## Archive for the ‘desargues’ Category

somewhere in all this “desargues” stuff

i claimed that i don’t have the skill to draw

the “symmetric” version on the diagonals of

a dodecahedron (or, equivalently, on pairs-

-of-opposite-faces of an icosa… as shown here).

well, now i can prove it. the notes are already

posted i guess. i’m going on not much sleep &

anyhow, the *really* awesome stuff is guitar-

-music mostly these days. lectures-without-

-words is just something to do in between

tabbing-out arrangements for simple tunes.

you know my methods, watson.

the other four ways can be found

by holding the page up to a mirror

(rotated 90 degrees from this

orientation).

this is a fairly old drawing.

here is my recent discovery

of the “five-way symmetric” version

of the same situation (the “ten

circles theorem” as i call it).

finding this stuff out… color

displays in projective geometry…

is my probably my proudest accomplishment

in mathematics-as-such. my classroom

work impressed me quite a bit, too,

but those days are over now most likely.

here’s fano’s cube from 2011.

i wish i’d stayed a grill cook.

by now, i’d’ve probably gotten

pretty good at it.

the second, smaller, sketch is from

desargues’ theorem in color (but let’s call it

*desargues’ rainbow* from here on

to match fano’s rainbow,

posted the next day).

in the first, newer, bigger sketch,

i’ve used my mystical “projective

geometry” powers to bend all the lines

into circles. so we now (as you can see)

have a red *circle* (at the “omega” point)

along with (arcs of) blue and yellow

*circles* (replacing the red, blue, and

yellow *lines* on the original [textbook]

drawing).

again, as one should expect from the

names-of-colors aspect of all this,

we find certain yellow-and-blue

point-pairs occurring on the arcs

of certain circles… and two such

circles meet in the *green* point

(green is the “blend” of yellow

and blue). and then likewise for

the other “secondary” colors: an

*orange* point at the intersection

of two red-and-yellow circles, and

a “purple” point where two red-and-

-blue circles meet.

desargues’ theorem is then that the

secondary colors are on one of the

“lines” of the system at hand.

[in this case, (an arc of) another

“wide circle” (our system consists of

ten circles; the “narrow circles”

appear as circles in the diagram

and arcs of three of the “wide”

circles are indicated by three-

-point arcs).]

the diagram is, essentially, traced from

bruce e.~meserve’s _fundamental_concepts_-

_of_geometry_ (dover reprint of 1983;

originally addison-wesley 1959). but

i added in the colors. with which, one

has as follows.

there’s a red line, a blue line, and a yellow line.

to start with. all sharing a point.

then two “triangles” are constructed:

each of these is to have a red, a blue, and a yellow

“vertex”.

(on the diagram, the “red”, “blue”, and “yellow”

vertices of one of the triangles is actually black;

this helps [maybe… it helps *me*] in determining

which vertices belong to which triangles.)

next we can think of the *edges* of these two triangles

as “blending two primary colors”; for instance we can

think of the red-and-yellow edge of either triangle as

determining an *orange* line… and go on to construct

an “orange vertex” at the intersection of these lines.

likewise, the yellow-and-blue edges of the two triangles

will determine a *green* point (at the intersection of

two green lines) and the blue-and-red edges will determine

a *purple* point.

desargues’ two-triangle theorem then says that the

orange, green, and purple points will “line up”.

proof: meserve (or any of *many* other sources)

footnote: alas, one has *exceptional* cases

in “euclidean two-space”, i.e., the ordinary

(two-dimensional) *plane* of high-school

geometry. one remedies this by working

instead in a (so-called) *projective* plane

(in such a plane, there are no “parallel” lines;

nevertheless, much of “ordinary” plane geometry

becomes *easier* in projective geometry [example:

this theorem]).

putting color names on things is a common trick in math,

but if anybody else is using blends-at-intersections in

anything resembling this way (elementary projective

geometry, i suppose i mean), i don’t know about it.

(priority claim; if you use this amazingly good idea,

remember where you got it. please.)

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.