### the mod-three projective plane

“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.

August 15, 2015 at 7:39 pm

https://plus.google.com/103958627037371901715/posts

August 16, 2015 at 10:01 am

so here’s some commentary.

the line labeled “x=2” is “really”

x+z=0.

that’s why the “point at infinity”,

010, has a green circle.

likewise y=1 only “works” in

the “finite” part (z=1); for the

“big picture” including the 100

point (on the line-at-infinity,

z=0), we have to modify this

to y+2z=0.

y=x comes through without

modification.

i wrote the labels the way i did

so as not to have to invoke

“mod three arithmetic” explicitly

in the graphics.

(to leave something to the commentary,

in other words.)