A Hollow Octahedron Made of Rhombic Dodecahedra, with Variations

hOLLOW oCTAHEDRON mADE OF Rhombic Dodeca

The original polyhedral cluster I built using Stella 4d (available here) is above. Below is its 29th stellation.

hOLLOW oCTAHEDRON mADE OF Rhombic Dodeca 29th stellation

And the 30th stellation, as well:

hOLLOW oCTAHEDRON mADE OF Rhombic Dodeca 30th stellation

This is the original polyhedral cluster’s dual:

hOLLOW oCTAHEDRON mADE OF Rhombic Dodeca dual

The next image is a variant of the original polyhedral cluster, rendered with only its edges, but not faces or vertices, visible. I wish I could remember exactly how I made this variant, but I simply cannot recall the exact methods I used.

hollow octahedron variant

This is the dual of the polyhedron shown immediately above, rendered in the same manner:

hollow object made of cubes -- dual of hollow octahedron variantThis is a compound of the two dual polyhedra right before this sentence.

hollow octahedron variant compound of it and dual

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Two Compounds of Six Tetrahedra Each

compound of six elongated tetrahedra

In the image above, which I stumbled upon using Stella 4d (available here), the tetrahedra are elongated. If they are regular, instead, the same arrangement looks very different:

Tetrahedra 6

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Selections from the Stellation-Series of the Strombic Icositetrahedron, Including Some Polyhedral Compounds

The strombic icositetrahedron is the dual of the rhombcuboctahedron, and has many interesting polyhedra in its stellation-series. Here are a few of them, starting with the 10th stellation.

10th stellation of Strombic Icositetra

Here’s the strombic icositetrahedron’s 16th stellation:

16th stellation of Strombic Icositetra

And the 19th:

19th stellation of Strombic IcositetraAnd the 21st:

21st stellation of Strombic Icositetra

And the 23rd:

23rd stellation of Strombic Icositetra

And the 25th:

25th stellation of Strombic Icositetra

And the 26th:

26th stellation of Strombic IcositetraNext, the 28th stellation. It isn’t colored as the other stellations above are colored, simply because it is also a compound of six off-center square-based pyramids.

28th stellation of Strombic Icositetra

The 34th stellation is even more interesting. It’s a symmetrical four-part compound, but the component polyhedra have irregular faces, and are much less symmetrical than the compound itself.

34th stellation of Strombic Icositetra

Here is the 37th stellation in this series:

37th stellation of Strombic Icositetra

And the 43rd:

43rd stellation of Strombic Icositetra

And the 44th:

44th stellation of Strombic IcositetraThe 59th stellation in this series is an octahedron, with each face excavated by short, triangle-based pyramids. It can also be seen as a compound of three shortened square-based dipyramids, but coloring it as a compound proved difficult, so it is presented here in rainbow-color mode:

59th stellation of Strombic Icositetra

Here’s the 61st stellation:

61st stellation of Strombic Icositetra

And the 68th:

68th stellation of Strombic Icositetra

And the 71st:

71st stellation of Strombic Icositetra

And the (quite different from the 71st) 72nd stellation:

72nd stellation of Strombic Icositetra

And the 73rd:

73rd stellation of Strombic Icositetra

And, finally, the 74th, which is an interesting two-part compound.

74th stellation of Strombic Icositetra

And the 79th:

79th stellation of Strombic Icositetra

And the 82nd stellation:

82nd stellation of Strombic Icositetra

The last one I’m showing here is the 93rd stellation, another four-part compound.

93rd stellation of Strombic Icositetra

All these images were created using Stella 4d:  Polyhedron Navigator, which you may try for yourself at http://www.software3d.com/Stella.php.

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Falling into a Vortex

Falling into a Vortex

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Compound of Three Eight-Faced Trapezohedra

compound of three eight-faced trapezohedra and 6th stellation of triakis octahedron

I made this using Stella 4d, which you can try right here. In addition to being a compound of trapezohedra, it is also the sixth stellation of the triakis octahedron, the dual of the truncated cube.

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Fort Smith? Pine Bluff? What’s the Difference?

Fort Smith is Pine Bluff

Unless you also live in the American state of Arkansas, you may not have even heard of Fort Smith and Pine Bluff. For those who grew up in this state, though, they’re considered (by our peculiar standards) to be major cities. Fort Smith is on the far West side of the state, on the Oklahoma border, while Pine Bluff is in the Southeastern part of the state, also known as the Mississippi Delta.

I didn’t learn Arkansas geography in school. Instead, when I was a child, my family traveled, mostly within the state — a lot. By the time I was ten years old, I’d been in all 75 counties of Arkansas, and knew quite a bit about where things are here . . . except for these two cities, Fort Smith and Pine Bluff.

If you’re from Arkansas, you know these two cities are nothing alike. What I noticed in childhood, above all else, was the fact that the two cities smelled so different from each other. The reason is simple:  Pine Bluff has a lot of paper mills, and the smell near paper mills is not entirely unlike being locked in a small closet with several dozen rotten eggs. Fort Smith, by contrast, is relatively odorless.

What perplexed my parents, though, is the fact that I would consistently confuse these two cities. I’d refer to the “horrible smell of Fort Smith,” or, if I knew we were going to Pine Bluff, I might ask if we’d be crossing the border into Oklahoma. My parents always corrected these mistakes, but I kept making them, repeatedly, which is not like me at all. When young, I never had more than a 50% chance of correctly identifying either of these cities. Once I figured out what I was doing, though — at around age twelve — this repeated error made perfect sense.

When I try to understand something, I examine it, and consider it, mathematically. Often, I’m not even conscious I’m doing that — it’s simply how I think. Both Fort Smith and Pine Bluff are two-word city names. To make matters even worse, the first word of each city-name has four letters, and the second word in each has five. Once I realized these parallels, though, it all made sense:  no wonder I couldn’t tell these places apart, with names which, examined through the lens of childhood mathematics, looked exactly alike.

To my knowledge, no one else has ever had a long-term problem confusing these two Arkansas cities. However, when those who know me well hear this story, they are never surprised that I would do such a thing.

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Sam Harris, on Paying Attention

Sam_Harris_01

Source:  Waking Up: A Guide to Spirituality Without Religion, p. 3.

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