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Meta
Category Archives: sciences
tours
How many Euler circuits are there in the complete directed graph on N vertices? More interesting to me, how many equivalence classes under reversal and relabeling? N circuits: n(n-2) (n-2)!n classes 3 3 1 4 256 6 5 972000 4089 … Continue reading
Posted in mathematics
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Taylor series of integral of exponential of Fourier series
I have experimented with integrals of exp(i·f(t)) where ”f(t)” is a polynomial. I express these integrals as Taylor series, which are not hard to generate. Now I’d like to try f(t) = a·t + Σbk·sin(k·t) and I have a horrid … Continue reading
Posted in curve-fitting
1 Comment
more bent frets
Some simplified meantone guitars (here showing only the first octave): pentatonic (GDAEB); seven naturals; three flats and two sharps; six flats and six sharps.
Posted in mathematics, music+verse
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31 frets
Here is the neck of a guitar that I’d like to have made someday, if I should ever develop the dexterity to make it worthwhile. The blue stripes show where standard frets would be, for comparison. The tuning is my … Continue reading
Posted in mathematics, music+verse
2 Comments
phase-packing
I just thought of a new kind of packing problem, a mutant extension of the Thomson problem. In this version, each particle has coordinates in two independent spaces; in each it is confined to a sphere (of some dimension). In … Continue reading
Posted in mathematics
1 Comment
O fairest of randomizers
On most numbered dice, opposite sides are complementary; on a cube, for example, they add to 7. As a result, if you have the skill to throw a die so that the {1,2,3} corner lands on the table, the upward … Continue reading
Posted in mathematics
4 Comments
shining eyes
Could an animal have eyes like a reflecting telescope, rather than with a lens? The back of the eyeball is a paraboloid mirror, and the retina is a small body on its focal plane. Because the retina must be small, … Continue reading