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.


  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”).

    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
    [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.

  3. somewhere between the last comment
    and now, i think, i’ll’ve started redrawing
    everything in color (7 colors, to be precise).

  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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