Higher Dimensional Polytopes

A guest post by Bobby Holmquist:

It is common in sci-fi literature to hear about higher special dimensions. In Star Trek and Star Wars, one is able to access faster-than-light travel by accessing warp speed or hyperspace, respectively. In both of these cases, space time is being bent or “warped” into the fourth dimension which allows for faster-than-light travel without actually breaking the speed of light. By adding a fourth special dimension to space-time, we are able to then “fold” space and instantaneously jump from one point to the next. Think about space-time like a piece of paper where you want to get from one point to the other. In normal space-time, the obvious answer is a straight line but by adding a third dimension of maneuverability, we are able to fold the paper and put the two points right next to each other and so are able to move from A to B instantaneously. This is how faster-than-light travel would work; we would bend space-time into the fourth dimension putting the points right next to each other.

Now we may find ourselves wondering more about the nature of this fourth dimension more than our original inquiry about faster-than-light travel. For instance, what would shapes in this fourth dimension look like? Are there dimensions higher than four? What happens to mathematics in these higher dimensions? These questions are far more interesting than our original sci-fi question so let us shift our focus to these more abstract topics. In Numberphile’s video “Perfect Shapes in Higher Dimensions,” Carlo Sequin talks about higher dimensional geometry. He begins his talk with a review of three-dimensional Platonic solids which are solid shapes where every face has the same number of edges, each edge is the same length, and each face has the same area. There are five of these displayed below.

Expanding on these, he delves into four dimensional regular polytopes, which are constructed from the “folding” of these shapes into each other in different ways. Because we have access to the fourth dimension, we are able to do what seems impossible by folding one of our shapes into another one. Because of the nature of higher dimensional geometry, it is impossible to visualize what the objects actually look like so we use wire projections to best represent them. Below is an image of a tesseract which is composed of eight cubes folded into the fourth dimension.

Even though it looks like a smaller cube within a larger one connected by six rhombuses all the cells are really cubes of the same shape. The weird shape is only due to our lower perspective. You can think of the inner cube as always being farther away than the outermost cube making our impression of the tesseract look the way it does because of our forced perspective. This kind of thought experiment could be expanded to include things like functions, vector fields, and all other manner of mathematical concepts. How strange do our day to day conceptions of mathematics get when we allow them play in higher dimensions?

In fact, there are six four-dimensional regular polytopes which we create from “folding” our platonic solids into the fourth dimension. In order to do this, we must conjoin at least three of our platonic solids around a shared edge and then we are able to “fold” them into our four dimensional polytopes. At this point it is rather difficult to explain in further detail without the aid of several dozen pictures so I will leave the creation of the other five regular four-dimensional polytopes to the video. The video also looks into normal polytopes in n dimensions and found that there are only three regular polytopes for all dimensions higher than four. It is in this sequence where I will continue my inquiry. Why only three? What about the nature of space restricts the number of regular polytopes to such a specific number? Usually when we discover the nature of a space to be some numerical value or idea, we obtain zero, one, or that there are infinitely many. It seems very strange that the nature of reality forces there to be exactly three regular polytopes for dimensions higher than four. This brings up some interesting philosophical questions as well about the epistemology of mathematics. How is it that we are able to know things in mathematics which are unable to be experienced? How can we know the nature of a space which cannot be found in our own universe but nonetheless follows universal laws? Perhaps most significantly, what is mathematics and what exactly is the relationship between mathematics and reality?

Citations:
Sequin, C. (2016, March 23). Perfect Shapes in Higher Dimensions. [Video file]. Retrieved from https://www.youtube.com/watch?v=2s4TqVAbfz4
Image 1; http://www.21stcenturysciencetech.com/images/fall2000/moon1.jpg
Image 2; http://www.daviddarling.info/images/tesseract.jpg