If you're a fan of this podcast, I'm betting that at some point in your life you had a picture by M.C. Escher up on your wall, probably before you got married. As you may recall, M. C. Escher was the 20th-century Dutch artist famous for his many woodcuts and lithographs that used geometrical tricks to depict objects that could not possibly exist in real life. While Escher didn't have any formal mathematical training, it is said that he was inspired by a trip to the Alhambra in 1936, and its unique use of many of the 17 possible "wallpaper group" symmetries in its designs. After this, he studied an academic paper by mathematician George Polya on plane symmetries, and started drawing different types of geometric grids to use as the basis of his art. He later studied works of other mathematicians, basing lithographs on mathematical concepts like representations of infinity and hyperbolic plane tilings. As a result, Escher's art has been continually popular among scientists, engineers, and mathematicians.
When you look at one of Escher's impossible illustrations, it's always amusing to think about what it would take for it to exist in real life. But an Israeli professor, Gershon Elber, has gone one step further, and actually created physical models and 3-D printable CAD files, that allow physical creation of impossible Escher objects. How can you create an impossible object, you might ask? The key is the power of optical illusion. Each of these objects only looks correct, in other words matching the original Escher picture, from certain viewing angles-- if you rotate it or look from the wrong direction, you'll see that it is seriously distorted. At this point, you might say that Elber cheated, but give him a break-- we're talking about truly impossible objects here. If you learn how to fully warp spacetime at some point and make them non-impossible, then you can freely scoff at him.
One simple example is what's known as a 'Necker cube'. This simple illusion is based on the pseudo-3-D line drawing of a cube that we all learn in elementary school: basically two squares connected by diagonals on the corners. The impossible part comes when you give the lines some thickness, and draw the cube so that when edges in the drawing meet, they cross each other in contradictory ways, forcing the 'far' corner to be closer than the 'near' corner in some instances. If you haven't seen this before, you can find it at the links in the show notes. In Elber's version, it does indeed look like this miraculous cube has been created in real life-- if viewed from exactly the right angle. If you rotate it a little, you'll see that the whole thing is a mess of curved and distorted pieces, nothing like a real cube.
But Elber doesn't only tackle simple geometrical shapes-- he also recreates complete Escher artworks, such as the "Belvedere", a famous depiction of an impossible two-story tower. The two floors appear to be right on top of each other, a fact required by the placement of several supporting pillars. Yet when you look more closely, the picture also requires the two floors to exist at perpendicular angles; if one is going north-south, the other must be going east-west. Since the floors can't be simulatenously right on top of each other and at perpendicular angles, it's a truly impossible object. Again, you can see the picture in the show notes if you don't recognize it from my description. And once again, what looks like a perfect 3-D construction from a certain viewing angle is a distorted mess of angled and curved support beams from any other angle. For it to look like Escher's real Belvedere, you have to be at the exact spot where the angled beams will look straight to you.
By the way, the urge for a real-life Belvedere is apparently not unique to Elber: I also found a link where another self-described online "professional nerd" named Andrew Lipson constructed a similar model, entirely out of Legos! Lipson's version is actually more complete, even including the minor details and people in Escher's picture. He acknowledges that he needed to cheat at certain points-- for example, quote, "In Escher's original, he's holding an 'impossible cube', but in our version he's holding an impossible LEGO square. Well, OK, not quite impossible if you've got a decent pair of pliers (ouch)."
On Elber's and Lipson's web pages you can find a number of other Escher re-creations. Now while it's pretty fun to say you have created an impossible object in real life, one might argue: if it only looks like the truly impossible object from a certain viewing angle, aren't you cheating? Isn't this roughly equivalent to just creating the 2-D painting anyway? To someone who asks that question, I can only answer that they probably lack the geekiness level required to appreciate the coolness of Elber's and Lipson's work. I suspect most fans of this podcast don't have that problem.
And this has been your math mutation for today.