### i saw the figure 5 in gold

yet another duality diagram.

until now, this post from 2010

had my favorite such: “duality in

the fano plane”; seven “lines” through

seven “points”.

this one is a much-more-symmetric

picture of the situation represented

in this early version: ten “lines”

through ten points.

it recently occurred to me that

the “desargues configuration”

(aka “the situation represented here”)

could, in some sense, *best* be

visualized as the (set of) *opposite

faces* of an icosahedron.

but before i’d even made a decent

picture of that (hard-to-draw because

3-dimensional), along came this idea.

how it escaped me till now, i’ll never know.

June 28, 2012 at 9:04 am

Please explain this. It’s pretty, and I’d like to understand what it represents.

July 4, 2012 at 12:03 pm

so. draw a big pentagram (“star”).

label the five lines (in any

order you like): 1, 2, 3, 4, 5.

(there are 120 [=5!] ways to

label the lines; every fifth-grader

should understand this but alas

most college freshmen don’t

and a great many never will.)

the intersections… the *vertices* of the

big pentagram… are now conveniently

labeled with *two* “co-ordinates”:

namely, the numbers associated to

the lines intersecting there.

{

15, 14, 13, 12,

25, 24, 23,

35, 34,

45

}, say.

there are 10 = \choose(5, 2)

( = 5 / [2! 3!] ) such co-ordinate

pairs; one easily checks that

there are indeed 10 vertices

for the Big Star (the 5 “points”

of the star and the 5 forming

the inner pentagon).

each point is then associated

in the drawing with a certain

“triangle”: three *other* points

of the Star.

for instance… *regardless of how

the labels 1, 2, 3, 4, 5 are chosen*…

the point labeled 12

is associated with the triangle

whose points are {34, 35, 45}.

this phenomenon arises “because”

the complement of a two-set

chosen from {1,2,3,4,5}

is a three-set: the triangle associated

with a given vertex (a two-set)

will be all the vertices whose

co-ordinates do *not* include

either co-ordinate of the given vertex.

(and it’s this “complementation” computation

that allows me to assert that the labelings

from {1,2,3,4,5} are irrelevant in making the

pairing between vertices and triangles;

the complement of {1,2} is {3,4,5} *regardless*

of where these objects occur on some diagram.)

i hope this is easily understood thus far.

now comes the “duality”… the pet concept

of this whole blog for a couple years.

consider {34, 35, 45} again

(the triangle “dual to” to point 12).

look at the duals of its vertices:

34… {12, 15, 25}

35… {12, 14, 24}

45… {12, 13, 23}.

ah-*ha*!

the *intersection* of these is (gee!)

12 again… the “starting point”.

and so for any starting point:

“my dual is the triangle whose

points have duals that include

me as one corner”.

the Vertices and Triangles of the diagram

at hand become Points and Lines from

another point of view (where “the same”

duality is maintained); for instance here:

https://vlorbik.wordpress.com/2011/07/06/executive-summary/

.

someday soon i’ve gotta get all this

organizized.