# Geometry : Non-composite polyhedra with convex regular faces

*Created on June 20, 2024. Last update on August 03, 2024.*

/!\ THIS IS A WORK IN PROGRESS /!\

With this website, I am trying to share my hobby on polyhedra through making interactive viewers but my main topic of interest, my personal quest, is to find all non-composite non-self-intersecting (convex and non-convex) regular-faced polyhedra. To me, this is the category that lies after the (convex but not strictly) regular-faced polyhedra with conditional edges and after the regular-faced polyhedra with parquet faces (or conditional vertices). It simply lets loose of the convexity constraints. This quest is infinite but I am exploring it, one step at a time. By the way, I never displayed any solid with conditional vertices on this website, this is a task for future me, good luck my friend.

The first idea is to reduce the numbers of solids concerned by this search. The idea developed by Zalgaller is to only look for non-composite solids. This website explains it all, though in my case they are non-convex most of the time. The second idea is to start looking for solids by their vertex number. The difficulty is not to miss any in the process so I decided to base my search on nets, filter out some that are not polyhedral nets then try to embed the rest in 3D and try to solve for the vertices positions so that the faces are flat and regular, and the solid itself does not self-intersects. In the end, there is still a lot of manual checking and discarding of wrong ones, sometimes even retrieving good ones from the garbage. The whole process is far for being proof safe.

What counts as self intersecting ?

- one surface goes through another : forbidden

- cube excavated with a tri prism (dihedral angle is zero) : forbidden

- ring of 8 cubes ? okay since the vertices are at the same position but the net is different

- higher genus are okay

What about conformity ? the plan is not to find them all but at least one of them. Moreover, if a conformal version is a composite solid then maybe we should not count it. The alternative is to count all the conformal versions.

- done: all the ones with v=4 to 11 + zalgaller

- todo: what about ivanov q1-6 ? and stewart toroids ? v=12+ ? the ones fromSupermag + other sections ?

- should the dodecahedron no longer be considered a non composite solids ? (can be made with stewarts 3)

The following table sums up my findings for the first few vertex numbers. Values with a > are incomplete.

Each model view is interactive, the controls are the following:

- Click and drag with the mouse right click to rotate the model.
- Click and drag with the mouse left click to pan the model.
- The mouse middle click zooms in or out.
- Press R to reset the model position or triple click.
- Press F to trigger fullscreen or double click.
- Press P to pause/resume the animation.

## 4 vertices

## 5 vertices

## 6 vertices

## 7 vertices

None