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 (that is, the coordinates of all points are real). However, in the space , 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 points | Real line segment | |

P-comtelon | 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 | p points | p line segments | Real P-gon | |

P-comtelal duoprism | p^{2} points
| 2p P-comtela | ||

P-comtelal duopyramid | 2p points | p^{2} line segments
| ||

Berylocomgon | 8 points | 8 tricomtela | The Möbius–Kantor configuration | |

Carbocomgon | 24 points | 16 tricomtela | ||

Dual carbocomgon | 16 points | 24 line segments | ||

Borocomgon | 24 points | 24 tricomtela | ||

Oxycomgon | 24 points | 24 tetracomtela | ||

Silicocomgon | 72 points | 48 tricomtela | ||

Dual silicocomgon | 48 points | 72 line segments | ||

Fluorocomgon | 96 points | 48 tricomtela | ||

Dual fluorocomgon | 48 points | 96 line segments | ||

Neocomgon | 96 points | 72 tetracomtela | ||

Dual neocomgon | 72 points | 96 tricomtela | ||

Calcocomgon | 120 points | 120 tricomtela | ||

Sulphocomgon | 120 points | 120 pentacomtela | ||

Scandocomgon | 360 points | 240 tricomtela | ||

Dual scandocomgon | 240 points | 360 line segments | ||

Chlorocomgon | 600 points | 240 pentacomtela | ||

Dual chlorocomgon | 240 points | 600 line segments | ||

Argocomgon | 600 points | 360 pentacomtela | ||

Dual argocomgon | 360 points | 600 tricomtela |