discussions with kenji ...

?? mention idea of ... invariantization of decoration by superposition of translated copies ... ???? .... ??as "proof" of "every subgroupcomes from some decoration depicting some feature" ... ??? feature concept as abstracted from decoration concept, sort of ???....

??maybe ... ??wallpaper patterns ... ??? ... ???...

??/equivalence relation up to which they're classified ... ???

?? "pathological wallpaper pattern" ... ???

??? "wallpaper pattern" as "feature" of wall ??? .... ??or perhaps as "decoration" "depicting" feature ???...

?? "symmetry-breaking" ... ???....

??so ... ??not actually too sure about what traditionally qualifies as "wallpaper pattern" ... ???famous 29, or is it 37 or 23 or something??? .... ??cocompact?? ...????discrete ??? .... ??? ... ??certainly for purposes of talking to kenji want to consider features like "point", "line", and so forth .... which, ok, would get into non-discrete stabilizers ... probably a good idea ... ???? ..... ???some confusion here ??? .... ???? ..."all possible decorations of wall" ...???? .... co-compactness, vs .... ???? ..... ??? .... ?? "frame" ... ????.....

??in pattern p picture (some typical p ...), wall "looks like" ... ???certain compact manifold ... same dimension as group ... ???... ???not exactly my usual mental picture of a wall ... ????? .... ???....

??? ??"blank canvas" ... spectrum of unwritten writings that can be written on it ... logician's existence ... ??? borges's universal library ... ???....

???"pixie dust in kaleidoscope" ... ?? section on coxeter groups in those on-line notes on group theory by milne ....

?? "picture yourself as one of the specks of dust ..." ...

??? "sort-of classifying all concrete groups by sort-of classifying all abstract groups and ..." .... ??? ... subgroups and their cosets .... ??? .....

??kenji's question about "cosets" of non-subgroups ... ??? hmm, so what subgroup _is_ showing up here ??? ??subgroup generated by "(??right??) differences" between elements of subset ?? no... just ... subgroup of all elements under whose corresponding translation action the subset is closed ... ?? ?? "periodicities" ... ??? ... ... ???proof that for a subgroup this gives itself ?? ...

??hmm, subsets of Z as good playground here ??? ....

?? "characteristic functions of subgroups as representative of all symmetry (?? ...) classes of functions" ... ???.... ???? ... "characteristic function of set of periodicities" as idempotent map on function space ... ess taking each function to its symmetry class ... ?? .... ???

(?????situation where ... ideal carving out closed subgroup is principal for some "automatic" reason ??? ... ????? .... ???)