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…

Photo on 8-5-20 at 7.49 AM.jpg

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 \Bbb{C};
also the not-quite-so-well-known (but still
essential!) inversion formula]); that guy…
and his concept of barycentric co-ordinates.
Photo on 6-18-20 at 9.26 PM 2

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—P^3(\Bbb{F}_2)
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.

Photo on 6-17-20 at 8.05 PM

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 \Bbb{H}—the quaternions
(or, if you prefer… as i do… in
\Bbb{Z}[h,i,j,k]—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.

Photo on 6-15-20 at 9.55 PM

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) \tilde= 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.

Photo on 6-3-20 at 9.17 AM

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.

Photo on 5-28-20 at 10.34 PM

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, {\Bbb F}_3^2).

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 {\Bbb F}_3^2) “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.

there’s an offer at hand for more grading
at the Bigstate U. it’s an upgrade in the
sense that i’ve been unemployed for a year.
a downgrade in the sense that it’s now an
“hourly” position and i’ll be e-filing “time-
cards” weekly.

if i can work the damn interface.

(POG)(RYB)

Photo on 11-24-15 at 6.14 PM

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.)

Photo on 11-23-15 at 6.18 PM

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.