Musings on the Impossible 1
Monument Valley, M.C. Escher and the Dimensions of Reality
A couple of years ago, I wound up playing a game called “Monument Valley”. I found it fascinating due to its environment: a virtual world where Princess Ida had to traverse constructions that would be impossible to create in a three-dimension environment. Im not a game player at all, so I’ll accept criticisms that Monument Valley isn’t very difficult and (relatively) short. There are many other games where players can immerse themselves in huge and complex virtual worlds where the game can take—sometimes literally—forever. To me, it’s brevity and spartan simplicity are benefits as they allow me to wonder about these impossible constructions, imbuing me with a sense of wonder, with a perspective into the impossible.
Clearly, Monument Valley’s, well, monuments are inspired by the art of the Dutch artist M.C. Escher—a good thing IMHO—yet takes them to the next level by taking the artist’s ideas and making them manipulatable. They are beautiful demonstrations that somethings are possible in two dimension, while impossible in three.
It’s a strange observation: you add an extra dimension but it restricts certain configurations. On the one hand, there are many things possible in 3D space that are impossible in two dimensions, yet there are also configurations that work in 2D space but are impossible in three dimensions.
M.C. Escher’s drawings and Monument Valley’s constructions are almost like a dream of three dimensions of entities who live in two dimensions, and these dreams are so fantastic that they are, literally, impossible in an actual reality of three dimensions. Another example that yes, sometimes less is more.
We find ourselves living in a three-dimensional world1, so do we sometimes dream of higher- (or even lower-) dimensional environments? Of course we do, with the well-known hypercube (or tesseract) as an example, while the concept of lower and higher dimensions has been extensively explored in science fiction. Theoretical mathematics can, quite easily, handle many more dimensions, yet struggled for centuries to exactly define the concept of dimension. Yet when it did, it developed—via Felix Hausdorff the notion that non-integer dimensions are also possible, a possibility utilised by Benoit B. Mandelbrot which he coined as ‘fractal’ and is demonstrated in his Mandelbrot set.
More interestingly, though, is the question if higher (or even lower) dimensions actually exist (or play a role) in our current reality. There are a few scientific hypotheses that explore different dimensions:
The holographic principle: a hypothesis proposing that our reality is a hologram projected from a 2D surface.
String theory: an attempt at a Theory of Everything that proposes that our particles are manifestations of the vibrations of higher-dimensional strings.
So is our reality merely a shadow of event(s) taking place in higher dimensions? The problem with String Theory is that requires such immensely concentrated energies that it seems well-nigh impossible to test some of its predictions with experiments. While there have been several proposals for testing—at least some of the implications of—String Theory, none have delivered any positive proof, so far:
Departures from Keppler’s Third Law of planetary motion, drifts from gravitational equilibrium zones known as Lagrange points and oscillations in orbital distances due to accelerations towards a third body—
these have never been measured because the variations involved are incredibly small—
yet Dr James Overduin’s team reckons that it should be measurable at the Saturnian moons Thetys and Dione (right now, no space craft is there to perform that measurement);
Observing if primordial gravitational waves show how much inflation has occurred in the early Universe and if that amount agrees with the trans-Planckian censorship conjecture—
there are yet too many conflicting versions of this to confidently predict an observable result;
Detection of cosmic strings might give more insights—
So far, no cosmic strings have been detected;
Proof of supersymmetry would bolster String Theory’s claims—
TL;DR: right now there is no direct evidence that higher dimensions exert influence on our three-dimensional reality.
Or the converse (see the holographic principle): is our reality a 3D projection from a 2D canvas? According to Space(dot)com’s article “Are We Living in a Hologram?”, the arguments for the holographic principle are, so far, not very convincing.
As such, do higher dimensions really exist, or are they—not unlike infinity (see next essay) and perpetual motion—merely tools that can help us understand, even utilise, phenomena in our current world? Driven by evolution and the way our consciousness works, humans are pattern recognition experts par excellence to the point where we sometimes see patterns that are not really there.
It’s also possible that different dimensions are concepts like perfect symmetry and unending fractality2 that do not truly exist, yet helps us in discerning patterns that might be manifestations of underlying fundamental laws.
Also, being able to imagine different dimensions, or even suspect the possibility of different dimensions is a manifestation of creativity—or lateral thinking—that current AIs do not have3. As such, researchers needed to imbue ‘gauge-equivalent convolutional neural networks’—gauge CNNs—with a new discipline called geometric deep learning to lift these CNNs out of flatland.
People in the science fiction community—and beyond—are aware of Arthur C. Clarke’s three laws (which aren’t really laws but rather conjectures), of which the second ones states:
“The only way of discovering the limits of the possible is to venture a little way past them into the impossible.”
That is exactly what many of M.C. Escher’s paintings and—by extension—a game like Monument Valley is doing. They depict the impossible, making us wonder why it is impossible. Then we’re not only aware where the limits of the possible are, but also if there are ways to move beyond them; that is, make the previously impossible possible. Or at least find out why we can’t overcome this particular impossibility.
And I’m not counting time as a fourth dimension, as time can be applied in 1D, 2D and multiple-D environments, as well;
The virtual mathematical Mandelbrot set keeps going on indefinitely, but in the real world the fractality stops at the atomic level;
Together with agency, a sense of narrative and sentience/consciousness;