Archive for the ‘Projective Spaces’ Category
almost-symmetric desargues’ theorem.
(7-color version)
5 “flat” and 5 “tall” triangles
arranged as the 10 “intersection points”
of a (so-called) pentagram.
thus far the black-and-white “underlying
diagram” (which i probably should have
photographed before coloring it in and
erasing it… it was the best b&w version
i’ve done so far, i think… oh well…).
the b&w *already* “tells the story” (if you
know how to look): the ten “triangles” are
in the ten “positions” on the “big picture”
diagram in such a way that, mutatis mutandis
(read “line” for “triangle” & “point” for
“position, to wit), one has the well-known
points-&-lines “duality” exhibited in
purely *graphical* form: not only a
“lecture without words”, but a “lecture”,
if you will, without *code itself*.
so it’d be a pretty awesome picture, done right,
&’d make a good art project for somebody patient
enough to make actual measurements & stuff.
as would the colorized version.
so, briefly.
the letters {a, b, … , j}
are posted in “positions” in such a way
that the when the “triangles”
{abc, ade, afg, bdh, … , ijk}
are “placed in” the appropriate positions
(as is done here)
one has each letter-triple in the position
of its *dual* letter (“a” to “ijk”, eg.;
one should probably look at some other
diagram at about this point).
now replace all the letters with colors
according to the scheme pictured at right.
(whose connection with the “fano plane”
[& “vlorbik’s 7 color theorem”] is solid
but we don’d have to go into it here.)
happy neighborhood of yr b’day.
thanks for your part in, you know,
teaching me to read & bringing me
back from the dead and all that.
here’s your “virtual lanyard”.
PS the *actually* symmetric version has
the ten “lines” modeled as *diagonals
of an dodecahedron* (the “diameters”
connecting opposite vertices; there
are of course 20 such…);
the “asymmetric” (textbook) version
has a *marked* point (and, of course,
a “marked” dual line…); never mind…
the color-scheme is inspired by one-or-the-other of
a hyperbolic plane coloring & the simple group
of order 168 (dana mackenzie; monthly of 10/95)
or
why is PSL(2,7) 
monthly of 10/09)…
okay, it was the mackenzie. but i want you
to look ’em both up. the brown-loehr i’ve
known longer and studied more. anyhow, enough
about the actual math. more about me.
the bits *not* in color show “the desargues
configuration”… the triangle-lookin things
somehow are supposed to depict the version
where the “lines” (sets-of-three “points”)
of the configuration are made to coincide
with triples-of-faces on an icosahedron.
it’s one of the coolest things i know.
there are versions somewhere colorized.
the tetrahedral group at left: A_4, to the group-theory geeks.
up top, the “yrb” labeling of the vertices of a cube,
with the bit-string digital code and a 2-D projection.
the seven-color theorem… concerning the simple group
of order 168 & MRBGPYO… is hinted at.
under that, as one can *kind of* read on the blurry photo,
is “desargues theorem in color” — ten “points”
(one Mud, two Yellow, two Blue, two Red, one Green,
one Purple, one Orange) in abstract “space”.
the best version… i’m not technically up to drawing it…
is to put the colors on the ten diagonals of a dodecahedron.
next best is the five-point star version taking up
the biggest part of the file-folder.
next to that on the right: the vertices of the cube
colorized again. pretty much the same way if memory serves.
the points-to-lines “duality” is colorized better here, i think.
at the bottom, several versions of the seven-point star version
of the MRBGPYO theorem… and other stuff about heptagons.
there’re some books in there, too. that’s it for today.
i busted the glass on this picture-frame
half an hour ago. the seven-color doodle—
one of a long series of such—is months if
not years old. the damn thing was just
sitting there in a closet full of junk
not even pretending to be displayed in
a safe place somewhere. out it goes now
obviously.
so here’s how to memorize the alphabetic
version of “ten point reverse tic-tac-toe”.
just learn to draw half an icosa and fill
in the letters: A in the middle, BC,
DE, FG around the edge, and HIJ
back the other way. simple.
(the “poles” are then in the rather
disturbing order JIHGEFDCBA.
something fishy here.)
with “pairs of opposite faces” of an icosahedron as “points”, the ten “lines” of the
desargues configuration have the
very pleasant form seen here.
stand by for another post on this.
ten poles, ten polars,
and ten pairs-of-triangles:
ten ways to use one drawing
(from MathEdZine, of course)
to illustrate one theorem.
(desargues’ theorem at w’edia.)
performing calculations using
actual blobs of color rather than
alphabetical symbols *standing*
for colors is time-consuming
(and demanding of special tools)…
but one literally “sees” certain things
*much more readily* than with
symbol manipulation.
hey, i’m a visual learner.
the three directions associated
with “arrows’… 001, 010, and 100
in the binary code… are here
colored yellow, red, and blue
(it didn’t reproduce as well as
one would’ve liked).
these so-called “primary” colors
(for pigments; for colored lights
the rules are different) blend
along the appropriate cross-sections
of the diagram (the easy-to-see
ones with the arrows attached)…
*or* the sides of the triangle:
yellow and red form orange,
yellow and blue form green,
red and blue form purple.
blending *opposite* colors…
the primary/”secondary” pairs
yellow/purple, red/green, blue/orange…
gives a muddy brown (“mahogany”
sez crayola) and again the appropriate
cross-sections of the cube (and lines
of the “fano plane” triangle) associate
the appropriate “blends” with the
pairs-of-pigments combining to form them.
finally, there’s the weird cross-section
(the 3-5-6 arc in the fano diagram)
consisting of all three secondaries.
on the cube diagram, this appears
as a tetrahedron-through-the-origin
rather than a plane-through-the-origin.
colored pencils (with erasers) rock.
we now return to zeerox-friendly B&W.
the livingston review 