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Topic: NCTM logo (talking points)
Replies: 1   Last Post: Aug 25, 2002 11:36 PM

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Kirby Urner

Posts: 803
Registered: 12/4/04
NCTM logo (talking points)
Posted: May 4, 1999 9:01 PM
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The nctm.org logo shows three regular tetrahedra stacked to form a
half-octahedral 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 mid-points.
It has a volume of 4, relative to each of the 4 unit-tets, 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 8-folding 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/Tet-oct.jpg

[4] http://www.teleport.com/~pdx4d/ncmtmemo.html





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