I have been experimenting with encoding shapes as integer polynomials with the following algorithm. take the number of n dimensional "points" or "segment". Where a 0D segement is a point, a 1D segment is aline or curve, a 2D segment is a 2D surface etc. count them, say you have q. and then add to your polynomial qx^r. Do this for all parts of the shape.

You can describe some pretty non conventional 'shapes' this way, and the shapes that map to the polynomial can be very different but also share interesting properties. In some cases there are also relations to the set of real numbers.

Some patterns:

a point times a point is a line..

a point times any shape is the shape

a line times a shape is a prism of the shape in the next dimension (e.g. a square times a line is a cuboid).

xⁿ represents an n dimensional infinite plane (xⁿ is analogous to Rⁿ⁾

n*x^n-1 xⁿ represtents a plane with positive coordinates. (eg. 1 x is anagolous to R_ and 2x x² is anagolous to R^{2 )}

Multiplying these polynomials (can) give some surprisingly sensical results. 2 infinite lines (x) multiplied will give you a plane. however two finite lines(x 2) multiplied will give you a finite square.

A circle is 'equivelent' to a line of infinite length.

Next I am going to explore the exact relation between multiplying shapes and their results, explore if the zeros of these polynomials tell us any informaiton about the shapes, and see if it is possible to make sense of fractional/negative powers/coefficients - if so some shapes could have 'inverses'. Imagine a shape with -1 sides! I would also like to see if taking the derivative or antiderivative of these polynomials relate to their shapes in anyway.

(For example of the last point, the polynomial for planes with positive coordinates can be written in the form d(xⁿ)/dx xⁿ )