Sunday, October 06, 2013

Space Filling Triads of Tetrahedrons

Do we say "tetrahedra" or "tetrahedrons" for the plural?  My spellchecker prefers the latter, but through long habit, I tend to use the "hedra" ending.

Tetrahedrons in the plural is what this video is about.

My technique was to code in the PyCharm IDE by JetBrains, to which I subscribe, while importing the visual package from VPython dot org.  Then I turned on QuickTimePlayer on the Apple Mac Air, which does a decent job of screen recording.

Finally, I pull that recording into iMovie and talked over it, before uploading to YouTube.  These are skills within range of a broad audience and are also increasingly the skills associated with academic studies.

David Koski provided most of the brain power in terms of providing the original insight I'm endeavoring to communicate.

What's somewhat interesting about this video is what's not shown, or what I leave out of the narration.

For example, I don't make it abundantly clear that the "inadvertent tetrahedron" with four red edges, one green and one purple, also has the very same volume as that of the two complements with which it is associated.

These three, the two complements plus the inadvertent tet, are what comprise the space-filling triad.  Any two will assemble an octahedron with two copies of each (for a volume of 4x whatever volume we're at), and the remaining tetrahedron will complete the space-filling, with a volume 1/4 that of the octahedron at all settings.

In the video, I use the term "isotropic vector matrix" somewhat loosely, as it's the topology of this simplicial complex that I'm focused on, whereas clearly not all the rods are the same length, as they are in the pure IVM scaffolding (as they are in the XYZ scaffolding).

In the IVM topology, every vertex has 12 rods emanating therefrom and tetrahedrons combine with their partner octahedrons in a ratio of 2:1 i.e. there are twice as many tetrahedrons.

Do the triangular book covers need to start out as equilateral triangles?  No.  In a future demo, I will start with 45-45-90 degree book covers lying flat to make a square and go through the same transformation.  A triumvirate of space-filling tetrahedra are made that way as well.  Indeed, we can make the pure IVM rather straightforwardly.

The demo I'm showing here does have the pure IVM within range.  When either complement is the regular tetrahedron, the inadvertent tet and complement are the same 1/4 "orange slice" of a regular octahedron (four wedges =  1 octahedron).  David and I call these wedges "icebergs".

In XYZ accounting (cube based), when the page tip is at 90 degrees, the octahedron and tetrahedron have a volume ratio of 4:1, as always, but the volume actually is 4, the tetrahedron 1.

I'm assuming red edges of 2, my value for D, the Diameter of the four unit radius spheres that might pack to create and all-red-edges tetrahedron when their centers were interconnected.

Synergetics accounts this as a model of D to the 3rd power, which is why the volume numbers differ by sqrt(9/8).  When the complements reach their highest volume at 90 degrees, that's sqrt(9/8) more than the regular tetrahedron volume (= that of its iceberg complement).