August | 2015 | the livingston review

Archive for August, 2015

the mod-three projective plane

August 15, 2015 in Finite Geometry, Graphics

“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.

econ 101

August 10, 2015 in Blogs

i’m browsing as hard as i can.
but i can’t find any images
as pleasing to my eyes as these.
and yet these are sloppy first takes.
it’s almost as if reality
were more beautiful than the net.
this belongs in the other blog.
who said life is fair.