing, for little other reason than to be able to graph figures which I find interesting, but I again, I'm not experienced with it and would like to get a good idea of what it's supposed to look like.

As for your question, no, I do not believe it would. Given the very structure of the hexaprisms, the sides would have to intersect each other in order to fold in such a way.

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Fun! So...if there's E8 in this, it's beyond my ken.

Looking at these things and canoodling them is one thing, but building their wireframes and handling them, I have found, is quite another. Wood dowels & tubing vertices. A cube so constructed will fold under its own weight without bending edges or breaking vertices, whereas an icosahedron will not fold. (No Deltahedrons can fold.) I suspect our honeycomb folds, but without a model in hand I don't know it.

If one builds this hexaprismatic truncated octahedrille with rigid vertices & faces as pictured, it will partition space into 2 disjoint, but everywhere adjacent, regions. See Penguin Book of Strange Geometry for an analog on a cubic honeycomb which definitely folds flat.

Stick a fork in me. Live long & prosper.

EDIT: I misremembered the book title. It is The Penguin Dictionary of Curious and Interesting Geometry. Review here > https://www.newscientist.com/article/mg13217925-600-review-the-penguin-dictionary-of-curious-and-interesting-geometry/

Edited by Turtle

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