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https://www.facebook.com/hoyle.anderson/videos/10155639583220910/

Can anyone say what this actually is?

Michael Hardy
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  • It kind of reminds me of whatever a "phaseur" is (you can hit the Donneés button (again, whatever that is) to have more mysterious sliders to adjust. I first saw this linked to in Jean Marie's ever-growing question :), the end, the "Geogebra animation" – pjs36 Mar 15 '18 at 01:40

3 Answers3

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Klitzing is correct about the 4D polytope. It is four triangular prisms glued together at four triangular faces, and then three cubes adjoined to corresponding square faces of the prisms.

We can visualize this as a direct product $\triangle\times\square$ where the symbol $\square$ is the vertex set of a triangle in the plane and $\square$ is the vertex set of a square in the plane. The projection onto our 2D screen is not a simple coordinate projection, since those would either give four triangles all on top of each other (and we'd only see one triangle) or else three squares on top of each other (and we'd see only one square). We need a different projection.

If we interpret $\mathbb{R}^4=\mathbb{R}^2\times\mathbb{R}^2$. Our projection is then $(x,y)\mapsto x+y$. Clearly the squares are bigger than the triangles in the picture. Moreover, the imaginary square connecting the triangle centers is the same size as the displayed squares, and the imaginary triangle connecting the square centers is the same size as the displayed triangles. Thus, make sure the square $\square$ is larger than the triangle $\triangle$.

To understand the rotations, make note of the following facts:

  • Triangles rotate clockwise
  • Triangles revolve counterclockwise
  • Squares rotate counterclockwise
  • Squares revolve clockwise

This can be understood better using $\mathbb{C}\times\mathbb{C}$, in which the rotation is $(u,v)\mapsto (e^{-i\theta}u,e^{i\theta}v)$. Every 4D rotation is a combination of two independent rotations in two orthogonal planes; in this case the two orthogonal planes are $\mathbb{C}\times\{0\}$ and $\{0\}\times\mathbb{C}$. Moreover, if the angles of the two rotations are the same, the rotation is called isoclinic, so we have a one-parameter group of isoclinic rotations.

anon
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This picture from wikipedia is pretty suggestive:

This shows the relationships among the four-dimensional starry polytopes. The 2 convex forms and 10 starry forms can be seen in 3D as the vertices of a cuboctahedron.

(But the OP's figure doesn't seem to be a cuboctahedron.)

https://en.wikipedia.org/wiki/Regular_4-polytope

enter image description here

Ethan Bolker
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The intermediate pictures of your movie clearly show 3 squares (in a triangular setup), respectively 4 triangles (in a square setup). Thus it might be considered some projection of the cartesian product of a triangle and a square, i.e. of the triangle-square-duoprism. Cf. https://en.wikipedia.org/wiki/3-4_duoprism.
--- rk

Dr. Richard Klitzing
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