## Archive for July, 2011

mathblogging.org’s weekly picks #22.

Consider a set of five objects.

In fact, consider the “canonical” set of five objects: {0, 1, 2, 3, 4}.

There are then ten ways to form *pairs* of these objects, as one sees from the following display:

04 03 02 01

14 13 12

24 23

34

.

(Note that by “pairs”, I refer to *unordered* pairs. For example, “40” doesn’t occur in the display because this represents the *same* pair of objects as “04”. In fact, since {4, 0} = {0, 4} as sets, one could formalize the notion of “pairs” as I’m using it here by referring to sets-of-two-objects from our original five-set. Indeed, I’d *do* it this way despite the extra typing if I wasn’t afraid it would then look harder to read.)

Now let’s call the objects of this ten-set

{01, 02, 03, 04, 12, 13, 14, 23, 24, 34}

the **points** of a “space” called D (for Desargues). “D” itself is then this set of points, together with a collection of certain (unordered) *triples* of points. The triples themselves are called the **lines** of D.

Here are the lines of D.

{01, 02, 12}

{01, 03, 13}

{01, 04, 14}

{02, 03, 23}

{02, 04, 24}

{03, 04, 34}

{12, 13, 23}

{12, 14, 24}

{13, 14, 34}

{23, 24, 34}

.

To form a line, take any *three* objects from {0, 1, 2, 3, 4} and form all three possible pairs.

One now has a geometry-free construction of the “space” I’ve been examining here in the blog. In particular, each point appears in exactly three lines (just as each line contains exactly three points).

I intended more of this today. But, to my delighted surprise, duty calls: I’m meeting a new tutee in a couple hours and have to go catch the bus.

yesterday’s more verbose version.

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.

a by-hand reduction

of my desargues-space poster,

with one less layer of redundancy

(and the colors of the triangles

reversed; also some cosmetic tweaks).

ten poles, ten polars,

and ten pairs-of-triangles:

ten ways to use one drawing

(from MathEdZine, of course)

to illustrate one theorem.

(desargues’ theorem at w’edia.)

performing calculations using

actual blobs of color rather than

alphabetical symbols *standing*

for colors is time-consuming

(and demanding of special tools)…

but one literally “sees” certain things

*much more readily* than with

symbol manipulation.

hey, i’m a visual learner.

whenever anybody says anything like

“this vision can only succeed if

we all get on board and do our parts”…

well, all i can hear is “this vision

is doomed to fail”. there’s *no way*

we’re all going to get on board…

we all never got on board with *anything*.

never have so far anyway. history is conflict.

and anyhow, what-you-mean-we, mister politician?

and this is true even for

solomon garfunkel on common core standards

(PDF; from the *notices*).

they screw you up, your uncle sam.

they ought not mean to, but they do.