## 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, ).

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.

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.

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.

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).

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.

“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

(the **integers**), using any of the sets

…

…

(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

and much else besides.

the value for me in replacing the so-called real field

with the finite fields 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 ”

(which the colorful image calls ).

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.