more news from nowhere

November 17, 2015 in Finite Geometry, Graphics

consider ordinary (x,y,z) space.
co-ordinatize a “unit cube” in the all-positive octant.
put “primary colors” on the axes. next slide.

from the edges of the cube, “cut out” the three that
pass through the Origin of our system—i.e., (0,0,0).
next.

distort the resulting diagram so that the “top face”
(and the “missing” bottom face) remain *square*. i’ve
shown this “flattening out” in two steps: once as a
truncated-pyramid in a “3-D” view, and then as a
fully-flattened 7-vertex “graph”.

meanwhile, introduce the secondary colors… in the
“natural way”.

all this is pretty old hat around here. you could
look it up.

the novelty here is the stick-figure iconography
(each of the “icons” has the “top of the cube”
represented by the square-in-the-middle; the
three nonzero vertices of the “bottom” of the
cube appear along the top and right-hand “sticks”
of a given icon).

each of these 7 icons now represents a
*linear equation*; these are precisely
the equations of the 7 2D-subspaces-
-through-the-origin of the vector-space
{(0,0,0), (0,0,1), … , (1,1,1)}
having exactly eight vectors.

one can calculate directly on the icons
(rather than the triples-of-numbers or
the colors) using “set differences”.

but that’s it for today.

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