There are sixty of the irregaular, pentagonal gaps. Also, the hexagons themselves are of three types, two of which are sixty in number, and one of which is thirty in number.
If the gaps are filled, and the color scheme changed to make each of the four polygon-types into its own color-group, this looks, instead, like this (click on it if you wish to see it enlarged). It has 210 faces.
(Images created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)
This zonohedron has fifty faces:
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It’s like the snub dodecahedron’s big brother.
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(Image created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)
Years ago, I split a dodecahedron into four panels of pentagons, rotated the pentagon-panels and moved them outward from the center, and did so just the right amount to create gaps that could be filled with triangles. Thus was named the tetrated dodecahedron, which you can read more about here: https://en.wikipedia.org/wiki/Tetrated_dodecahedron
The choice of word “tetrated” was somewhat unfortunate, for tetration already exists in mathematics, as a means of expressing very large numbers, and which I shall not explain here. I didn’t learn this until much later, though, and by that time, it was too late to turn “tetrate” into something else. It had come to mean an operation one does on a polyhedron: break it into four multi-face panels, move them out and rotate them just enough, and fill the resulting gaps with triangles.
As such, “tetrate” can, in the geometrical sense, be modified for differing numbers of panels of multiple faces from a polyhedron. Consider the pentagonal icositetrahedron, the dual of the snub cube. Here, it has been split into six panels, and then each panel moved out from the center and rotated, with triangles filling the gaps. The triangles differ between color-groups slightly, but are close to equilateral, except for the ones shown in green, which simply are equilateral.
(Image created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)
This is the dual of the polyhedron seen in the last post. It appears to be an interesting blend of the snub cube and an icosidodecahedron.
(Image created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)
(Image created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)
The triakis octahedron, a Catalan solid, is the dual of the truncated cube. When stellated six times, the triakis octahedron yields this polyhedral compound with three parts. The parts themselves appear to be unusual, irregular, dipolar octahedra with eight kites for faces, each in sets of four, with their smallest angles meeting at one vertex. However, given that these vertices are, in each case, hidden under the other parts of the compound, there is uncertainty in this.
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A few minutes ago, I wondered how to write a function whose graph would be a sine curve, but one that undulated above and below the diagonal line y=x, rather than the x-axis, as is usually the case. How to accomplish such a 45 degree counterclockwise rotation?
Well, first, I abandoned degrees, set Geometer’s Sketchpad to radians, and then simply constructed plots for both y = x and y = sin(x). Next, I added them together. The result is the green curve (and equation) you see above.
This only half-solves the problem. Does it undulate above and below y=x? Yes, it does. However, if you rotate this whole thing, clockwise, one-eighth of a complete turn, so that you are looking at the green curve going along the x-axis, you’ll notice that it is not a true sine curve, but a distorted one. Why? Because it was generated by adding y-values along the original x-axis, not by a true rotation.
I’m not certain how to correct for this distortion, or otherwise solve the problem. If anyone has a suggestion, please leave it in a comment. [Note: an astute follower of this blog has now done exactly that, so I refer the reader to the comments for the rest of the story here.]
The last post made me curious about other trigonometric functions’ graphs, in a polar coordinate system. They were not what I expected. Here they are.
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