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A prism is a shape that can be constructed from taking the Cartesian product of some number of lower-dimensional shapes.

But wait, what is a Cartesian product?

The Cartesian product of two sets is the set of all ordered pairs of elements of those sets. As a simple example, {0, 1} × {0, 1} = {(0,0), (0, 1), (1, 0), (1, 1)}.

This is useful for shapes because a shape is just a set of points. Say you have a square, defined between (0, 0) and (1, 1). And you have a line segment, defined between (0) and (1). The cartesian product gives you a cube, defined between (0, 0, 0) and (1, 1, 1).

Cubes

You can make a cube by stacking lots of lines, or by stacking lots of squares.

One way to visualise this is by turning every point on shape A into a copy of shape B. A cube is like a square, but every point on the square has been upgraded into a line (or like a line, where every point has been upgraded into a square).

These Cartesian products are also equipped with something called the product topology. The product topology essentially means (non-rigorously) that you can move between points on shape A as you would normally, and you can move along each copy of shape B as you would normally. Take the stacked squares example: if you're inside a cube, you can move up and down the line segment as you normally would with a line segment, and you can also move along each square as you would a normal square.

Tell me more about this upgrading points thing

When you take the Cartesian product of two polytopes, not just points are upgraded; all subfacets are upgraded.

Take the simple example of multiplication by a line segment. All the points are upgraded into line segments, but all the line segments are upgraded into squares, and all the squares are upgraded into cubes, and all the disks are upgraded into cylinders, and so on.

Multiplication by a square will upgrade your points into squares, but it will also upgrade your line segments into

In general, a subfacet will itself be upgraded into the cartesian product of that subfacet and shape B.

One way of getting this done is to express each shape as a polynomial in its subfacets. Take the example of the cartesian product of a cube and a pentagon. You can write the subfacets of a cube as

8 + 12l + 6 l^2 + l^3,

where l is a line segment. You can write the subfacets of a pentagon as

5 + P^2,

where P2 is a pentagon. You'll notice that when written like this, the order of each term is equal to the dimensionality of the subfacet, and that these are basically just polynomials and you can do operations on them in the normal way.

So, the subfacets of the Cartesian product of these two is easy. It's just

\left( 8 + 12l + 6 l^2 + l^3\right) \left( 5 + P^2 \right)

= 40 + 60l + 8 P^2 + 30 l^2 + 12 l P^2 + 5 l^3 + 6 l^2 P^2 + l^3 P^2,

which is the correct subfacets for a five-dimensional pentagonal-cubic duoprism!

Okay, you can multiply any two shapes together. What about sums?

Adding together two shapes adds together their subfacets. This operation would correspond to having the disconnected sum of the two shapes, such as with the imaginative compound of two tetrahedra.

Multiplying by a number?

That's just repeated addition. 2 × cube is the disconnected sum of two cubes.

Adding a number?

That's interesting. In this system, adding a number just increases the number of vertices, which is the same as adding some disconnected points somewhere nearby.

This also means that a number corresponds to the discrete point set with the same cardinality. 2 is a point pair. 1 is a point. 3 is a point triplet.

Raising it to the power of a number?

A shape to the power of a number is just that shape multiplied by itself that many times. For example, \left( 2 + l \right)^3 = 8 + 12l + 6 l^2 + l^3; if you cube a line segment, you get a cube.

Extracting roots?

Kind of. \sqrt{4 + 4l + l^2} is geometrically meaningful; the square root of a square is a line segment, as you can determine by factoring it and doing the rest. \sqrt{5 + P^2} might not be, because that's not a perfect square.

Exponentation?

e^\left( 2 + l \right) can be done with the power series, maybe, probably not, don't even try.

See if you can assign meaning to \left( 2 + l \right)^{\left( 2 + l \right)} yourself. By the normal rules of set exponentiation, this is probably an ℵ1-dimensional space of functions mapping between two line segments, or something. I don't think the result counts as a shape.

Other functions?

If the natural polynomials are closed under a function, then it's probably generally meaningful. If they're not, it's probably not. If you can do a function to one and get another natural polynomial, it has meaning in that specific case. If you can generalise, that's awesome, please leave a comment, I would love to know what the square root of a pentagon is.

What about the hypervolumes?

The process is exactly the same, except you have the polynomials represent the hypervolumes of the shapes. For example, with a pentagonal-pentagonal duoprism, the hypervolumes can be extracted from

\left( 5 + 5a + \frac{5}{4} \sqrt{ 1 + \frac{2}{ \sqrt{5} } } a^2 \right) \left( 5 + 5b + \frac{5}{4} \sqrt{ 1 + \frac{2}{ \sqrt{5} } } b^2 \right),

with the 0th order terms being vertex count, the 1st order terms being edge length, the 2nd order terms being surface area, the 3rd order terms being surcell volume, the 4th order terms being surteron bulk, and so on.

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