she wears an iron vest

November 9, 2015 in Finite Geometry, Graphics

{1, 2, 3, 4, 5, 6, 7} (“barred” inside
the “big circles”… as they ought to
be throughout) are here identified with
{
001,
010,
011,
100,
101,
110,
111
}
in the usual way (“binary arithmetic”).

these objects — triples of zeros-and-ones —
can of be considered as (so-called) 3-vectors:
{
(0,0,1),
(0,1,0),
(0,1,1),
(1,0,0),
(1,0,1),
(1,1,0),
(1,1,1)
}.

these can be pictured… exercise (or you could
look it up… warning: the earlier drawing
uses a different assignment of the colors)…
as the “nonzero” vertices of a cube.

by considering this set, together with (0,0,0),
one forms a (so-called) finite abelian group.
renaming objects (again), this group—call it (G,+)—
can conveniently be displayed as
G = {
(E,E,E),
(E,E,O),
(E,O,E),
(E,O,O),
(O,E,E),
(O,E,O),
(O,O,E),
(O,O,O)
}.
the group’s operation, here called “+” (i’d’ve
“barred” this sign in handwritten work, probably), is
defined by “add E-for-even and O-for-odd in the usual
way, three times (once for each “column”)” (or “add
component-wise, mod two”).

thus (E,O,E) + (O,O,E) = (E,E,E), etc.

(G,+) now has the structure of a —
three copies of the “parity group” .
this happens to be the simplest example of a 3-dimensional
vector space. the simplest projective plane,
S = (P, L), is then
(1) a set of seven Points
P = {(0,0,1),…,(1,1,1)}
(or P = {[1],[2],..,[7]}—
easier to type, but harder to
compute with), together with
(2) a set of seven Lines
L={
{[1],[2],[3]},
{[1],[4],[5]},
{[1],[6],[7]},
{[2],[4],[6]},
{[2],[5],[7]},
{[3],[4],[7]},
{[3],[5],[6]}
}.

one now places “colors” on the (nonzero)
“corners of the cube”— or, equivalently,
on the “points of S”— in such a way that
the “blends”, the “blurs”, and the “ideal”
all “line up”… which is to say, “occur
as triples-in-L”.

(“adding in” [0] = (0,0,0)
to each of the “lines” of S
one has
{
{[0],[1],[2],[3]},
…
{[0],[3],[5],[6]}
}:
the (7) “planes through the origin”
in (Z_2)^3… the front, side, and bottom,
for example, and four others harder to
describe without the handwaving; the
“bottom”, for example has z=0 (in the
usual (x,y,z) interpretation of an
ordered-triple. each of these planes
is a subgroup of ( (Z_2)^3, +)
[or a (vector) subspace].
now back to the 2-D versions.)

our diagram today has
M-R-B-G-P-Y-O (in this order:
mud, red, blue, green,
purple, yellow, orange)
identified with
[1]-[5]-[7]-[4]-[2]-[3]-[6].

this ordering is one of the (168) ways
to cause the “lines of rainbow space”
(blends, blurs, and ideal) to coincide
with the algebraically-defined “lines
of S”.

each of the dots-and-sticks
(at each colored-circle
“point” of today’s selfie)
gives a *graphical* representation—
a “figure”, let’s say—
of a Line. so at lower-right, (the
Green point), we have [4]= (O,E,E),
associated to the set
{Mud, Yellow, Purple}—
one of the “blurs”—,
and also to
{[1],[3],[2]}… i.e., to
{
(E,E,O),
(E,O,O),
(E,O,E)
}; each triple {[A],[U],[X]}
(in any order) satisfies
[A]+[U]=[X], i.e., for
A=(a,b,c),
U=(u,v,w),
X=(x,y,z),
one has a+u=x, b+v=y, and c+w=z
(all “mod two” of course; recall
that we are “adding” Evens and Odds).

one now checks—over and over—
that the drawing at hand “satisfies
pro-planarity”. in terms of colors-and-figures,
this means that the blends-blurs-and-ideals
coincide with the 7 “shapes” of the figure.
in vectorspace terms, with the planes-
-through-0 (of (Z_2)^3). in the
“fano plane” (w’edia),
S=(P,L) of “abstract” Points and Lines,
with lines. so on.

the rest is commentary. for a while.
eventually you want to get to know more
about those 168 ways, for example.

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