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NCTM logo (talking points)
Posted:
May 4, 1999 9:01 PM


The nctm.org logo shows three regular tetrahedra stacked to form a halfoctahedral void. This is the beginning of a space frame known to NASA engineers as the octet truss. Alexander Graham Bell explored this structure, as did R. Buckminster Fuller, as the Smithsonian Institution makes clear (useful American heritage which every USA citizen should know about).
You will find this structure frequently deployed in architecture, as well as in models for space stations.[1]
Early gradeschoolers will be learning that this half octa has a volume twice that of the surrounding tetrahedra, i.e. a volume of 2 if we take the tet's volume as unity.[2]
What's implicit in the logo is the possibility of a fourth tetra hedron, behind the other three, which would complete the octahedron and show it as internal to a larger tetrahedron.[3]
The octahedron's edges connect the larger tet's edge midpoints. It has a volume of 4, relative to each of the 4 unittets, meaning the total volume of the larger tetrahedron is 4+4=8.
In order to promote flexibility in thinking and emphasize the cultural underpinnings of any mathematics, students will be learning that a growing tetrahedron demonstrates third powering as effectively as does the cube.
The initial tetrahedron has a volume of 1, the larger tet with edges twice as long, has a volume of 8. Doubling the initial edge length results in an 8folding of volume. More generally, scaling edges by a factor of N scales the volume by N^3, and the area by N^2.
The tetrahedron and triangle, topologically simpler than the cube and triangle respectively, amply demonstrate this "power rule"  which applies to all shapes (shape being defined by central and surface angles, scale by modular increment along a reference edge).
Instead of saying 2^3 as "2 cubed" or 3^2 as "three squared", a Martian might say "2 tetrahedroned" or "3 triangled"  and from a mathematical point of view, this would be just as correct.
For more information re these spatial geometry number facts, you might check my memo of Feb 13, 1997 to the NCTM. That was over two years ago, of course, and a lot has transpired since then to ensure that 21st century math students are not left in the dark re these basic facts re spatial geometry.[4]
Kirby Curriculum writer Oregon Curriculum Network http://www.inetarena.com/~pdx4d/ocn/
[1] http://www.teleport.com/~pdx4d/octet.html
[2] http://www.teleport.com/~pdx4d/volumes.html
[3] http://www.teleport.com/~pdx4d/images/Tetoct.jpg
[4] http://www.teleport.com/~pdx4d/ncmtmemo.html



