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Topic: Tetravolumes
Replies: 1   Last Post: May 11, 2009 11:48 AM

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Kirby Urner

Posts: 4,713
Registered: 12/6/04
Posted: May 11, 2009 10:22 AM
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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 sphere-packing 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 4-ball defined tetrahedron (edges
going center-to-center) this shape have a volume of
precisely 6 (six).

Another shrink-wrapper 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 inter-convert, no problemo.

Our volumes chart also includes the cube of volume 3
(defined by unit volume intersecting itself -- tetrahedron
is self-dual), 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
on-line. 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 K-12 level. Students are quite
willing to accept this, as are parents. In moving to
on-line resources for a lot of our best curriculum
writing, we save money, invest it in computers,
projectors, and the school intranet instead.


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 tri-rectangular space-filler 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 all-6th-grade
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 non-standard 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:

Date Subject Author
Read Tetravolumes
Kirby Urner
Read Re: Tetravolumes
Kirby Urner

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