u/BadJimo

A cevian is a line that connects the vertex of a tetrahedron to somewhere on the opposite face of a tetrahedron. The somewhere is usually a mathematically interesting point such as the centre of the face. Now something interesting about triangles is that there are many different types of centre of a triangle. Here (not made by me; just found in a Google search) is a wonderful Desmos graph that shows the 10 most interesting/useful centres of a triangle. I will use the Gergonne triangle centre in this project (which I wasn't aware of until today).

Mostly, the cevians of an irregular tetrahedron do not intersect. I wanted to make a tetrahedron where the cevians do intersect.

Apparently if the tetrahedron is an inspherical tetrahedron then the cevians that extend from the Gergonne centre of each face of the tetrahedron to the opposite vertex do intersect.

It turns out that if you make a tetrahedron with a sphere centred at each vertex and the spheres are tangent to each other (as I made recently in another project) then this is the right kind of tetrahedron. Yay.

Anyway, here is a link to the graph (the green dots are at the Gergonne centres):

https://www.desmos.com/3d/ngyznuch9u

I recently discovered the marvelous Van Aubel's theorem. If you have a quadrilateral with a square on each edge, then the line segments from centres of opposite squares will be perpendicular and the same length.

I wondered if there was a 3D extension of Van Aubel's theorem. I couldn't find one, so I made one. My first thought was a cube at every face of a hexahedron, but obviously that won't work. So instead I thought tangent spheres are kind of similar.

Start with 6 spheres, each sphere tangent to 3 adjacent spheres. That is, spheres at the vertices of a hexahedron (think a distorted cube).

Then add a sphere tangent to each group of 4 spheres. The connect the centres of these opposite spheres to make three lines, and voilà, the three lines are orthogonal and two are the same length... under some conditions. Specifically, only when the 'centre of inversion' is only modified in one axis. I used the spherical inversion technique I mentioned in my previous post, but this time using a cube.

When the 'centre of inversion' is modified in two axes, you still have two of the lines being orthogonal.

I've called it Van Baubel's theorem because, well, I think you can figure it out.

https://www.desmos.com/3d/letm5p0sfh

Four tangent (kissing) spheres.

https://www.desmos.com/3d/0fw8xdiedu

I started with a regular tetrahedron with spheres of the same size at the vertices and then applied a 3D Möbius transformation known as Spherical Inversion.

​A spherical inversion turns 3D space inside-out through a "lens" (a sphere of inversion). Things close to the lens get blown up and pushed far away, while things far away get shrunk and pulled inside.

​An important property of this transformation is that it preserves tangency. If two spheres kiss before the inversion, they will kiss after the inversion.

To change the sizes of the sphere you move the "Center of Inversion" (using the sliders o,p,q).

I realized a corner cube reflector only requires a single line to define the ray of light.

https://www.desmos.com/3d/3vb2nni5j4