you’ve done it, vlorbik… cracked it wide open

June 4, 2020 in binary tetr'al, Graphics, Number Theory

here are 2 (of 4) panels from the newly-created
binary tetrahedral rosetta stone:
the code… what the hell…
{+000,-000,0+00,0-00,00+0,00-0,000+,000-}…
represents {1,-1,i,-i,j,-j,k,-k} (the “familiar”
unit quaternions; one has i^2 = j^2 = k^2 = ijk = -1
[per w.r.~hamilton; the margin is too small]).
then things get (slightly) messy… for
{—-,—+,–+-,–++,-+–,-+-+,-++-,-+++,
+—,+–+,+-+-,+-++,++–,++-+,+++-,++++}
we have weird sixth-roots of one; for example
++++ denotes h = (1+i+j+k)/2. (the “h” is for hurwitz).
mutatis mutandis for the rest, eg,
-+-+ = (-1+i-j+k)/2;
this is corresponds to “(mgy)(nrp)”
in the permutations-notation version,
as one can see from the photo.
together, the 24 trit-strings ({-,0,+} are
the *trits* in question…) represent the
*hurwitz units* U(Z(h,i,j,k)): the 24
invertible elements of the set of sums
(and differences) of i, j, k, & h.

the other two panels are the “matrix” version…
SL_2(F_3) so called…
and the “semi-direct product” version
where the “hi = jh” relation is made explicit
in the code. six graphical isomorphisms all told.

this would be the best lecture i ever gave
if i ever gave it. here it is for the internet.

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