## Archive for the ‘Projective Spaces’ Category

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.

here’s another picture of a dualization

of the projective plane on the field

of four elements: .

in the first of two tweaks since the last version

published here, i reversed

the positions of the roots of x^2 + x +1…

i’ve been calling ’em alpha and beta…

reversed their positions, i say,

for the vertical axis (as opposed

to the horizontal:

0a 1a aa ba

0b 1b ab bb

01 11 a1 b1

00 10 a0 b0

as opposed to the earlier

0b 1b ab bb

0a 1a aa ba

01 11 a1 b1

00 10 a0 b0

).

this had the pleasing effect that

*all* the “lines” now had naked-eye

symmetry. the earlier drafts had

some twisty-looking “lines” once

the alphas and betas got involved;

this was in some part due to the

artificial “symmetry breaking” that

i’d indulged in by *putting alpha

first*. the new version had beta

close to the origin just as often

as alpha… which better describes

their relation in .

anyhow, the second, more radical

“tweak” involved swapping (0,0)

and (1,1)… and causing “lines”

actually *looking like lines* to

appear as *broken* lines

(but simultaneously causing the

alpha-points and beta-points

to behave better still).

this looks like a pretty good board

for pee-two-eff-four tic-tac-toe.

(or, ahem, “21-Point Vlorbik”…

he who blowet not his own horn,

that one’s horn shall never be blown).

Tic Tac Toe

Official Rules (all rights reserved).

“The Board” is the 21-line

array of letters:

ABCDE

AFGHI

AJKLM

ANOPQ

ARSTU

BFJNR

BGKOS

BHLPT

BIMQU

CFKPU

CGJQT

CHMNS

CILOR

DFLQS

DGMPR

DHJOU

DIKNT

EFMOT

EFLNU

EHKQR

EIJPS

.

two players alternate turns.

in the first turn the first player

chooses any letter from A to U

and “colors” all five copies of

that letter on the board;

the second player then chooses

any *other* letter and “colors”

all five copies (in some other

color… X’s and O’s can be made

to do in a pinch if colored pencils

aren’t available).

players continue alternating turns,

each coloring all five copies of

some previously uncolored letter

at each turn.

play ends if one player… the winner…

has colored *all five letters* of any row.

otherwise play continues until all (21)

letters are used.

if one player… the winner… now has more

“four in a row” lines than the other, so be it;

otherwise the game ends in a draw.

i actually played this for the first time

this morning on the bus with madeline:

a four-to-four draw.

in the thus-far-imaginary computer version,

upon selection of a letter, five dots…

all the same relative position in their

respective “crosses”… light up and

“five in a row” becomes “an entire

cross lights up”. this might be worth

learning to write the code for.

better if somebody else did it, though,

i imagine. i just want a “game designer”

credit and a small piece of every one sold.

i’ve finally put the “points at infinity”

where they’ve belonged all along.

this’ll probably be the last draft

as far as “where do the points go”.

the color scheme is still almost

entirely up in the air. all i know

so far is that nobody not already

interested in maths will look

at the damn thing in b&w.

i did this before rearranging the points.

now i suppose i oughta do it again.

anyhow there sits the plane on

the field with *two* elements

in red right inside the plane

on the field with *four* elements.