Three-Dimensional Thinking, Part 2

The last time I wrote about three-dimensional thinking, I discussed impossible figures. They are fun ways to challenge our brains to see things in a different way. But to me they signify more than just artwork.

Different Angles

Looking at objects from different angles helps us understand their spatial structure.

Looking at a given subject from different angles is a requirement for creativity. But eventually, in your mind, you realize that reality itself is malleable, and this is the domain of dreams. And dreaming is good for creativity because it helps us get out of the box of everyday experience and use our vision in a new way.

The key

Once I asked myself a question about impossible objects: what is the key to making one?

The key trick which is used in impossible figures is this: locally possible globally impossible. In the case of a Penrose triangle (also called a Reutersvärd triangle because Oscar Reutersvärd was the first to depict it) local corners and pieces of objects are entirely possible to construct but the way they are globally connected is spatially impossible.

I have constructed another impossible figure which is included above. This figure contains several global contradictions, yet remains locally plausible. However, there are two global levels of impossibility in this figure. Let's consider what they are.

First off, there are plenty of locally plausible geometries depicted in the figure. For instance, the M figure is a totally real and constructible object in the real world.

My original drawing didn't actually have M's at the three corners. It was a Penrose triangle. To make the figure compact, I added the M's on each of the three corners of the Penrose triangle. This doesn't make the figure any more possible though. It just adds a little salt and pepper to the mix; it helps confuse the eye a bit.

The next part shows the three strands connected to the three loops that wrap around the Penrose triangle.

There is really nothing about this strong figure that is impossible either. It can be totally constructed in real space.

Actually, it is a nice figure by itself, standing alone. You can see each block sliding by itself through the set of blocks.

And further, I this this figure would make a good logo. It feels like an impossible figure even though it's perfectly realizable. And it can be depicted from any angle because it is an honest three-dimensional construction. I have an idea to construct one out of lucite or another transparent material.

The next part of the figure is the loop. Each loop wraps around one of the sides of the Penrose triangle and creates an interlocking impossible figure, a concept I have shown examples of before in this blog. For instance, there is the impossible Valknut.

But this is the first level of impossibility. Such a loop is not really constructible without bending the top face. In this way, it is related to the unending staircase of M. C. Escher's Ascending and Descending.

The second level of impossibility is, of course, the Penrose triangle itself. When it comes to levels of impossibility and a clean depiction of impossibility, consider Reutersvärd. Pretty much all of Reutersvärd's art contains this illusion as a key. Though, I would encourage you to look at all of his work, because individual pieces can be both stunning and subtle simultaneously.

The next impossible figure is another modification of the Penrose triangle, showing what happens when the blocks intersect each other.

Any two blocks may certainly intersect each other, but to have all three intersect each other in this way is a clear impossibility.

It would probably have been more striking to make the triangular space in the center a bit larger.

Impossible objects take imagination out of the real world and into a world that maybe could be. Perhaps it's the world of flying cars, of paper that can hold any image and quickly change to any other, or of people whose thoughts are interconnected by quantum entanglement. In such a world, imagination can fly free.