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.
the livingston review 
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.