Primary references:
http://www.inetarena.com/~pdx4d/ocn/
http://www.inetarena.com/~pdx4d/ocn/calculus2.html
http://www.teleport.com/~pdx4d/makeover1.html
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Philosophy for young people (curriculum notes):
The focus here is getting "beyond flatland" early, meaning
returning to those "scribing in sand" demonstrations by coming
to "Euclid's world" from space [1], i.e. from spatial geometry,
wherein surfaces are clearly aspects of volumetric topologies.
Classes focus on Sun->Earth economy with energy transfer from
fusion to "big cycle" weather/climate and biomass formation
(e.g. agriculture, wilderness). The geometry of electromagnetic
waves linked to Archimedes screw (irrigation) as precursor to
latter phase-in of cross-product (div). Field trip to local
alternator farm (e.g. Bonneville dam) if feasible.
Use Earth as link to spherical geometry, projecting polyhedra
from inside as "shadow-castings" e.g. spherical tetrahedron
gives 4 equal regions on the surface, each of area equal to
spherical cross-section.[2] Projecting icosahedron gives 20
spherical triangles setting stage for world map projection and
overview of resources (emphasize sufficiency, if intelligence
in the wings).[3]
Zoom in on a beach in Greece and show ancient greek
philosophers with ruler and compass using 'Elements'-style
postulates and constructing theorems (do some). Relate
"points" to grains of sand, "plane" to sandy beach. Emphasize
locally flat aspect of Earth, given relatively tiny dimensions
of humans, and Greek propensity to use post-and-lintel (e.g.
Parthenon) as precursor of western civ fixation on "orthogonal"
as "normal" even though it isn't, outside human designs.[4]
De-emphasize points as "zero dimensional" or planes as
"infinitely expansive" (important for historical context and
backward compatibility, but not for drilling).[5]
Zoom in further on beach sand and idealize as close-packed
spheres. Explore packing 12 around 1 (2 ways) one way giving
cuboctahedron. Use to set stage for concentric hierarchy w/
tetrahedral and octahedral voids in 2:1 ratio (population)
and 1:4 volume. Cuboctahedron = 8 tetrahedra + 6 1/2 octahedra
for volume 20. Jitterbug as bridge to 5-fold (e.g. icosahedron,
rhombic triacontahedron). Do volumes again with cube as unit
and compare. Discuss cube as model of 3rd powering. Do again
with tetrahedron as 3rd powering model. Emphasize philosophy
of math as self-consistent games, some tightly coupled, some
relatively isolated.[6]
Regarding axioms, note their definitional nature without
dwelling too much on "self-evidence" (not really a necessary
stipulation). Look at "real numbers" as "rationals" plus
"irrationals" then revisit through eyes of another culture,
e.g. a so-called "Persian math" with different categories:
symbols as "terminal" versus "algorithmic" (bears a family
resemblance to "rational" and "irrational", certainly).[7]
Discuss definitions as cultural artifacts, not "cast in stone
for all time" (perhaps concluding with Ozymandius poem).[8]
NOTES:
[1] picturing from Al Gore's L1 webcam for example
[2] Exercise: 8000 miles diameter w/ pi r^2 gives approx
16,000,000 x 3 = 48,000,000. Earth surface area roughly
48,000,000 x 4 = 192,000,000 square miles.
[3] computer algorithm by Robert Gray, global data
from WorldSat http://www.teleport.com/~pdx4d/dymax.html
[4] sections of virus, buckyballs (C60), stickyballs
(C36) relevant (good intro to 5-fold symmetry segment)
[5] genesis of '3D' might be retold using the tetrahedron: show
3-edge zigzag "snake" defines complement, and therefore concept
of volume. http://members.xoom.com/Urner/images/genesis3d.gif
[6] [Aldous Huxley quote in Stuart Kaufmann book goes here]
[7] http://members.xoom.com/Urner/images/persian.gif
[8] http://www.lgmicro.com/ozymandius.htm
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