Mathematics & Development

On this page we will state a few very basic properties of a regular polygon with $n$ sides. We assume that each side is of length $a$. The next figure shows a typical example, in this case a heptagon, a seven-sided regular polygon. The blue circle is called the circumcircle of radius $r_u$, passing through all the vertices. In analogy, there is a unique incircle of radius $r_i$ touching all sides from within, here drawn in red.

About the process of folding a net made of polygons into a three-dimensional polyhedron.

New version of fillygons code allowing us to specify the edge length. This is how we can build a triangle with a side length of $\sqrt{2}$ units. Example of these new fillygons include 3 gon sqrt2 (normal) and 4 gon sqrt2 (normal) which differ from models like 3 gon double (normal) and others. (The model 3 gon double (normal) is technically even a hexagon with three angles of $\frac{\pi}{3}$ and three angles of $\pi$.

Starting with a regular $n$-gon in the center, we attach a square on each side. Next we fill the gaps between the squares with equilateral triangles. Then we finish the solid by inserting a regular $2n$-gon in the bottom. The next image shows some cupolas for different $n$ but without the closing bottom face. In this post I’d like to study some of the geometrical properties of cupolas like height and volume.