Friday, March 26, 2021

Cultural Illiteracy?

Despite the two volume Synergetics (Macmillan) from the late 1970s, the Wikipedia page on the Tetrahedron has no links to Bucky Fuller's philosophy as of March, 2021.  

Why this would seem an omission is that, alone among the philosophers, Fuller undertook to recast the regular tetrahedron as his unit of volume, even while preserving edges one.

Here's a volume formula, derived subsequently to Synergetics' publication, by Gerald de Jong, which takes the six edges of any tetrahedron and returns its volume in terms of "tetravolumes".  

The regular tetrahedron of edges 1 has volume 1.  The regular tetrahedron of edges 2 has volume 8.  

However, irregular tetrahedrons have tetravolumes too.  Indeed, Fuller defined a common conversion constant for going back and forth between cubic and tetrahedral volumes, named S3.

One discovery stemming from this use of the tetrahedron, is the result that any tetrahedron with all four vertexes at the centers of spheres in a CCP packing, has a whole number volume.  

Likewise, any Waterman Polyhedron* turns out to have a whole number volume as well.

Fuller twisted our customary concept of "dimension" around to make it mean something in terms of the tetrahedron, which is the topologically minimum volume in terms of V, F and E (vertexes, faces, edges).  

He described res extensa (the physio-spatial world), as well as res mensa (the imaginary world), as "4D" meaning primitively and minimally a container with four facets and four corners.

Why don't I add a link from Tetrahedron to Bucky Fuller myself?  I concede that Tetrahedron page to others, having learned the hard way (e.g. Math Forum) what wars to skip. I did add a link to Tetrahedron from Quadray Coordinates today, and likewise to Caltrop.

* defined as a maximal convex polyhedron centered at a CCP ball with its vertexes defined as all those CCP centers at radius R or less.