13-point projective space

the left-hand photo shows
a nine-point plane: an “ordinary
two-dimensional plane” over the
field with three elements (and its
label is, therefore, {\Bbb F}_3^2).

such a plane is ordinarily co-ordinatized as
(0,2) (1,2) (2,2)
(0,1) (1,1) (2,1)
(0,0) (1,0) (2,0):
the set of (x,y) such that
x & y are both elements of
the set {0, 1, 2}.
one could convey the same information more
concisely as
02 12 22
01 11 21
00 10 20.
it’s useful for our purpose here, however,
to consider our plane as belonging to a
*three*-dimensional space… (x, y, z)-
-space, let’s say… and as having a
*non-zero* “third” (i.e., “z”)-co-ordinate.
thus, in the photo, our plane is represented by
021 121 221
011 111 211
001 101 201.

the colors come into play in displaying the
solution-sets for various (linear) equations.
the reader can easily verify that the Green
equation—x=2— is “true” for the points of
the vertical line at the right… i.e., for
{ (2,0), (2,1), (2,2) } (old-school), i.e. for
{ 201, 211, 221} (“our” version).

likewise “y=x” (the Red equation) gives us
{(0,0), (1,1) (2,2)}… i.e.,the “Red line”
{001, 111, 221}.

now for some high-theory. by Algebra I, one has
a well-developed theory of Lines (in the co-ordinate
Plane). the usual approach there is to use the
(so-called) Slopes. the (allegedly intuitive) notion
of “rise over run” allows one to calculate—for any
*nonvertical* line—a number called the Slope (of that
line). vertical lines are said to have “undefined”
slopes. one might also say that they have an “infinite”
slope… though this invites confusion and is usually
best left unmentioned.

y = Mx + B
x = K
are then our “generic” *equations of a line*.

any particular choice of numbers M & B will
correspond to the a set of solutions lying
along a (nonvertical) line having the slope
of M (an passing through (0,B)… the so-
-called “y-intercept” of the line); each vertical
line (likewise) is represented by some particular
choice of K.

now. having different “forms” for vertical and for
nonvertical lines can be devilishly inconvenient,
so, also in algebra I, one sometimes instead uses
the “general form” for an equation of a line in the plane:
Ax + By = K
(with A & B not both zero).
likewise (but typically *not* in algebra-i)
Ax + By + Cz = K
(with A, B, & C not all zero)
is the equation of a *plane* in *three*-dimensions.

thus far i’ve been as precise as i know how.
but now i’m going to start waving my hands around
and making leaps-of-faith all over the place.
in the second photo, four new “points” have been
added into our framework (namely
{010, 120, 110, 100}—the “line at infinity”).

the upshot… without the details… of this move is
that one now has an algebraic theory of “Lines” in a
“Plane” containing precisely our 13 “Points”. more-
over, this theory is “structurally” very similar to
“ordinary” linear theory. in particular, we dealing
with solutions to
Ax + By + Cz = 0—
the “K=0” case of the “general form” for (the 3D case
of the “ordinary” theory).

the Green equation—which must now be written without
its “constant term” (x = 2 is “the K=2 case” of x = K)—
becomes x – 2z = 0;
similarly, rather than (the three-point “line”
of {\Bbb F}_3^2) “y = 1” (concentric black-and-
-white circles), the (“homogeneous”—for us, right now,
this can be taken as meaning “having no constant term”)
equation is “y – z = 0” (and, again, we pick up a “new”
point at 100).

when the smoke clears… which won’t be here and now…
we’ll have a *very nice* geometry. just as in “ordinary”
space, two distinct points determine a unique line.
but… *unlike* “ordinary” space, it’s also true that
(*any*) two lines determine a unique point.

the same “trick”—homogeneous co-ordinates in finite
fields—converts any plane having p^2 as its number
of points to a *projective* plane having
p^2 + p^1 + p^0
as its number of points. thus there are PP’s having
7, 13, 31 = 5^2 + 5 +1, 57 = 7^2 + 7 + 1, ….
as their number-of-points. there are also some others.
but the margin is too small.

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