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

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” (\hat A—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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