### desargues theorem for poets

the MEdZ logo indicates where

the ten lines of the desargues diagram

fall. one has ten such lines.

also ten points. each line

can be considered as a set

of three points; similarly each

single *point* belongs to three *lines*.

in fact, we have a “duality” here…

theorems about points-and-lines

remain true when the words

“point” and “line” are interchanged.

for example, in a self-dual space

having the property… which this one

does *not*… that “any two points

determine a unique line”, one would

also have “any two lines determine

a unique point” (and one makes

adjustments to plain-english like

“any two points *lie on* a line”

becoming-replaced-with

“any two lines *meet at* a point”

or what have you).

it turns out in desargues-space

(let’s say, in D) each line has

three parallel lines, all meeting

at a point. the point is said to be

the **pole** of the line and

the line is said to be the **polar**

of the point. a choice of pole-and-polar

for the diagram is a *polarity*.

ten polarities are displayed in color here.

each white dot represents the pole

for its diagram; the polar is colored

with the three “secondary” colors

(green, purple, and orange).

the three lines through the pole

have matching “primary” colors:

a red line, a blue line, and a yellow line

(if you will).

it turns out that the primary-color points

can be arranged… in exactly one way…

into *two* red-yellow-blue “triangles”

(whose “edges” are along lines of D).

now we come to the payoff.

the two triangles are said to be

“perspective from the pole”

(“p”, say; call the polar line “l”

while we’re at it if you please):

one imagines shining a light

held at p through the vertices

of one triangle to produce the

other triangle… like a slideshow.

and what happens is that now

*either* red-blue line will “hit”

the purple point… and *either*

red-yellow line will hit the orange

point… and *either* blue-yellow

line will hit the green point:

the colors “mix like pigments”.

recall that the secondaries…

the “mixed” colors… all fall on

the polar of p. recall that this

is a *line* of D.

when, as in this case, the three

points-of-intersection for the three

corresponding-edge-pairs

of a pair of triangles happen to

lie on a single line, the triangles

are said to be “perspective from

the line”. in our RBY metaphor,

perspectivity from a line means

we color the vertices of the triangles

and form the secondary colors

by intersecting the lines.

perspectivity from a line means

that the secondaries all line up.

now

if P is a “space” (a set of points

together with certain subsets

called lines) satisfying certain

axioms (those of a **projective
space**), then we have

**desargues’s theorem**

any two triangles perspective

from a point are perspective

from a line; any two triangles

perspective from a line are

perspective from a point.

(d’s theorem is *almost* true in

the ordinary euclidean plane…

but alas special cases must be

written in to account for

parallel lines. parallels are

banned from projective spaces

which makes ’em easier to work with

algebraically but harder to visualize.

luckily “ordinary” planes can be

made to “sit inside” projective planes

so we can recover all of euclidean

geometry in a more-convenient-for-

-abstract-symbol-manipulation form.

PS

i still haven’t “solved”

ten-point reverse TTT,

by the way. but it’s very likely

only a matter of time.

somebody skilled in computer coding

could probably knock it out in a few hours.

July 23, 2011 at 1:26 am

Nice! Love the title.

July 23, 2011 at 1:31 am

One of these days, I’ll have to put all of this into InDesign for you–after we sit for about three days and you explain it all to me in detail.

July 24, 2011 at 11:48 am

sounds great. of course, i’m still

working *out* the details.

this stuff is tantalizingly always

just beyond my ability to just look

and see everything there *is* to see.

“elementary” stuff gets complicated

in a heck of a hurry. color is a great

gimmick: this primary-versus-secondary

trick has helped me considerably

in holding the relevant info in my head

all at once.