Category:Tesseract torus rgby

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lower and upper half

Raytraced with POV-Ray. The source can be found on GitHub.

tesseract and torus
The graph of the tesseract is equivalent to that of a 4×4 torus. In the tesseract image below, that torus can be seen as the surface of the four horizontally outward facing cells. The torus shown is the dual of that one. (The 4×4 torus is self-dual like a square tiling.) Its squares correspond to the tesseract's vertices (and thus to the tetrahedral cells of a 16-cell).

closer look at the four squares in front
The four squares in front of the torus are 0, 1, 4 and 5. The following images show the corresponding elements in the 16-cell, i.e. tetrahedral cells ~ torus squares, triangles ~ torus edges, edge ~ torus vertex:
cell 4		cell 5

cell 0		cell 1

closer look at the parallel circles

In the corresponding 4 sphere Venn diagram (16-cell) each border between a set and its complement is a sphere - i.e. contiguous.
But in the torus, the same border is split in two separate circles. The images below show the half spheres (of four triangles) corresponding to each circle (of four edges).
The borders shown in each of the two rows form a 2 set Venn diagram - like parallel circles in the torus.

left green circle

backward red circle

right green circle

front red circle

lower yellow circle

inner blue circle

upper yellow circle

outer blue circle

The image on the left shows the rings where spheres in complementary colors intersect
They do not have an equivalent in the torus, where circles in complementary colors do not intersect.
The image on the right shows the remaining 16 of 24 edges, that do correspond to vertices in the torus.
E.g. there are four edges dashed in red and yellow, that correspond to the four intersections of red and yellow circles in the torus.

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Venn spheres rgby; 32 triangles; half spheres corresponding to torus circles‎ (8 F)

Pages in category "Tesseract torus rgby"

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Template:Tesseract torus rgby

Media in category "Tesseract torus rgby"

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Tesseract Schlegel rgby.png 4,000 × 4,240; 730 KB
Tesseract torus 0.png 4,000 × 2,680; 1.4 MB
Tesseract torus 1.png 4,000 × 2,680; 1.54 MB
Tesseract torus 2.png 4,000 × 2,680; 1.66 MB
Tesseract torus 3.png 4,000 × 2,680; 1.81 MB
Tesseract torus half lower.svg 916 × 916; 52 KB
Tesseract torus half upper.svg 916 × 916; 54 KB
Tesseract torus matrix.svg 1,033 × 1,023; 553 KB
Tesseract torus.png 4,000 × 2,680; 1.54 MB

Category:Tesseract torus rgby

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Pages in category "Tesseract torus rgby"

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Media in category "Tesseract torus rgby"

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