Working in Higher Dimensions
Most people are pretty familiar with the first three dimensions. The first dimension would be a line, the second a flat plane, and the 3-d world is what we can see right now. Most people may think that this is all there is, but this barely scratches the surface of what mathematicians are now studying.
Mathematicians can represent different objects or points in each of those planes with coordinated. One coordinate for 1st, two for 2nd, and three for the 3rd. And until the 1700 century, this was all there was. But as mathematics and physics were progressing and theories became more complicated, the suggestion of higher dimensions began to arise. In the suggestion of higher dimensions came the possibility of different shapes. You may recognize the shape below as a 4-dimensional cube or a tesseract well known for being in science fiction movies.
This tesseract is even further devloped in literature pieces, like in Heinlein’s short story “And He Built a Crooked House.“ This work includes an architect who builds a house that folds into the 4th dimension. There are 8 cubical rooms in the 3d space of one room. It is a pretty fun read if you want to give it a shot and try visualizing the plotline yourself.
Another common shape in the fourth dimension is the Klein Bottle which is the equivalent of the 4th dimensions Mobius strip. But how can shapes like this exist if we have never seen them and why is it so important?
First, let's answer the how. Before creating any shape in higher dimensions, there still has to be a set of rules. Just like said above, points can be used to describe the shape. In the above image, it would take four coordinate to represent this cube. Beyond describing the location of the shape, the shape itself has rules that govern how it can exist in that higher dimension.
Let's take the cube. A "cube" in one dimension would be a line and would have two "vertices." Then in two dimensions, it would have 4, in three a cube has 8. Moving into higher dimensions this same doubling pattern for vertices is applied to the cube above having 16 vertices. Along with 16 vertices, this shape also has 32 edges and 24 faces. These might all seem like random numbers but they actually follow a really important rule. Euler's Characteristic is used to determine the relationship between faces, edges, vertices, and cells (in higher dimensions) in a shape. In the third dimension, it applies to Polyhedreas with equation V - E + F = 2. The equation is slightly different in different dimensions but it is important that shapes in the same dimension exist. A cell is used to describe shapes in the higher dimensions and is a three-dimensional object part of the higher dimension object. In the 4-d cube's case, it has 8 cells.
The formula including the number of cells is now:
Vertices - Edges + Faces - Cells = 0
If we plug in our numbers
16 - 32+ 24 - 8 = 0
Which is correct!
There are many more in-depth explanations especially for dimension lower than 10 and how different unique shapes are formed. Even though this same equation if slightly different for each dimension, the important thing is that there is a relationship between this shape and others. This is just one example of how even though there is this somewhat looking random shape, it still has to follow basic rules and formulas that we operate around today.
Now let's ask why.
After the concept of higher dimensions was formed, mathematicians took to this concept for proving old or making new theorems. They applied the concepts in higher dimensions to probability, point relationships, and applications for hyperspace. It can also be used in computer science for problems and structure to programs. In the 1800's, books began to churn out explaining a new theory that would now work in the 7th or 23rd dimensions. The main reason behind the theory of 4th or 98th dimension is to better understand our own 3d world. Mathematics is forced to be proved when working in dimensions you can't visually prove something. Rather than just relying on visual proof, concepts are driven and worked out to mathematical representations. Mathematics has to be truly understood to even attempt to work in these dimensions. Even if there wasn't a concrete use for the fourth dimension or four thousandth dimension yet, the concept helps to push mathematics further and drive ideas applicable to our world.