### E^2(F_3) considered as a tape doughnut marked with nine (unordered) order-pairs

(the “typed” image will be munged by nearly any interface,

i suppose… it looks lousy enough in mine, i know…

i never thought i’d miss DOS boxes *then*…)

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the interior of our square

can be conveniently co-ordinatized as

I = { (x,y) | 0 < x < 1, 0 < y < 1}.

the *boundary*, , then

consists of four line segments

L = {(0,y) | 0 =< y =< 1} &

R = {(1,y) | 0 =< y =< 1}

("left" & "right"), and

T = {(x,1) | 0 =< x =< 1} &

B = {(x,0) | 0 =< x =< 1}

("top" & "bottom").

(convince yourself of this

if you can. hint: pencil

& paper.)

the Torus, T^2, can then be modeled as

"the union of I with its boundary, mod 1".

but never mind the fancy terminology.

instead we can imagine the situation

by thinking of the inside of the square

as if it were a video screen…

with the condition that when we

"move the mouse"

upwards and get to the Top (say),

we will "vanish off of" the Top

boundary and "re-emerge" at

(the corresponding point—

"having the same x-co-ordinate"—of)

the Bottom (still going upward).

& likewise for L & R

( (0, y) ~ (1, y)… i.e., for any

particular "y", we consider these

two boundary-points-of-the-square

to represent *one point* of T^2 ):

moving the mouse rightward, one

will, as it were, "go through"

the Right edge of the screen and re-

-emerge on the Left

(still going rightward).

thus far, so familiar, i hope.

using "identification diagrams" of this

kind is one of the coolest tricks

i ever learned in a math department.

now we just replace the "field of Real Numbers"

with "the" field with three elements (F_3).

all of (x,y)-space over this field

consists of *nine* ordered pairs

(and we "calculate mod 3"…

1+1+1 == 0 in this world…

but otherwise "ordinary algebra"

works…

(1, 2) + (1, 2) == (2, 4) == (2, 1),

for example…

the upshot is that we need only the

three elements of {0, 1, 2} to represent

*any integer pair* (x, y) of our

good old-fashioned "euclidean" space

):

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

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

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

i'll call this E^2(F_3)…

"euclidean two-space over the three-element field".

mod three arithmetic produces

"the video game effect" on this

space… one might imagine starting

with (a very flexible copy of)

(0,3) (1,3) (2,3)

(0,2) (1,2) (2,2) (3,2)

(0,1) (1,1) (2,1) (3,1)

(0,0) (1,0) (2,0) (3,0),

and then "wrapping it around"

to form a tube with each (x, 0)

covering up its corresponding (x, 3);

next (here's where "very flexible"

comes into it) "wrap around" again

the other way to produce the

(doughnut shaped) Torus by overlapping

each (3, y) with its corresponding (0, y).

now just replace the *ordered pairs of numbers*

with (unordered) pairs of "direction icons"

& draw the whole thing on the nearest

tape-doughnut to hand; voila. a pleasant

time not grading papers.

PS

ancient footnotes are everywhere:

April 25, 2015 at 3:51 pm

i used to routinely call the torus-

-modelled-by-identifications-on-

-the-square by the name of

“the pac-man topology”…

pretty much whenever i had

a chance, in fact… in front of

groups of students.

how many of them had any idea

what the devil i even meant by that

i’ll’ve had pretty much no idea…

the 80s was a long time ago it now

begins to seem… so of course

after about the turn of the century,

the rest of my remarks would be put

in terms of some more general

“video game”, more or less as

i did in the post.

still, it’ll always be the pac-man

topology to *me*. god, i hated

that game.

April 25, 2015 at 3:55 pm

one frequently *gets* such chances. or used to.

consider the graph of y = tan(x), for example.

April 28, 2015 at 3:15 pm

now, “cross” the donut with (two other copies of) itself

and “homogenize” (technospeak alert); voila, P^2(F^3):

https://vlorbik.wordpress.com/2011/02/01/start-one-dimension-up-and-form-a-quotient-space-or-start-with-ordinary-n-space-and-add-infinite-points/