While trying to formalize basic Year 8 geometry, I came up with the following:

Definition 0. A shape is a CW-complex $S$ together with a topological embedding $f_S:S \rightarrow \mathbb{R}^n$ such that for all natural numbers $n$, writing $S_n$ for the union of the $n$-dimensional cells of $S$, we have that $f_S(S_n)$ has finite $n$-dimensional measure. Two shapes are equal iff there exists a proper rigid transformation that turns one shape into the other.

Examples. The line segment of length $a$, the $(a\times b)$-rectangle, the circle of radius $r$, the two-dimensional disk of radius $r$, etc.

We can also take cartesian products of shapes; for example, the $a\times b$-rectangle can be expressed as the cartesian product of a line segment of length $a$ and another of length $b$. Similarly, we can get the cylinder by taking the cartesian product of a circle and a line segment.

The definition of a shape is rigged so that to each shape $S$, we can assign a polynomial $S(x)$ as follows.

Definition 1. If $S$ is a shape, write $S(x)$ for the formal polynomial in the symbol $x$ with coefficients in $\mathbb{R}_{\geq 0}$ such that the coefficient of $x^n$ is the $n$-dimensional measure of $f_S(S_n)$.

For example:

  1. If we write $[a]$ for the line segment of length $a$, then $[a](x) = 2+a x$. Notice the polynomial tells us both the number of vertexes, and the length of the interval.

  2. If $S=[a] \times [b],$ then $S$ is the rectangle of size $a \times b$. Therefore: $$S(x) = 4+(2a+2b)x+abx^2,$$ because $4$ is the number of vertexes, $2a+2b$ is the perimeter, and $ab$ is the area.

  3. If $S_r^2$ is the circle of radius $r$ (that's not filled in), then $S_r^2(x) = 2\pi rx$, because the coefficient of $x$ is the perimeter of the circle. Similarly, if $B_r^2$ is the two-dimensional ball of radius $r$, then $B_r^2(x) = 2\pi rx+\pi r^2x^2.$ The $2\pi r$ is the perimeter, and the $\pi r^2$ is the area.

  4. If $S$ is the point, then $S(x) = 1$.

Now. the whole reason we want to be viewing these as polynomials (rather than just sequences) is that it seems to be a general principle that $$(S \times T)(x) = S(x) \cdot T(x).$$

We can use this to derive dot point $1$ from dot point $2$. In particular: $$([a] \times [b])(x) = [a](x) \cdot [b](x) = (2+ax)(2+bx) = 4+(2a+2b)x+abx^2.$$

Its also possible to take sums of shapes, defined by disjoint union. We have: $$(S + T)(x) = S(x)+T(x)$$

Anyway, geometry has been around for a long time. Surely I'm not the first person to have this idea, so:

Question. What are "shapes" and/or the polynomials associated to them really called?

  • $\begingroup$ This looks a lot like some version of Steiner's formula to me. $\endgroup$ – Xander Henderson Apr 22 at 22:03

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