“dual” elements identified with “perpendicular directions”

April 24, 2015 in Finite Geometry, Lectures Without Words

yet another sketch from the
“lectures without words” run
of MEdZ. here improved with
colored inks (and spoiled by
flouting the “without words” rule).

“binary arithmetic” is here exploited
to assign *number values* to the corners;
the symbol “xyz” chosen from
000, 001, 010, 011, 100, 101, 110, 111
corresponds on this model to
4x + 2y + z.

(this follows the usual “place-value”
conventions typically used in the
context bases-other-than-ten
[in base ten, the same symbol “xyz”
would denote 100x + 10y + z].)

the “front face” of our cube (for example)
is now {000, 001, 100, 101}.
these number triples share the feature
y=0…
and are the *only* triples with this feature.

now, we can think of “y=0” as meaning
“don’t move in the y direction at all”
(the “y direction” here is “front to back”…
going [as it were] from the 000 point “back”
toward the 010 point is the only way to get
a y=1… that’s why “y=0” gives us the “front face”.

but the point 010 is not itself a “direction”…
so another notation is introduced: [0:1:0].

the diagram shows (or hopes to) that similarly
[1:0:0] “is perpendicular to”
the left-hand face {000, 001, 011, 010} and
[0:0:1] {000, 100, 010, 110}.
(excuse me my “joy of ” here;
has what i hope is its obvious
meaning.)

anyhow… there’s real work to be done
(getting to campus and back; the hardest
part of the job some days)… that’s *almost*
it for today.

it remains only to remark that
[0:1:1], [1:0:1], [1:1:0], and [1:1:1]
can also be considered as “perpendicular”
to the other four rainbow-space “lines”
(certain cross-sections of the cube
on the 3-D model)… giving us a
full-blown *algebraic* model of
fano-space duality.
[exercise. hint: binary arithmetic.]

feed me!

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