Sunday, October 06, 2013

IVM 1-2-3

Given how I wrote the code for these demos, spreadsheet style, with governing globals up top, it wasn't hard to stretch the spine of the book, to make the two book covers make a square, instead of a rhombus.

In the previous "book covers" video, two equilateral triangles lay flat against a plane, with a triangular "page" flapping between them.  In this one, it's two right triangles laying flat, and when the page reaches 90 degrees, the half regular octahedron shows up, each of the complementary tetrahedrons a quarter of same.

Then there's the "inadvertent tet" made from the purple and green rods, others red.  Right when the complementary tets are equal, it turns regular (they're produced together) and the "octet truss" is born (the pure IVM).

Let's be clear though:  the IVM was there last time too, with the equiangular book covers.  The regular tet's complement, the "iceberg tet" is a quarter octahedron, just like the two "iceberg tets" forming here in complement (see Fig. 987.210D).

So this time the inadvertent is the regular tet and both complements are icebergs.  Last time the IVM formed when both the inadvertent tet and one of the complements were icebergs, with the other complement a regular tet.  So two views of the same thing.  A little dance.

Here again, even with the different book covers, you have the option to pair the inadvertent tet with an iceberg (1/4 oct) to get an oblate octahedron of volume 4 + another iceberg to fill space. That's not the focus, but is a consequence of the generalization in the earlier video, that any two of the three may be chosen to build the octahedron, leaving the third tetrahedron to complete the "IVM-like" space-filling matrix.

It's not hard to see that the IVM gets to "waver" in some affine ways (to become "IVM-like").  Picture a layer of squares, like a checkerboard, then another layer above, but with its squares offset to have its corners above the others' centers.  Keep stacking that way, corners over centers, and connect each center to the four corners below.  The distance between layers is just right such that these slanted intra-layer members are also all the same length, the length of the square edges.  That's your IVM.  No shortage of squares.

Now picture the squares "wavering" to become rectangles as the distance between layers also wavers.  All the rods have become stretchy but we're keeping the layers parallel and no rods are disconnected, so the same 12 from every hub.  The familiar topology.