Archive for the ‘Finite Geometry’ Category

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.

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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.

zoom

Photo on 11-17-15 at 10.44 AM

while you still can.

Photo on 11-17-15 at 10.43 AM

consider ordinary (x,y,z) space.
co-ordinatize a “unit cube” in the all-positive octant.
put “primary colors” on the axes. next slide.

from the edges of the cube, “cut out” the three that
pass through the Origin of our system—i.e., (0,0,0).
next.

distort the resulting diagram so that the “top face”
(and the “missing” bottom face) remain *square*. i’ve
shown this “flattening out” in two steps: once as a
truncated-pyramid in a “3-D” view, and then as a
fully-flattened 7-vertex “graph”.

meanwhile, introduce the secondary colors… in the
“natural way”.

all this is pretty old hat around here. you could
look it up.

the novelty here is the stick-figure iconography
(each of the “icons” has the “top of the cube”
represented by the square-in-the-middle; the
three nonzero vertices of the “bottom” of the
cube appear along the top and right-hand “sticks”
of a given icon).

each of these 7 icons now represents a
*linear equation*; these are precisely
the equations of the 7 2D-subspaces-
-through-the-origin of the vector-space
{(0,0,0), (0,0,1), … , (1,1,1)}
having exactly eight vectors.

one can calculate directly on the icons
(rather than the triples-of-numbers or
the colors) using “set differences”.

but that’s it for today.

Photo on 11-17-15 at 10.46 AM

saturday night i colored in the corners of this cube.
the underlying black-and-white is based on a work of
the great dutch artist m.~c.~escher. the cardboard
cut-out version is from a collection by the american
mathematician doris schattschneider.
(_m.c._escher_kaleidocycles_).

anyhow, i’ve had the whole “5 platonic solids” set
from this work on display in the front room at home
for a while. the others are in color already, right
out of the book. i’ve had *another* set of these,
too: it’s a great “book” and might still be in print
for all i know. i had two editions, from years apart,
years ago.

i took this one to church on sunday and used it in my talk.
there wasn’t time to explain why i’d colored it the way i
did. but i *did* count the symmetry group of the cube,
two ways. any talk by me should have a theorem in it;
i’m happy to count that as a theorem.

24 because any of the 6 faces can be “face down”,
and each such choice-of-face allows for any of 4
remaining faces then to “face front” (all but the
“face-up” one *opposite* to our “chosen” face).

but also 24 because
1 identity
6 180-degree “flips” that fix two edges
(one for each pair of opposite edges)
8 120-degree “corner-turns”
(fix a pair of opposite corners;
there are 4 such; one may “turn”
right or left)
6 90-degree “face turns”
(fix opposite faces—3 ways; again,
one can go “right” or “left”)
3 180-degree face turns.

and this messy version is actually quite clear when
one is actually holding up an actual cube and pointing
at the drawing on the board. or in this case, at one’s
own sweatshirt. canvas makes a good whiteboard.

you can do it all without even mentioning “group theory”…
and i *sure* didn’t get to prove that this set of 24
“moves” gives a version of “the symmetric group on 4
objects”. anyway, part of the point is that one need
not have introduced any “math code” into the discussion
at all to arrive some some *very* useful results.

i learned the symmetry-groups-of-solids trick from an
old master
. i mentioned this book in the service.

but mostly i talked about stuff like bible studies and music.

stuff that people actually show up at church *for*. bring what
you love to church and share it. food and money are particularly
welcome.

Photo on 9-22-15 at 8.31 AM

here’s the page where i decided
i’d been looking at things inside-
-out… that the “primary color”
points of Rainbow Space should be
displayed as (midpoints of) *sides*
of the “color triangle”, rather
than as *vertices* (as i’ve been
doing for a couple of years or more
now… how long *was* it since i
discovered “color”, again?).

anyhow. *four* (rather than one)
of the lines on this display
now appear to the eye as “circles”:
the line-at-infinity (still) and
the three “blends” (newly). also
the line at infinity shows up
*outside* the rest of the figure
instead of inside.

the primaries R, B, Y can now be seen,
as it were, “emerging from the Mud”
(the “ideal” point in the center),
with the secondaries then
“following” (as it were in *time*;
evolving or big-banging or what have
you) by “forming the blends” (ROY,
BPR, & YGB of course; if this isn’t
clear one should break out the finger-
paints).

the “blurs” take us back to the Mud
(and the whole thing starts over again…
or not… the metaphysics isn’t clear
at this point…)

yesterday’s post follows this (and, as i type,
doesn’t have the “finite geometry” category-tag on it;
there’s *much* more on this topic to be found by clicking
that category).

then love died

Photo on 9-21-15 at 3.09 PM

product endorsement:
triangle-graph paper.
better than “hex”, even.
anyway, for my purposes now.
you scribble away contentedly
for a couple hours. *then*
(maybe) you might as well
get online to get *in* line
and make a digital copy & post.
which is increasingly difficult.
and not *only* because i’m getting
stupider and more stubborn…
though both of these faults of mine
do play their parts in the whole mess.

Photo on 8-15-15 at 6.53 PM

“modular arithmetic” was introduced to me
somewhere around 1968 as “clock arithmetic”
(among other things). it turns out—
very interestingly, as it turns out—
that one can do Additions and Multiplications
very much like the familiar operations on
{\Bbb Z} = \{ 0, \pm1, \pm2, \ldots\}
(the integers), using any of the sets
{\Bbb N}_2 = \{0, 1\}
{\Bbb N}_3 = \{0, 1, 2\}
{\Bbb N}_4 = \{0, 1, 2, 3\}
{\Bbb N}_5 = \{0, 1, 2, 3, 4\}

{\Bbb N}_{12} = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 \}

(the last-named, of course, gives
“clock arithmetic” its name).

suchlike “arithmetics”… having an addition
and a multiplication (with the addition assumed
commutative and with the multiplication “distributive
over” the addition) are (for some reason) called
rings. the rings we care about today are
called fields; examples include whichever
“clock arithmetics” you care to name that have
a prime number of elements.

fields are “good” because we can *divide* in them.
(not just add, subtract, and multiply). the “prime”
criterion ensures this by ruling out the possibility
of solutions to “xy=0”
having “x” and “y” both *non*-zero.
for example, “on a mod-12 clock”, one has 3*4=0,
so we have a zero-product formed by two nonzero factors.
this is a “bad” thing and doesn’t happen in fields.

once we can do divisions, we can carry over most of
the “analytic geometry” from intro-to-algebra into
any of the settings {\Bbb N}_2, {\Bbb N}_3, {\Bbb N}_5, {\Bbb N}_7, {\Bbb N}_{11}, \ldots
and much else besides.

the value for me in replacing the so-called real field {\Bbb R}
with the finite fields {\Bbb N}_2, {\Bbb N}_3, {\Bbb N}_5, \ldots is inestimable:
one can show *every single instance* of a certain phenomena
directly, without appeal to such weird notions as “continuity”.

in the example at hand, we have
“projective space over {\Bbb N}_3
(which the colorful image calls {\Bbb F}_3).
each line has four points. four of the lines…
including the “line at infinity”… have been
color-coded (and given equations as names).
the “finite points” here have z=1;
the line-at-infinity has z=0;
the “point at infinity” has x=z=0.

the rest is commentary.