for hannah on the back of an envelope

August 5, 2020 in Cool Tricks, Finite Geometry, Projective Spaces

almost-symmetric desargues’ theorem.
(7-color version)

5 “flat” and 5 “tall” triangles
arranged as the 10 “intersection points”
of a (so-called) pentagram.

thus far the black-and-white “underlying
diagram” (which i probably should have
photographed before coloring it in and
erasing it… it was the best b&w version
i’ve done so far, i think… oh well…).

the b&w *already* “tells the story” (if you
know how to look): the ten “triangles” are
in the ten “positions” on the “big picture”
diagram in such a way that, mutatis mutandis
(read “line” for “triangle” & “point” for
“position, to wit), one has the well-known
points-&-lines “duality” exhibited in
purely *graphical* form: not only a
“lecture without words”, but a “lecture”,
if you will, without *code itself*.

so it’d be a pretty awesome picture, done right,
&’d make a good art project for somebody patient
enough to make actual measurements & stuff.

as would the colorized version.
so, briefly.

the letters {a, b, … , j}
are posted in “positions” in such a way
that the when the “triangles”
{abc, ade, afg, bdh, … , ijk}
are “placed in” the appropriate positions
(as is done here)
one has each letter-triple in the position
of its *dual* letter (“a” to “ijk”, eg.;
one should probably look at some other
diagram at about this point).
now replace all the letters with colors
according to the scheme pictured at right.
(whose connection with the “fano plane”
[& “vlorbik’s 7 color theorem”] is solid
but we don’d have to go into it here.)

happy neighborhood of yr b’day.
thanks for your part in, you know,
teaching me to read & bringing me
back from the dead and all that.

here’s your “virtual lanyard”.

PS the *actually* symmetric version has
the ten “lines” modeled as *diagonals
of an dodecahedron* (the “diameters”
connecting opposite vertices; there
are of course 20 such…);
the “asymmetric” (textbook) version
has a *marked* point (and, of course,
a “marked” dual line…); never mind…

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