## 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) GL(3,2)?(ezra brown & nicholas loehr;

*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.