### “dual” elements identified with “perpendicular directions”

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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