?? in discussion with kenji yesterday (unfortunately windowa journall malfunctioned and the notes were lost) ... kenji pointed out (ess ...) that "halving" structure on 4-point set amounts to "square" structure .... as the corners (??or equivalently sides??) of a square ... ?? the two halves being the two opposite diagonals ....

?? consider tetrahedron as "join" (is that what it's called?) of two opposite edges ... mapping down to interval with fiber over midpoint being sqaure ... ??? .. holding the tetrahedron so that that fiber is square in your visual field, so to speak ... ?? then where/how do the corners lie in visual field ... observer at infinity ... ??? .... ??? hmmm, i guess that they're lying at the midpoints of the sides (of an enlarged version of the square ...) .... ??? so maybe better to think of the halves as the "opposite-side pairs" than as the "diagonals" ....
?? because you can think of the "opposite-side pairs" as the two "dimensions" ("height" and "width", say ...) of the square ... ??which is sort of what's going with the "tetrahedron as join of two mid-line axises of middle square" picture ... ?????....

??switching between "diagonals" and "mid-lines" pictures as corresponding to an outer automorphism here ???..... .... ???just curious; does that outer automorphism become inner in 4! ??? .... ?? perhaps not, since seems to be order 2 outer automorphism ... ???? ..... ?? so what _is_ the whole outer automorphism group ??? ... ??? getting confused with "g / center(g)" ... ???conceptual interpretation of which is _what_ ??? ...

?? dihedral group always has "side/corner duality" outer involution ??? .... ??actually confuses me a bit at moment ... semi-direct product ... ??? .... ???






??is this square maybe of some interest in connection with quartic formula ??? .....


??came as bit of a cryptomorphism to me ... ??? maybe look for similar such ??

?? maybe this involved me failing to sufficiently appreciate "one person's decoration as another's graffiti" ??? .... which i thought of mentioning to kenji and probably should have but didn't ...


??where for example is ... ??does it seem like the quaternion unit group ought to be represented faithfully on a 4-elt set ??? ..... ??or no, maybe it's sort of clear that it can't ... "downward normalization" of subgroups as identity process here since famously all subgroups are nprmal here ... ?? vaguely reminds me of something baez sometimes says about lie algebra e8 (i think ...) ... "can't be faithfully represented on anything smaller than itself" ... ??? ....