Penrose Triangle and Umbilic Torus

Question

Here are two images - one of each. It seems to me that they are the same object from the topological perspective, that one is just a smoothed-out version of the other. I think this because it is clear each has one side face, an enclosed body, triangular cross-sections etc. I guess I'm thinking loosely in terms of simplicial homology.

The Penrose Triangle

https://upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Penrose-dreieck.svg/2000px-Penrose-dreieck.svg.png

The Umbilic Torus

http://scgp.stonybrook.edu/wp-content/uploads/2012/10/UmbilicTorus-aerialview-web.jpg

To be fair, I cannot determine by just inspection if the Penrose triangle has a single edge or not - and if this would be enough to distinguish them topologically - and it's not like I can go out and look at one to determine if it's really the same as the umbilic torus on campus.

Can anyone with a good visualization ability or maybe just more knowledge of homology help me determine the homology groups of the Penrose triangle? Or if not, is there perhaps some easier way to see if they are distinct?

Answer 1

These are topologically both the same. Indeed both are homeomorphic to a solid torus $S^1 \times D^2$, and hence are homotopy equivalent to a circle: you can imagine shrinking each cross-section down to a point.

The edges/faces thing is irrelevant from the point of view of topology. For example the unit square $[0, 1]\times [0, 1]$ is homeomorphic to the disc $D^2$ despite having different numbers of corners/sides.

The Penrose triangle has two edges and a square cross-section.

Answer 2

If we were to imagine cutting the surface of both shapes along the "edges" (where in the Penrose triangle we ignore the triangular bends), then the resulting surface is one continuous piece in both cases, but there will be a different number of half-twists (three for the triangle, one for the torus). So in that regard, the two surfaces are not equivalent.

Answer 3

I'll make a brief contribution here because topology is not my forte. Nevertheless, I've developed a 3-D program that can create Möbioids (my terminology) of arbitrary planiform and number of twists. A twist consists of a single edge-to-edge turn. 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 cross-section and also a single twist. Each possesses a single surface.

Finally, the figure at the bottom shows a 3-D printing of the Penrose triangle. You can find more images and some animations at A New Twist on Möbius.