Polytopes
Welcome! Polytopes are beautiful mathematical and geometrical objects. I have started a collection (still growing) of 3D printed polytopes that I have constructed at home. There is a lot of information about polytopes online and by no means am I trying to be exhaustive here, just a brief introduction to the figures I have printed. The readership is encouraged to dig deeper online for any nagging questions that are not answered here and to dive into the mathematics behind these wonderful objects.
TODO - Add image of overall collection of polytopes
Let us introduce some needed terminology.
The Schläfli symbol is a notation, expressed as \(\{p, q, r,\dots\}\), that defines both regular polytopes and tessellations. It provides a recursive description, starting with \(\{p\}\) for a convex, \(p\)-sided regular polygon. For instance, \(\{3\}\) represents an equilateral triangle, \(\{4\}\) a square, \(\{5\}\) a convex regular pentagon, and so on.
However, the Schläfli symbol extends beyond convex polygons to include regular star polygons, which are non-convex. The symbols for regular star polygons take the form \(\{p/q\}\), where \(p\) denotes the number of vertices and \(q\) represents their turning number. These symbols connect the vertices of \(\{p\}\) every \(q\) steps, resulting in fascinating shapes. For example, \(\{5/2\}\) represents a pentagram, and \(\{5/1\}\) represents a pentagon.
Moving to three dimensions, a regular polyhedron that features \(q\) regular \(p\)-sided polygon faces around each vertex is denoted by \(\{p, q\}\). For example, a cube, which has 3 squares around each vertex, is represented by \(\{4, 3\}\).
Expanding further into four-dimensional space, a regular 4-dimensional polytope with \(r\) \(\{p, q\}\) regular polyhedral cells around each edge is symbolized by \(\{p, q, r\}\). For instance, a tesseract, denoted as \(\{4, 3, 3\}\), consists of 3 cubes, \(\{4, 3\}\), surrounding each edge.
In general, a regular polytope represented by \(\{p, q, r,\dots, y, z\}\) has \(z\) \(\{p, q, r,\dots, y\}\) facets around every peak. A peak refers to a vertex in a polyhedron, an edge in a 4-polytope, a face in a 5-polytope, and an \((n-3)\)-face in an \(n\)-polytope.
The Schläfli symbol’s flexibility and concise representation make it an invaluable tool for describing and understanding the structures of regular polytopes and their higher-dimensional counterparts.
Crystallography and Mineralogy
This particular subsection focuses on polytopes relevant to the fields of crystallography and mineralogy. While many of the aforementioned polyhedra represent various Brillouin zones found within crystals and/or the crystal forms themselves, one might argue that this subsection could be omitted on that basis. However, the inclusion of this subsection provides a distinct emphasis and concentration on the subject matter at hand.