And now I have learnt it. I understand orbifolds and the group theory necessary to prove Conway's theorem. I have proved the theorem myself, and the corollary that there are only seventeen periodic tessellations of the plane. I still like the subject a lot and I am happy to have written my bachelor thesis about it.
Schlagwort: Mathematik
What academia should always be and feel like.
Equilibrium Pages.
Good and new ideas should always fit on two one page.
Musical Doughnut.
The topology of the syntonic temperament's tonnetz is generally cylindrical. Any syntonic "equal" tuning (i.e., a tempered width of the perfect fifth which divides the octave into a number of equally-wide intervals) snaps this cylinder into a torus(the shape of a ring doughnut, a hula hoop or an inflated tire), showing that it has a topology equivalent to S1×S1.
42.
For example the field of real numbers forms a structure R whose elements are the real numbers, with signature consisting of the individual constant 0 to name the number zero, a 1-ary function symbol - for minus, and two 2-ary function symbols + and . for plus and times. At first sight we can't add a symbol to express 1/x, since all the named functions have to be defined on the whole domain of the structure, and there is no such real number as 1/0. But on second thoughts this is not a serious problem; any competent mathematician puts the condition ‘x is not zero’ before dividing by x, and so it never matters what the value of 1/0 is, and we can harmlessly take it to be 42. But most model theorists are uncomfortable with any kind of division by zero, so they stick with plus, times and minus.
Mathfail.
All this fussing and primping about which “topics” should be taught in what order, or the use of this notation instead of that notation, or which make and model of calculator to use, for god’s sake — it’s like rearranging the deck chairs on the Titanic! Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion — not because it makes no sense to you, but because you gave it sense and you still don’t understand what your creation is up to; to have a breakthrough idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it. Remove this from mathematics and you can have all the conferences you like; it won’t matter. Operate all you want, doctors: your patient is already dead.