aristotle organized elements into four groups based on properties.
aristotle organized elements into four groups based on properties.
who organized elements into four groups based on properties?
aristotle
though boethius is drawing from aristotle's topics, "differentiae" are not the same as topics in some ways. boethius arranges "differentiae" through statements, instead of generalized groups as aristotle does. stump articulates the difference. they are "expressed as words or phrases whose expansion into appropriate propositions is neither intended nor readily conceivable", unlike aristotle's clearly defined four groups of topics. aristotle had hundreds of topics organized into those four groups, whereas boethius has twenty-eight "topics" that are "highly ordered among themselves." this distinction is necessary to understand boethius as separate from past rhetorical theories.
boethius
rank four groups also defined the 3-dimensional line groups:









































section::::rank four groups.:duoprismatic group.









































rank four groups defined the 4-dimensional duoprismatic groups. in the limit as p and q go to infinity, they degenerate into 2 dimensions and the wallpaper groups.









































section::::rank four groups.:wallpaper groups.









































rank four groups also defined some of the 2-dimensional wallpaper groups, as limiting cases of the four-dimensional duoprism groups:









































subgroups of [∞,2,∞], (*2222) can be expressed down to its index 16 commutator subgroup:









































section::::complex reflections.
coxeter notation