Thursday, October 24, 2013

Book of Rhombs

Glenn Stockton was regaling me with stories this morning, as we hiked around Mt. Tabor, stemming from all the Megalithic Math he's been studying.  He's been devouring Keith Critchlow's new book Time Stands Still.  At one point in our conversation he mentioned finding it very British to hear diamonds described as "rhombs" as if this latter word were so familiar.

Meanwhile, David Koski has been pushing this triangular book covers demo from several angles.  Start with any rhombus really, but some have more interesting properties.  We started with the two book covers being equilateral triangles of edges D, then right triangles with edges D, and now, in this latest video, the long diagonal of the rhomb is D, while the short diagonal is sqrt(2).

This D is the diameter of the unit-radius sphere.

I'd actually written quite a bit about these two rhombs defining a Coupler when placed at 90 degrees, but it took David's nudging for me to finally realize I was again covering this same territory, now with the "triangular book covers and two oppositely flapping pages".  Putting the Coupler at the XYZ origin is a great way to build a bridge to the IVM and Synergetics way of thinking more generally.

In massaging the source code for this demo, I realized that my code for the inadvertent tetrahedron was hard coding around all edges being D except the green and magenta, so needed to fix that for this video to have the right volumes.

Towards the end, I start mentioning the Rite, though it might not be clear that's actually the name of a specific tetrahedron.  The Rite and quarter Rite are both space-filling tetrahedrons.  Aristotle said tetrahedrons fill space and is often criticized on the theory he meant regular tetrahedrons.  However irregular tetrahedrons do fill allspace with identical copies of themselves and without left and right handedness, the Rite is one of these, as is the Mite.

To recap a theme of the last three "triangular book covers" videos:  the flapping triangular page defines two equal volumes, with a 3rd "inadvertent tet", again of equal volume, supplying a space-filling complement to the other two.  Indeed, any two of the three tetrahedrons formed, may be used to build an octahedron (two and two needed), with the third tetrahedron playing the role of the complementary space-filler ala the isotropic vector matrix model, but skewed and/or stretched (same topology).

In this case, starting with the rhombus of the rhombic dodecahedron, when the page is at 90 degrees, all three tetrahedrons are Rites and the octahedron formed by any two is the Coupler, of unit volume in Synergetics.

The rhombic triacontahedron hovers as tantalizingly relevant.  A next video might get into five-fold symmetric space-filling more, David's forte.  The page tip needs to click stop at "4/8" on the way to its vertical at 9/8, where 8/8 is the regular tetrahedron.  Length-determining volumes are the 2nd roots of these fractions.  That's back to when our rhombic book has edges 2 (i.e. D).

Link to source code on Github.