### eye candy

the three directions associated
with “arrows’… 001, 010, and 100
in the binary code… are here
colored yellow, red, and blue
(it didn’t reproduce as well as
one would’ve liked).

these so-called “primary” colors
(for pigments; for colored lights
the rules are different) blend
along the appropriate cross-sections
of the diagram (the easy-to-see
ones with the arrows attached)…
*or* the sides of the triangle:
yellow and red form orange,
yellow and blue form green,
red and blue form purple.

blending *opposite* colors…
the primary/”secondary” pairs
yellow/purple, red/green, blue/orange…
gives a muddy brown (“mahogany”
sez crayola) and again the appropriate
cross-sections of the cube (and lines
of the “fano plane” triangle) associate
the appropriate “blends” with the
pairs-of-pigments combining to form them.

finally, there’s the weird cross-section
(the 3-5-6 arc in the fano diagram)
consisting of all three secondaries.
on the cube diagram, this appears
as a tetrahedron-through-the-origin
rather than a plane-through-the-origin.

colored pencils (with erasers) rock.

1. this appears to be my greatest-hit drawing.
i give these away pretty readily.

[most of the rest of my math zines
provoke undisguised horror and
the prospect won’t *touch* the
damn thing (never mind take it home
and study it or look up the URL or
pass it on to some random math-head
or what have you). just… yick.]

it seems even more certain that my greatest
*prose* hit is the “set of natural numbers”
piece from my mid-nineties lecture notes,
re-run here in the early days (of “VME”).
https://vlorbik.wordpress.com/2008/07/16/11-the-set-of-natural-numbers/

if these things are so, they content me.
student work though both pieces are.

2. let S := {000, 001, … ,111} be the vertex set
for the cube of our illustration.

*******************************************************************
the points of P^2(F_2)… the colored dots
of the diagram in the lower right…
are then obtained as the sets

[001] = {000, 010, 100, 110}
[010] = {000, 001, 100, 101}

[111] = {000, 011, 101, 110}

obtained by

$xyz \in [ABC] \Leftrightarrow Ax + By + Cz = 0 (mod 2)$

[
essentially this calculation amounts
to certain vector-pairs of F^2 having
*zero* as their “dot-product”.
the property of “dotting to zero” is
associated in certain geometries
with “being at right angles”…
for example in ordinary “real 3-space”…
and it’s useful to keep this in mind
from time to time here as well.

the “dual space” of a vector, v, is then
the set of vectors *perpendicular to* v:
dual(v) = {w : w \dot v = 0}.
this notion gives rise to the phenomenon
whereby every
*line* through the origin
is associated to a *plane dual*
(the *plane* through the origin
at a right angle to the given line).

i’ve made some reference to this idea
in my illustration in the parts labelled
by 1:0:0, 0:1:0, & 0:0:1.
in *these particular cases*, the mod-2
arithmetic *still produces right angles*
(visible as such to the ordinary eye
when certain dot-products are zero.
]
***********************************************************
the *complements* (in S) of the 7
“generalized cross sections” (GCS’s)
of our illustration…
considered as subsets of the vertex set
V := {000, 001, … ,111} …
are themselves also GCS’s.

moreover, each is of the same “type” as
its complements: there are
6 Sides (of the cube),
6 Cross-sections (rightly so-called), &
2 Tetrahedra
when we consider all 14 GCS’s
(and the “types” i mentioned are then
S, C, and T).

the 7 GCS’s that *include* 000
form “the 2-dimensional projective
space over the field with 2 elements”.
this space.

it’s recently come to my attention that this
same same P^2(F_2) is an example…
the *simplest* example… of a so-called
“Steiner System”. and that other examples
include *other* geometries i’ve drawn pictures of.

i’ve published shots of several other
two-dimensional projective spaces
over small finite fields.
P^2(F_4) in particular
(“21-line TicTacToe”, e.g.).

also some of *affine* spaces like (F_5)^2.
if i recall correctly, the original inspiration
for *all* this work came from me finding
an ancient homework from the one geometry
course i ever taught and noticing that i’d
created a pretty cool format for homework
and exam exercises on suchlike easily-generated
diagrams and deciding to follow up on it
even in the absence of any students.

upon recently discovering that i’d been publishing
*pictures of steiner systems* all this time without
even knowing it, i set out to make some more.

and the punchline to this whole part of this discussion

the 14 GCS’s are a model of an S(3,4,8).

running on reserve battery power.
this will require editing that it won’t get now.

3. somewhere between the last comment
and now, i think, i’ll’ve started redrawing
everything in color (7 colors, to be precise).
https://vlorbik.wordpress.com/2015/03/28/fanos-rainbow/
https://vlorblog.wordpress.com/2010/11/23/now-heres-something-youll-really-like-that-trick-never-works/
(e.g.)

1. 1 two triangles six ways | the livingston review

[…] classroom work impressed me quite a bit, too, but those days are over now most likely. here’s fano’s cube from […]

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Vlorbik On Math Ed ('07—'09)
(a good place to start!)