## Archive for the ‘Finite Geometry’ 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…

there’s four directions on the map.

i’ve called ’em Up, Down, Equal, and Op—

{U, D, E, O} more affectionately (or when

actually *writing things down*).

never mind why for now; these are just their

names. call ’em table & beermug if you like.

anyhow, the title of this display is “barycentrics”.

it owes this name to the great a.~f.~möbius

(he of the immensely famous non-orientable surface

[and the merely very-famous transformations of ;

also the not-quite-so-well-known (but still

essential!) inversion formula]); that guy…

and his concept of barycentric co-ordinates.

the drawing underlying all this mess was done

freehand by me a few years ago. the idea was

to be sure all 1+2+4+8 points of the tetrahedron

in question— if you must know—

were distinguishable one-from-another. you can

easily look up similar drawings in textbooks and

so on.

anyhow, here the face-centers (of the tetra) are labelled

Yellow, Blue, Red, and Mud (or {Y, B, R, M}—

you know the drill—); the vertices opposite

these points are the “secondaries”

Purple, Orange, Green, and Neuter.

the “four directions” (U, D, E, & O) then correspond

to the (opposite) color-pairs Y-P, B-O, R-G, & M-N.

i hope this is all completely obvious from the drawing.

because it’s very useful for the math.

here’s A-four-hat three ways.

but really *four* ways; like we agreed upthread,

the trit-string version is inherent in the very

positioning of the table entries, to wit.

consider

-+___++

– -___+-

—the “quadrants” of beginning algebra…

iterate: each of the three “versions” of our group

has each of its entries in one of the (16) “positions”

-+-+___-+++___++-+___++++

-+- -___-++-___++- -___+++-

– – -+___- -++___+- -+___+-++

– – – -___- -+-___+- – -___+-+-

; now just remember that, e.g.

“+- -+” in this context means

(1-i-j+k)/2—a “hurwitz unit”

in —the quaternions

(or, if you prefer… as i do… in

—the *integral*

quaternions). where was i.

the matrix notation is “mod 3”;

the generators-and-relators version

requires one to work with “relators”

like “hi = jh”—(this is, like, the

very *textbook example* of a

“semi-direct product”, if you want

my opinion… anyhow, this is quite

close to the actual way *i* actually

got it if i can be said to have it now]).

finally, the “permutation notation” version

is very much the easiest to work with (and you

should learn right away how to work with these

if you haven’t already; i had to be dragged

slowly and painfully into accepting this stuff

but maybe you’ll be one of the lucky one in

a million): one readily sees which elements

have order six, for example.

anyhow, this is one of the coolest things i ever

put on one sheet of paper or so it seems to me now.

the seven black triangles are

the blends

Mud Yellow Purple

Mud Red Green

Mud Blue Orange

the blurs

Yellow Blue Green

Yellow Red Orange

Blue Red Purple

and

the ideal

Purple Orange Green.

the “theorem” in question is then that

when the “colors” MRBGPYO are arranged

symmetrically (in this order) around a circle

(the “vertices”of a “heptagon”, if you wanna

go all technical), these Color Triples will

each form a 1-2-4 triangle.

but wait a minute, there, vlorb. what the devil

is a 1-2-4 triangle. well, as shown on the “ideal”

triple (center bottom), the angles formed by these

triangles have the ratios 1:2:4. stay after class

if you wanna hear about the law of sines.

note here that a 1-4-2 triangle is another beast altogether.

handedness counts. (but only to ten… sorry about that.)

anyhow, then you can do group theory. fano plane.

th’ simple group of order 168. stuff like that.

all well known before i came and tried to take

the credit for the coloring-book approach.

with, so far anyway, no priority disputes.

okay then.

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.

the left-hand photo shows

a nine-point plane: an “ordinary

two-dimensional plane” over the

field with three elements (and its

label is, therefore, ).

such a plane is ordinarily co-ordinatized as

(0,2) (1,2) (2,2)

(0,1) (1,1) (2,1)

(0,0) (1,0) (2,0):

the set of (x,y) such that

x & y are both elements of

the set {0, 1, 2}.

one could convey the same information more

concisely as

02 12 22

01 11 21

00 10 20.

it’s useful for our purpose here, however,

to consider our plane as belonging to a

*three*-dimensional space… (x, y, z)-

-space, let’s say… and as having a

*non-zero* “third” (*i.e.*, “z”)-co-ordinate.

thus, in the photo, our plane is represented by

021 121 221

011 111 211

001 101 201.

the colors come into play in displaying the

solution-sets for various (linear) equations.

the reader can easily verify that the Green

equation—x=2— is “true” for the points of

the vertical line at the right… *i.e.*, for

{ (2,0), (2,1), (2,2) } (old-school), *i.e.* for

{ 201, 211, 221} (“our” version).

likewise “y=x” (the Red equation) gives us

{(0,0), (1,1) (2,2)}… *i.e.*,the “Red line”

{001, 111, 221}.

now for some high-theory. by Algebra I, one has

a well-developed theory of Lines (in the co-ordinate

Plane). the usual approach there is to use the

(so-called) Slopes. the (allegedly intuitive) notion

of “rise over run” allows one to calculate—for any

*nonvertical* line—a number called the Slope (of that

line). vertical lines are said to have “undefined”

slopes. one might also say that they have an “infinite”

slope… though this invites confusion and is usually

best left unmentioned.

y = Mx + B

x = K

are then our “generic” *equations of a line*.

any particular choice of numbers M & B will

correspond to the a set of solutions lying

along a (nonvertical) line having the slope

of M (an passing through (0,B)… the so-

-called “y-intercept” of the line); each vertical

line (likewise) is represented by some particular

choice of K.

now. having different “forms” for vertical and for

nonvertical lines can be devilishly inconvenient,

so, also in algebra I, one sometimes instead uses

the “general form” for an equation of a line in the plane:

Ax + By = K

(with A & B not both zero).

likewise (but typically *not* in algebra-i)

Ax + By + Cz = K

(with A, B, & C not all zero)

is the equation of a *plane* in *three*-dimensions.

thus far i’ve been as precise as i know how.

but now i’m going to start waving my hands around

and making leaps-of-faith all over the place.

in the second photo, four new “points” have been

added into our framework (namely

{010, 120, 110, 100}—the “line at infinity”).

the upshot… without the details… of this move is

that one now has an algebraic theory of “Lines” in a

“Plane” containing precisely our 13 “Points”. more-

over, this theory is “structurally” very similar to

“ordinary” linear theory. in particular, we dealing

with solutions to

Ax + By + Cz = 0—

the “K=0” case of the “general form” for (the 3D case

of the “ordinary” theory).

the Green equation—which must now be written without

its “constant term” (x = 2 is “the K=2 case” of x = K)—

becomes x – 2z = 0;

similarly, rather than (the three-point “line”

of ) “y = 1” (concentric black-and-

-white circles), the (“homogeneous”—for us, right now,

this can be taken as meaning “having no constant term”)

equation is “y – z = 0” (and, again, we pick up a “new”

point at 100).

when the smoke clears… which won’t be here and now…

we’ll have a *very nice* geometry. just as in “ordinary”

space, two distinct points determine a unique line.

but… *unlike* “ordinary” space, it’s also true that

(*any*) two lines determine a unique point.

the same “trick”—homogeneous co-ordinates in finite

fields—converts any plane having p^2 as its number

of points to a *projective* plane having

p^2 + p^1 + p^0

as its number of points. thus there are PP’s having

7, 13, 31 = 5^2 + 5 +1, 57 = 7^2 + 7 + 1, ….

as their number-of-points. there are also some others.

but the margin is too small.

that 7-space has seven-way symmetry is obvious

(rotate the drawing by 2\pi/7—about 51.43\degrees—).

but the *three*-way symmetry isn’t so obvious.

here it is in its 7-color wonder.

we’ve chosen to fix the “Mud” point at the top.

we then chose the “secondaries” line

{Green, Purple, Orange} and permuted;

the “primaries” permute accordingly;

voila.

*************************************************

a 2-way symmetry can be displayed by “swapping”

each primary with its “opposite”:

(RG)(BO)(YP).

i call this one “reflection in the Mud”.

(one may also… of course… reflect

in any of the other colors;

for each (there are three)

“line” through a given color,

interchange the positions for

the other two colors.

the lines-on-blue are

{bgy, bmo, bpr}, so

“reflection in the Blue” has

(GY)(MO)(PR)

as its permutation-notation.)

or, the fano plane presented symmetrically.

each of the three triangle-edges

found along any of the “long lines”

(joining vertex-to-vertex

on the biggest 7-point “star”)

is a “line” of rainbow-space.

check it out. the “points” are

Mud Red Blue Green Purple Yellow Orange

the lines are MRG, RBP, BGY, GPO, PYM, YOR, OMB.

here’s a version from last year.

this new one’s much cooler.