### 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.

we now return to zeerox-friendly B&W.

November 22, 2012 at 12:32 am

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.

December 10, 2012 at 11:09 am

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

[

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

[trained in reading perspective drawings])

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”.

and i’ve made quite a bit of fuss about

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

is that i got one for free.

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.

March 28, 2015 at 5:20 pm

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/

updates

https://vlorblog.wordpress.com/2010/11/23/now-heres-something-youll-really-like-that-trick-never-works/

(e.g.)

November 9, 2015 at 4:35 pm

https://vlorbik.wordpress.com/2015/11/09/she-wears-an-iron-vest/