## Conway's 3D Mandala

Instructions

Up/Down Right/Left arrows and a/z keys rotate the object along the x,y,z axis.

I once had the great pleasure of meeting the legendary mathematician John Conway at a Graph Theory symposium. I asked him a question that had been on my mind for a while: is it possible to arrange n points in 3D in such a way that the complete graph Kn (joining every vertex to every other with edges) has no edge-collisions. He thought for a second and gave a lovely inductive proof (*) showing that it is indeed possible to do so, adding his famous phrase "there is a lot of room in 3D". When asked what this configuration might be, he said he did not know but would think about it. We never met again but this demo is my solution to the collision-free Kn configuration puzzle: Create one cycle (360-degree turn) of a spiral and evenly distribute n points on it. Join every point to every other point. I believe in this extendable configuration no two edges will ever intersect. In other words on the spiral, no four points are ever coplanar and every point is visible from every other point. Which may be one of the reasons why the helix shape is so favored in nature.

(*) Conway's inductive proof went something like this:
Base case- we can clearly join 3 points without edge-collisions in 3D.
Inductive case- Assume that we already created a configuration where m points have been arranged in such a way so that there are no edge-collisions. We want to place the m+1 th point in such a way so that edges from it to all the other m points will not collide. That is this last point will not be coplanar with any other 3 points. With the existing m points we can create mC3 (m choose 3) planes. If the m+1 th point is on any of these planes there will be an edge-collision. Therefore, we must not place this point on any of these planes. 3D has "a lot of room" -infinitely many locations-so we will be able to find a location off these mC3 planes.
Thus a collision-free configuration for the complete graph Kn exists.

Notes and Credits

My thanks to legendary mathematician John Conway for this idea.

The 3D interface is a based on a projection algorithm I developed some years ago to make Geometer's Sketchpad display 3D curves.
I learned a tremendous amount from @Mr_Pyro_ 3D Wireframe Engine project about optimizing rendering and keyboard effects.

Shared: 27 May 2016 Modified: 15 Feb 2017

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