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One thing I've just read about today are complex polytopes. Polytopes are some collection of points, lines, faces and so in in the space $\mathbb{R}^X$ (that is, the coordinates of all points are real). However, in the space $\mathbb{C}^X$, you get a whole bunch of new polytopes! As a plus side, you can recover your original real polytopes by constraining your imaginary parts to 0.

Polycomtela

Just as a polytelon is two connected points on the real line, a complex polytelon is a group of connected points on a complex line. Represented on an argand plane, these will look like polygons (and become polygons, if you split the complex number into real and imaginary parts), but make no mistake: every point on a complex polytelon can be represented by a single complex number.

Name Schlafli Symbol Vertices Notes
Dicomtelon $_2 \{ \}$ 2 points Real line segment
P-comtelon $_p \{ \}$ p points

Polycomgons

A polycomgon is made up of multiple connected polycomtela. The naming scheme used identifies each one with an element, which is based off the Shephard-Todd number of its complex symmetry group.

 Name Schlafli Symbol Vertices Edges Notes P-gon $_2 \{ p \} _2$ p points p line segments Real P-gon P-comtelal duoprism $_p \{ 4 \} _2$ p2 points 2p P-comtela P-comtelal duopyramid $_2 \{ 4 \} _p$ 2p points p2 line segments Berylocomgon $_3 \{ 3 \} _3$ 8 points 8 tricomtela The Möbius–Kantor configuration Carbocomgon $_3 \{6\} _2$ 24 points 16 tricomtela Dual carbocomgon $_2 \{6\} _3$ 16 points 24 line segments Borocomgon $_3 \{4\} _3$ 24 points 24 tricomtela Oxycomgon $_4 \{3\} _4$ 24 points 24 tetracomtela Silicocomgon $_3 \{8\} _2$ 72 points 48 tricomtela Dual silicocomgon $_2 \{8\} _3$ 48 points 72 line segments Fluorocomgon $_4 \{6\} _2$ 96 points 48 tricomtela Dual fluorocomgon $_2 \{6\} _4$ 48 points 96 line segments Neocomgon $_4 \{4\} _3$ 96 points 72 tetracomtela Dual neocomgon $_3 \{4\} _4$ 72 points 96 tricomtela Calcocomgon $_3 \{5\} _3$ 120 points 120 tricomtela Sulphocomgon $_5 \{3\} _5$ 120 points 120 pentacomtela Scandocomgon $_3 \{10\} _2$ 360 points 240 tricomtela Dual scandocomgon $_2 \{10\} _3$ 240 points 360 line segments Chlorocomgon $_5 \{6\} _2$ 600 points 240 pentacomtela Dual chlorocomgon $_2 \{6\} _5$ 240 points 600 line segments Argocomgon $_5 \{4\} _3$ 600 points 360 pentacomtela Dual argocomgon $_3 \{4\} _5$ 360 points 600 tricomtela