the mod-three projective plane

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.


  1. so here’s some commentary.

    the line labeled “x=2” is “really”
    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

    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.)

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