Trolls and Tesseracts: Playing dice in four dimensions

My question for today is, why are there three dimensions? If there were more, would the universe be a better or worse place? What would it be like?

We currently live in a world with four dimensions, but one handles movement through time and therefore doesn’t affect the shape or area of objects. This is why, for example, you can get a train to Swansea, but not to 1973. There may be many more dimensions, according to string theory. But that, again, is a different thing.

For the purposes of this post, we’ll be looking solely at what things would look like if we added more dimensions in regular, unstring-ey space. Turns out, the best first question to ask to understand this is, how many dimensions are optimal for playing Dungeons and Dragons? Of course, that makes this one of the most nerdy blog posts I can imagine. But I’m okay with that.

A 2D representation of 3 dimensional Convex Regular Polytopes
A 2D representation of 3 dimensional Convex Regular Polytopes

Dungeons and Dragons is a game played with convex regular polytopes, or, as most shops insist on calling them – dice. There’s three things you need to know about them:

  1. They’re convex: every point inside them can be joined to every other point in them by a straight line.
  2. They’re regular: They are as symmetrical as possible, to put it simply and perhaps wrongly, they are made up of a repetition of a lower dimensional shape. So, for instance, a six sided 3D dice is made up of 2D squares, and all the squares have to be of the same size.
  3. They are polytopes – a object made up of points connected by flat lines.

Dungeons and Dragons is traditionally played with the five regular convex polytopes available in three dimensions, as well as one irregular convex polytope (ten sided dice). The fact that we have to use irregular shaped dice shows clearly that 3 dimensional geometry is not ideal for the game. Other numbers of dimensions give us more options.

In a two dimensional universe, there would be an infinite number of possible dice because they’re just shapes where every side is the same length. The only problem is, without an up and down, the dice can’t role. So we’re going to need more dimensions.

In four  dimensions, there are six possible dice. I can’t show you direct picture of one, because it would break the laws of physics. But we can put one together, and get some idea how they might look. Here’s how we would construct a 4D cube, or tesseract:

Building a tesseract

1. Draw a line

2. Add another line, join them together and make a square.

3. Add another square, join them together and make a cube.

4. Add another cube, join them together and make a tesseract.

tessearct flattened into 3 dimensions

If we then pulled the cubes apart, it would make a kind of cross shape.

Because they’re beautiful, here’s the other dice, and what they may look like to you if you saw one being rolled from a three dimensional vantage point.

 

 

1. Tetrahedral Hyper-Pyramid (10 sided)

pentachoron

2.  Tesseract (24 sided)

tesseract

3. Regular Hexadecachoron (32-sided)

16 cell

4. Polyoctahedron (96 sided)

24 cell

5. Hyperdodecahedron (720 sided)

120 cell

6. Hexacosichoron (1,200 sided)

600 cell

Dungeons and Dragons fans will note that while in our dimension, the best dice roll you can get is a natural twenty (a twenty on a twenty sided die), in 4D D&D, it is possible to roll a natural 1,200!

So 4D D&D is better than 3D D&D, so 5D D&D will be better still, right? Strangely, no. In 5D there could only be three regular dice: the Hexateron, the Pentacross, and the Penaract. After that, no matter how many dimensions you add, the number of possible regular dice stays the same.

Why is this? Well, the simple answer is that a convex regular polytope needs to have angles equalling less than 360° at its corners, anymore and the shape would bend inward and it wouldn’t be convex. When a dice is made up of sides that are the same shape this is only possible with 5 3D shapes and 6 4D shapes, and three shapes in higher dimensions.

This is one example of 3 and 4 dimensional space being in some ways more complex than higher dimensions. This is interesting, given we live in a universe with 3 dimensions plus 1 dimension for time. Maybe this complexity is a good thing, many people believe that life is only possible in a universe like ours.

Certainly it helps with playing Dungeons and Dragons.

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