before you throw anything away, zap it so you’ll have a copy

June 9, 2020 in binary tetr'al

here’s the other image of a-hat from w’edia.

this one looks quite a bit more like the diagrams
i’ve been drawing in pursuit of a geometric “feeling”
for—what i’ve been calling—the “binary tetrahedral”
group; what i’ll be calling “a-hat” (—this rhymes
with “A-flat”, not “the cat”) henceforth. this is
short for a-four-hat. the idea here (as i found
on some webpage) is that our group
is a “covering” of A_4—th’ “tetrahedral group”
aka the “alternating group on four objects”.

(
the four objects in question, in the unlikely
event that anyone is following so far, “are” the
vertices of the tetrahedron in question…
the group
A_4 = { (),(01)(23),(02)(13),(03)(12),
(012),(021),(013),(031),(023),(032), (123), (132 }
(typing is fun and easy)…
then is made to represent the three 180-degree
“flips” and the eight 120-degree “rotations”
that permute the vertices of a tetr’on [while
preserving its “orientation”—if we allow for
reversals we get the cube-group S_4].
the vertices i call up, down, equal & opp.
in the “covering”, they become the color-pairs
mud-neuter yellow-purple, blue-orange, & red-green.
get it?
)

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