Jul 5, 2020 · As another additional bonus point, although it's not possible to embed the Penrose triangle into normal, flat, Euclidean 3D space, it is possible to embed it into curved three …

May 22, 2022 · The image of the fundamental unit is a topological loop with its "initial" cube abutting the "final" cube, i.e., a Penrose triangle. While a Penrose does exist in this locally …

Jul 3, 2017 · Penrose has an article on this here: On The Cohomology of Impossible Figures. I find this interesting because I find cohomology interesting and the fact there is such an …

The way Penrose used $\tau$ is to identify a 1-cocycle (the "distance cocycle") which is not a coboundary, the key fact being $\tau : C^1(X) \to \Bbb R$ being 1 on coboundaries. I suppose …

Feb 12, 2024 · This seems tricky, since in a simpler sense every 2D figure is a projection of a 3D figure, and we can even make 3D models that look like the Penrose Triangle from certain …

The figure on the left below depicts the umbilical torus with a triangular cross-section (deltoid, really) and a single twist. The figure on the right shows the Penrose triangle with a square …

$\begingroup$ The area of paper that the impossible figure occupies is what we're talking about. The non-trivial cohomology guarantees that you can't "fill in the void" in the middle of the figure …

Particular parts of Penrose tilings exhibit local 5-fold rotational symmetry (Penrose stars, Penrose suns, etc.), but since the tiling as a whole is non-periodic, how do you describe the overall …

If you calculate the coordinates of a triangle with sides 1,3,5 or 3,4,8 you get complex numbers. In ordinary real-coordinate Euclidean geometry this means there is no such triangle. If complex …

Penrose graphical notation seems to be a convenient way to do calculations involving tensors/ multilinear functions. However the wiki page does not actually tell us how to use the notation. …