Rather than deliver yet another π/pie pun based post for π Day, we decided to take a more geometric approach.  This time looking at something completely different, yet still round.  One might even say, hyper-round.

Initial observation:

Higher spacial dimensions in science fiction is a commonly recurring concept.  One such occurrence that stands out is the classic Twilight Zone episode Little Girl Lost, where a girl is lost in the forth dimension through a portal under her bed and her dog goes chasing after her.

Most attempts to visualize higher dimensions focus on projecting the entirety of an object into three dimensional space, such as the common depiction of a hypercube as a smaller cube within another cube.  That approach however, doesn’t explain what a higher dimensional object would look like to a three dimensional creature (un)fortunate enough to find themselves in four dimensional space.  A recent video by The Action Lab does a good job of demonstrating what a four dimensional sphere moving through our three dimensional space would look like.

Question:

What would it actually be like to visit four dimensional space?  Additionally, would the dog have had any hope of actually finding the little girl?

Expectation:

Given the odd properties of higher dimensional systems, four dimensional space would be very confusing.  As a result, it is suspected that the dog would have a very hard time finding the girl.

Equipment & Materials:

• Hyper-dimensional ray-tracer
• Compatible elastic collision simulator
• Time and Computronium

Data:

To get a better understanding of how things in higher dimensions look and move, a hyper-dimensional ray-tracer was implemented.  A simulation of hyperspheres bouncing around inside a hypercube was then rendered.

First, to validate that the elastic collisions were being simulated in a reasonable fashion, a 3D scene with spheres bouncing around inside a cube scene was rendered.

Then a 4D scene with hyperspheres bouncing around inside a hypercube was rendered.

Conclusion:

Higher dimensional space may look deceptively normal at first, but motion is very bizarre looking to a three dimensional creature.

Given the short period of time that any visible hyperspheres are in view, and that most of them are not in the field of view at all, it seems unlikely that the dog would have been able to locate the girl by sight.  Perhaps the girl could have been located by smell or other senses though.

Future Questions:

Now that we have a rough idea of what one type of motion looks like in four dimensions, what about other types of motion, such as Brownian motion or four dimensional Boids.  What do commonly discussed 4D shapes (e.g., hypercube, klein bottle, etc.) look like from various angles.  How would 4D analogs of common 3D objects move, for instance, what would a Rubik’s Tesseract look like?