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A130795
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Multiaxial coordinate vectors normalized at Theta=0 and Phi=0 and rounded to the nearest integer ( "n" factor is added to make the integers show up better): based on cyclotomic angles for solving polynomials of the type x^n-1=0.
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1
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1, 2, 2, 3, 1, 1, 4, 0, 4, 0, 5, 0, 3, 3, 0, 6, 2, 2, 6, 2, 2, 7, 3, 0, 6, 6, 0, 3, 8, 4, 0, 4, 8, 4, 0, 4, 9, 5, 0, 2, 8, 8, 2, 0, 5, 10, 7, 1, 1, 7, 10, 7, 1, 1, 7
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OFFSET
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1,2
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COMMENTS
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The multidimensional coordinates give generalized cylinders in n dimension. The 3-dimensional example is a right cylinder : {x, y, z} = {Cos[p] Cos[t], Cos[p + (2 Pi)/3] Cos[(2 Pi)/3 + t], Cos[p + (4 Pi)/3] Cos[(4 Pi)/3 + t]}
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REFERENCES
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A torus based on the n=3 version of these coordinates was an MAA sticker by Paul Bourke: http://local.wasp.uwa.edu.au/~pbourke/surfaces_curves/tritorus/index.html
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LINKS
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FORMULA
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a(theta,phi,i,n)=Cos[theta + 2*i*Pi/n]*Cos[phi + 2*i*Pi/n]; t(n,i)=Round[n*a(0,0,i,n)]
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EXAMPLE
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{1},
{2, 2},
{3, 1, 1},
{4, 0, 4, 0},
{5, 0, 3, 3, 0},
{6, 2, 2, 6, 2, 2},
{7, 3, 0, 6, 6, 0, 3},
{8, 4, 0, 4, 8, 4, 0, 4},
{9, 5, 0, 2, 8, 8, 2, 0, 5},
{10, 7, 1, 1, 7, 10, 7, 1, 1, 7}
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MATHEMATICA
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a[t_, p_, i_, n_] = Cos[t + 2*i*Pi/n]*Cos[p + 2*i*Pi/n]; Table[Table[Round[n*a[t, p, i, n]], {i, 0, n - 1}], {n, 1, 10}] /. t -> 0 /. p -> 0; Flatten[%]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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