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Tetravolumes
Posted:
May 11, 2009 10:22 AM


One of the standard things we teach in the newer digital mathematics curriculum is this technique of altering the unit of volume. This corresponds to analogous techniques when measuring length, as we don't want students too fixated on any particular unit.
Length may be measured in radians around a circle, or the equivalent central angular degrees. Area might be measured in unit squares, but also unit rectangles or triangles, something naturally occurring.
For example, if the area is tiled with hexagons (like a bathroom floor), just get the area of one hexagon and multiply by their number (perhaps this is given).
The unit tetrahedra we use for volume get defined at the outset in terms of balls crammed together. Four ping pong balls make a tetrahedron, with the fourth in the valley defined by the other three. Other polyhedra likewise get defined in terms of this spherepacking matrix or lattice. We tend to use the CCP (same as FCC) as our "home base" (yes, I've written about this a lot already).
Let's talk about polyhedra that "shrink wrap" a sphere, in the sense of having faces tangent thereto. In the CCP, the rhombic dodecahedron, a zonohedron, is the signature shrink wrapper, as this is the shape that fills space without gaps such that each contained sphere ends up in the CCP arrangement, i.e. is tangent to 12 other balls. Relative to the 4ball defined tetrahedron (edges going centertocenter) this shape have a volume of precisely 6 (six).
Another shrinkwrapper is the rhombic triacontahedron, a zonohedron with 30 diamond faces. The one with the same radius as a CCP sphere has a volume just a tad more than 5 which leads us to wonder what radius would give a volume of precisely 5 (using tetravolumes as usual). The answer seems ugly at first: ((sqrt(2) * (2 + sqrt(5)))/6)^(1/3).
However, we have another approach to that same shape that has more pedagogical value. We get out Jay Kappraff's 'Connections: the Geometric Bridge between Art and Science' and note how he divides his chapters around the theme of sqrt(2) versus phi, in terms of the different geometries they suggest (periodic vis aperiodic). Phi is the golden ratio here.
It turns out that a rhombic tricontahedron with a radius of phi/sqrt(2) has a volume of 7.5 tetravolumes, plus its edges intersect those of the shrink wrapping rhombic dodecahedron of volume 6. Those are easy mnemonics. Then we remind ourselves (or our students as the case may be) that to scale the linear dimensions of something without changing its shape is to change its area by a 2nd power of that scale factor, and its volume by a 3rd power (good review). Ergo, to shrink a rhombic triaconta hedron of volume 7.5 to volume 5, we need to multiply its radius by the third root of 2/3 (because 7.5 is 3/2 the volume 5).
So our previously ugly ((sqrt(2) * (2 + sqrt(5)))/6)^(1/3) now becomes (2/3)^(1/3)*(phi/sqrt(2)) which is much easier on the eyes.
As I've many times posted, this tetravolumes approach is not meant to exclude the more traditional XYZ approach of using only unit cubes. We have a conversion formula for going back and forth, just as we'd expect based on our experience with areas, lengths, temperatures, currencies i.e. it's the usual thing to have multiple options for units with ways to interconvert, no problemo.
Our volumes chart also includes the cube of volume 3 (defined by unit volume intersecting itself  tetrahedron is selfdual), the octahedron of volume 4, and the cuboctahedron of volume 20. This easy and memorable framework anchors our ball packing exercises in terms of rule defined sequences (polyhedral numbers  just like figurate numbers but in space). These sequences give an easy on ramp to our computer language (whichever one we choose). In Python, we'd use generators at this point, per my many slide shows.
Of course none of this stuff is in the textbooks, is only online. Jay's book is a trade book, not a textbook.
However, the information is important and relevant, as it connects to a lot of science and architecture, so we use the fact that it's not included in the textbooks as more evidence that textbooks aren't where the action is these days, including at the K12 level. Students are quite willing to accept this, as are parents. In moving to online resources for a lot of our best curriculum writing, we save money, invest it in computers, projectors, and the school intranet instead.
Kirby
Note: when you divide a volume 5 rhombic triacontahedron into 120 tetrahedral slivers, 60 left and 60 right, each of those will have a volume of 1/24. This is the same volume as our A and B modules, with 2 As (left *and* right) and one B (left *or* right) forming the MITE or minimum tetrahedron, the trirectangular spacefiller on page 91 of Coxeter's 'Regular Polytopes' (not called a Mite though, nor is the dissection into As and Bs part of this work). The regular tetrahedron of unit volume is made of 24 As, as we discussed in that all6thgrade assembly at Winterhaven (Portland Public). This is our flagship geek hogwarts and I was invited in to give some up to date geometry content, helping to put Winterhaven on the map as ahead of the curve (most of the curriculum is Math Learning Center materials, not the obsolete textbooks the rest of the district uses). Given all these easy fractions you'd think this would be a standard approach to spatial geometry but spatial geometry itself is nonstandard in underpowered schools (the ones that only use calculators  a thing of the past, but still prevalent, in the Lower48 especially, an educational backwater in today's global economy, with some exceptions of course (Portland rocks)).
Related reading: http://www.grunch.net/synergetics/volumes.html http://www.youtube.com/watch?v=CPNhLpeK6o



