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Compasses and straightedges were their attempt to distill the nature of plane geometry down to its essential and simplest form (lines and circles).
by whatshisface 5y ago
Compasses and straightedges were their attempt to distill the nature of plane geometry down to its essential and simplest form (lines and circles).
- abecedarius 5y agoI wonder, though, is it not a coincidence that they're practical tools to do rather precise multi-step constructions? (E.g. Durer wrote a whole book about type design by those methods.) And with all the geometric algebra in Euclid, did they ever use them for calculations that aren't originally geometrical? Would we know?
- whatshisface 5y agoI believe that ancient geometry was used for governance and engineering. However the compass and straightedge had an element of abstractness or deliberate simplified impracticality even back then: they had rulers and strings, and could have practically used them as well.
- riidom 5y agoA note to the circles: It is not so much about circles, but rather "Some points which share the same distance to another point X" If you follow a manual of how to construct something with compass and straightedge, the job of the circles is often only to intersect with something else, and these points of intersection are of actual interest (as far as I remember).
- gilleain 5y agoExactly this. There are many patterns that can be constructed by drawing a regular array of circles, then connecting various intersection points with lines, then erasing the original circles. As it happens, I sometimes find that the same drawing can be achieved by a simpler construction path that involves (say) only midpoints of squares, which makes life a lot easier.
- thaumasiotes 5y agoHow do you find the midpoint of a square without drawing circles?
- whatshisface 5y agoDraw the diagonals.
- thaumasiotes 5y agoThat will work, but you need to have the entire square already marked. (As opposed to e.g. one or two of the sides marked and representing a hypothetical square.)
- ogogmad 5y agoIt can even be reduced to Clifford algebra very cleanly: https://en.wikipedia.org/wiki/Conformal_geometric_algebra https://en.wikipedia.org/wiki/Conformal_geometric_algebra There is something very "right" about it.
- gilleain 5y agoAgreed, I am sure that the drive for axiomisation of geometry drove a lot of this interest. All I really mean is that actually using these tools for an artistic, constructive purpose gives me a feel for why these problems might of been of interest. Of course, without knowing much about the history of mathematics this far back, I cannot be sure.