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Gnu Math Construction
Posted:
Jul 25, 2006 5:11 PM


NOTE TO READER: these lesson plans or fragments thereof were logged to mathteach in service of our various computer labs scattered about [waving], currently showcasing alternatives to some of the more established K12 curriculum models (e.g. Chicago's, Saxon's and/or Singapore's).
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Picture the XYZ coordinating apparatus, consisting of three intersecting square fragments [1], namely: XY, YZ and XZ.
Consider the diamond face of a CCP ball casing, our old friend, and friend of Kepler, the spacefilling rhombic dodecahedron. Apply that diamond in plane XY with its center at the origin (0,0,0) and with a long diagonal stretching along the X axis.
Now take another copy of the diamond and position it at 90 degrees to the first in the YZ plane, such that its short diagonal fuses together with the first one's, and its long diagonal runs in the (Z,Z) direction.
You now have four pointy devices with sharper tips on the directed vectors X, X, Z, Z, plus two less sharp points at Y and Y, for six corners in all.
The sharper tips all interconnect around an equator to form the square (X, Z, X, Z)  moving clockwise as viewed from the positive Y camera and pointing back at (0,0,0).
The square so formed may be considered the interface between neighboring IVM cubes, we find a neighboring cube on each side. These have volume 3 apeice.
The 3equator octahedron we've just constructed, named the Coupler, has a volume of 1. Each cube contains a halfcoupler, apexing at the cubes' centers, to fill one sixth its total volume (1/6 of 3 = 1/2).
The diamond faces we started with may likewise be considered interfaces, this time between two ball casings, voronoi cells in the CCP packing. The line from ball center to neighboring ball center is right through (0,0,0), the K point where they kiss.
These rhombic dodecahedral ball casements each have volume 6, twice the IVM cube's. The volume 3 cube inscribes as its short diagonals, the volume 4 octahedron as its long ones (we often use the colors green and red respectively, if trying to color coordinate consistently  blue for the rhombic dodecahedra).
Note that we so far have two notions of the halfcoupler:
(a) slicing to provide a square interface of diagonal 2R (ballcenter to ballcenter control length), and
(b) slicing to provide a diamond interface of long diagonal 2R (= IVM octahedron edge), short diagonal 2nd root of 2 (= IVM cube edge).
We gave two ways to slice to the latter (two diamond equators), only one way to slice to the former (one square equator).
There's a 2nd implied square however, orthogonal to the first, with face diagonals 2nd root of 2.
This time the plane of slicing would correspond to the square defined by the neighboring IVM cube centers (Y and Y in this construction), and opposite midedges on their Coupler's square interface (e.g. at (X,Z)/2 and (Z,X)/2 or else at (X,Z)/2 and (X,Z)/2).
So that's two more cuts through the Coupler for a total of five so far:
(i) *two* diamonds with unequal diagonals, orthogonal to one another,
(ii) *one* 2Rdiagonal square,
(iii) *two* 2ndrootof2R diagonal squares, also mutually orthogonal
(note: R = CCP ball radius).
Every such cut through the Coupler leaves volume 1/2 on either side.
Consider how this nonregular octahedron or Coupler, centered at (0,0,0), fragments into eight tetrahedra, each in an XYZ octant, each with a 909090 degree angle at the XYZ origin.
These spacefilling MITEs (MInimum TEtrahedra) of volume 1/8 in turn fragment into our old friends the A & B Modules (left and right handed, each of volume 1/24), per lesson plans already filed.[2] MITE = (A,A,B), +MITE = (B,A,A), although outwardly they're indistinguishable.
Possible lesson:
Have students storyboard the above narrative as a sequence of comic book frames. Crude drawing is OK, i.e. we're not aiming for graphical perfection, although gifted students are welcome to show off.
Optionally transition to a computer graphics approach for those wishing to visualize in this way e.g. using Python + POVRay (standard in most computer labs).
Testing objective:
Students should be familiar with the various edge lengths and angles associated with this storyboard, given CCP ballcentertoballcenter as either 2R (2 radii) or 1D (1 diameter) as the case may be.
Commercial objective:
More interesting open source DVD clips for our collection.
End notes:
[1] Euclidean view: "...of infinitely extended, infinitely thin planes"  optional greek metaphysics. Staying with a more finite/discrete Democrituslike approach is also OK, even encouraged. Emphasize to students that geometry is not about studying *just one* set of axioms.
[2] e.g. search here in mathteach for more links to rational number arithmetic, based on A & B module assemblies.
Please submit errata to kirby.urner@gmail.com, subject line: [mathteach errata]: whatever. Please include the URL and a description of the error.
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