## Archive for November, 2015

{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) sub*space*].

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.