module ConvexHull.Examples where import Data.List hiding (permutations) import Data.List.Split (chunksOf) import Math.Combinat.Permutations import System.Random sixhundredCell :: [[Double]] sixhundredCell = [[i, j, k, l] | i <- pm, j <- pm, k <- pm, l <-pm] ++ [[0, 0, 0, i*2] | i <- pm] ++ [[0, 0, i*2, 0] | i <- pm] ++ [[0, i*2, 0, 0] | i <- pm] ++ [[i*2, 0, 0, 0] | i <- pm] ++ [ permuteList p [i*phi, j, k/phi, 0] | p <- permutations4, isEvenPermutation p, i <- pm, j <- pm, k <- pm] where permutations4 = permutations 4 phi = (1 + sqrt 5) / 2 pm = [-1,1] hexadecachoron :: [[Double]] hexadecachoron = [ [1,0,0,0] , [-1,0,0,0] , [0,1,0,0] , [0,-1,0,0] , [0,0,1,0] , [0,0,-1,0] , [0,0,0,1] , [0,0,0,-1] ] runcitruncated5cell :: [[Double]] runcitruncated5cell = [[2/sqrt 10, -2/sqrt 6, 1/sqrt 3, 3] ,[2/sqrt 10, 2/sqrt 6, -1/sqrt 3, 3] ,[2/sqrt 10, -2/sqrt 6, 4/sqrt 3, 2] ,[2/sqrt 10, 2/sqrt 6, -4/sqrt 3, 2] ,[2/sqrt 10, -2/sqrt 6, -5/sqrt 3, 1] ,[2/sqrt 10, 2/sqrt 6, 5/sqrt 3, 1] ,[2/sqrt 10, sqrt 6, 0, 2] ,[2/sqrt 10, -sqrt 6, 0, 2] ,[2/sqrt 10, sqrt 6, sqrt 3, 1] ,[2/sqrt 10, -sqrt 6, sqrt 3, 1] ,[7/sqrt 10, -1/sqrt 6, 2/sqrt 3, 2] ,[7/sqrt 10, -1/sqrt 6, -4/sqrt 3, 0] ,[7/sqrt 10, 3/sqrt 6, 0, 2] ,[7/sqrt 10, 3/sqrt 6, sqrt 3, 1] ,[7/sqrt 10, 3/sqrt 6, -sqrt 3, 1] ,[7/sqrt 10, -5/sqrt 6, 1/sqrt 3, 1] ,[7/sqrt 10, -5/sqrt 6, -2/sqrt 3, 0] ,[-3/sqrt 10, 1/sqrt 6, 1/sqrt 3, 3] ,[-3/sqrt 10, 1/sqrt 6, 4/sqrt 3, 2] ,[-3/sqrt 10, 1/sqrt 6, -5/sqrt 3, 1] ,[-3/sqrt 10, 5/sqrt 6, 2/sqrt 3, 2] ,[-3/sqrt 10, 5/sqrt 6, -4/sqrt 3, 0] ,[-3/sqrt 10, -7/sqrt 6, -1/sqrt 3, 1] ,[-3/sqrt 10, -7/sqrt 6, 2/sqrt 3, 0] ,[-8/sqrt 10, 0, 0, 2] ,[-8/sqrt 10, 0, sqrt 3, 1] ,[-8/sqrt 10, 0, -sqrt 3, 1] ,[-8/sqrt 10, -4/sqrt 6, -1/sqrt 3, 1] ,[-8/sqrt 10, 4/sqrt 6, 1/sqrt 3, 1] ,[-8/sqrt 10, -4/sqrt 6, 2/sqrt 3, 0] ,[-8/sqrt 10, 4/sqrt 6, -2/sqrt 3, 0] ,[2/sqrt 10, -2/sqrt 6, 1/sqrt 3, -3] ,[2/sqrt 10, 2/sqrt 6, -1/sqrt 3, -3] ,[2/sqrt 10, -2/sqrt 6, 4/sqrt 3, -2] ,[2/sqrt 10, 2/sqrt 6, -4/sqrt 3, -2] ,[2/sqrt 10, -2/sqrt 6, -5/sqrt 3, -1] ,[2/sqrt 10, 2/sqrt 6, 5/sqrt 3, -1] ,[2/sqrt 10, sqrt 6, 0, -2] ,[2/sqrt 10, -sqrt 6, 0, -2] ,[2/sqrt 10, sqrt 6, -sqrt 3, -1] ,[2/sqrt 10, -sqrt 6, -sqrt 3, -1] ,[7/sqrt 10, -1/sqrt 6, 2/sqrt 3, -2] ,[7/sqrt 10, 3/sqrt 6, 0, -2] ,[7/sqrt 10, 3/sqrt 6, sqrt 3, -1] ,[7/sqrt 10, 3/sqrt 6, -sqrt 3, -1] ,[7/sqrt 10, -5/sqrt 6, 1/sqrt 3, -1] ,[-3/sqrt 10, 1/sqrt 6, 1/sqrt 3, -3] ,[-3/sqrt 10, 1/sqrt 6, 4/sqrt 3, -2] ,[-3/sqrt 10, 1/sqrt 6, -5/sqrt 3, -1] ,[-3/sqrt 10, 5/sqrt 6, 2/sqrt 3, -2] ,[-3/sqrt 10, -7/sqrt 6, -1/sqrt 3, -1] ,[-8/sqrt 10, 0, 0, -2] ,[-8/sqrt 10, 0, sqrt 3, -1] ,[-8/sqrt 10, 0, -sqrt 3, -1] ,[-8/sqrt 10, -4/sqrt 6, -1/sqrt 3, -1] ,[-8/sqrt 10, 4/sqrt 6, 1/sqrt 3, -1] ] cantellated5cell :: [[Double]] cantellated5cell = [[4/sqrt 10, 0, 0, 2] ,[4/sqrt 10, 0, 3/sqrt 3, 1] ,[4/sqrt 10, 0, -3/sqrt 3, 1] ,[4/sqrt 10, -4/sqrt 6, 2/sqrt 3, 0] ,[4/sqrt 10, 4/sqrt 6, -2/sqrt 3, 0] ,[4/sqrt 10, -4/sqrt 6, -1/sqrt 3, 1] ,[4/sqrt 10, 4/sqrt 6, 1/sqrt 3, 1] ,[-6/sqrt 10, -2/sqrt 6, -2/sqrt 3, 0] ,[-6/sqrt 10, 2/sqrt 6, 2/sqrt 3, 0] ,[-1/sqrt 10, -1/sqrt 6, -4/sqrt 3, 0] ,[-1/sqrt 10, -5/sqrt 6, -2/sqrt 3, 0] ,[-1/sqrt 10, 3/sqrt 6, 0, 2] ,[-1/sqrt 10, -1/sqrt 6, 2/sqrt 3, 2] ,[-1/sqrt 10, -5/sqrt 6, 1/sqrt 3, 1] ,[-1/sqrt 10, 3/sqrt 6, 3/sqrt 3, 1] ,[-1/sqrt 10, 3/sqrt 6, -3/sqrt 3, 1] ,[-6/sqrt 10, -2/sqrt 6, 1/sqrt 3, 1] ,[-6/sqrt 10, 2/sqrt 6, -1/sqrt 3, 1] ,[4/sqrt 10, 0, 0, -2] ,[4/sqrt 10, 0, 3/sqrt 3, -1] ,[4/sqrt 10, 0, -3/sqrt 3, -1] ,[4/sqrt 10, -4/sqrt 6, -1/sqrt 3, -1] ,[4/sqrt 10, 4/sqrt 6, 1/sqrt 3, -1] ,[-1/sqrt 10, 3/sqrt 6, 0, -2] ,[-1/sqrt 10, -1/sqrt 6, 2/sqrt 3, -2] ,[-1/sqrt 10, -5/sqrt 6, 1/sqrt 3, -1] ,[-1/sqrt 10, 3/sqrt 6, 3/sqrt 3, -1] ,[-1/sqrt 10, 3/sqrt 6, -3/sqrt 3, -1] ,[-6/sqrt 10, -2/sqrt 6, 1/sqrt 3, -1] ,[-6/sqrt 10, 2/sqrt 6, -1/sqrt 3, -1]] bitruncated5cell :: [[Double]] bitruncated5cell = [ [0, 4/sqrt 6, 4/sqrt 3, 0] , [0, -4/sqrt 6, -4/sqrt 3, 0] , [0, 4/sqrt 6, -2/sqrt 3, 2] , [0, 4/sqrt 6, -2/sqrt 3, -2] , [0, -4/sqrt 6, 2/sqrt 3, -2] , [0, -4/sqrt 6, 2/sqrt 3, 2] , [5/sqrt 10, 1/sqrt 6, 4/sqrt 3, 0] , [-5/sqrt 10, -1/sqrt 6, -4/sqrt 3, 0] , [5/sqrt 10, 1/sqrt 6, -2/sqrt 3, 2] , [5/sqrt 10, 1/sqrt 6, -2/sqrt 3, -2] , [-5/sqrt 10, -1/sqrt 6, 2/sqrt 3, -2] , [-5/sqrt 10, -1/sqrt 6, 2/sqrt 3, 2] , [5/sqrt 10, 5/sqrt 6, 2/sqrt 3, 0] , [-5/sqrt 10, -5/sqrt 6, -2/sqrt 3, 0] , [5/sqrt 10, 5/sqrt 6, -1/sqrt 3, 1] , [5/sqrt 10, 5/sqrt 6, -1/sqrt 3, -1] , [-5/sqrt 10, -5/sqrt 6, 1/sqrt 3, -1] , [-5/sqrt 10, -5/sqrt 6, 1/sqrt 3, 1] , [5/sqrt 10, -3/sqrt 6, 0, 2] , [5/sqrt 10, -3/sqrt 6, 0, -2] , [-5/sqrt 10, 3/sqrt 6, 0, -2] , [-5/sqrt 10, 3/sqrt 6, 0, 2] , [5/sqrt 10, -3/sqrt 6, sqrt 3, 1] , [5/sqrt 10, -3/sqrt 6, sqrt 3, -1] , [5/sqrt 10, -3/sqrt 6, -sqrt 3, 1] , [5/sqrt 10, -3/sqrt 6, -sqrt 3, -1] , [-5/sqrt 10, 3/sqrt 6, -sqrt 3, -1] , [-5/sqrt 10, 3/sqrt 6, -sqrt 3, 1] , [-5/sqrt 10, 3/sqrt 6, sqrt 3, -1] , [-5/sqrt 10, 3/sqrt 6, sqrt 3, 1] ] rectified5cell :: [[Double]] rectified5cell = [ [-3/sqrt 10, -3/sqrt 6, 0, 0] , [-3/sqrt 10, 1/sqrt 6, -2/sqrt 3, 0] , [-3/sqrt 10, 1/sqrt 6, 1/sqrt 3, 1] , [-3/sqrt 10, 1/sqrt 6, 1/sqrt 3, -1] , [2/sqrt 10, 2/sqrt 6, 2/sqrt 3, 0] , [2/sqrt 10, -2/sqrt 6, -2/sqrt 3, 0] , [2/sqrt 10, 2/sqrt 6, -1/sqrt 3, 1] , [2/sqrt 10, 2/sqrt 6, -1/sqrt 3, -1] , [2/sqrt 10, -2/sqrt 6, 1/sqrt 3, 1] , [2/sqrt 10, -2/sqrt 6, 1/sqrt 3, -1] ] sircope :: [[Double]] sircope = let a = (1+ sqrt 2)/2 in [ [-a, -0.5, -0.5, -0.5], [a, -0.5, -0.5, -0.5], [-a, 0.5, -0.5, -0.5], [a, 0.5, -0.5, -0.5], [-a, -0.5, 0.5, -0.5], [a, -0.5, 0.5, -0.5], [-a, 0.5, 0.5, -0.5], [a, 0.5, 0.5, -0.5], [-a, -0.5, -0.5, 0.5], [a, -0.5, -0.5, 0.5], [-a, 0.5, -0.5, 0.5], [a, 0.5, -0.5, 0.5], [-a, -0.5, 0.5, 0.5], [a, -0.5, 0.5, 0.5], [-a, 0.5, 0.5, 0.5], [a, 0.5, 0.5, 0.5], [-0.5, -a, -0.5, -0.5], [0.5, -a, -0.5, -0.5], [-0.5, a, -0.5, -0.5], [0.5, a, -0.5, -0.5], [-0.5, -a, 0.5, -0.5], [0.5, -a, 0.5, -0.5], [-0.5, a, 0.5, -0.5], [0.5, a, 0.5, -0.5], [-0.5, -a, -0.5, 0.5], [0.5, -a, -0.5, 0.5], [-0.5, a, -0.5, 0.5], [0.5, a, -0.5, 0.5], [-0.5, -a, 0.5, 0.5], [0.5, -a, 0.5, 0.5], [-0.5, a, 0.5, 0.5], [0.5, a, 0.5, 0.5], [-0.5, -0.5, -a, -0.5], [0.5, -0.5, -a, -0.5], [-0.5, 0.5, -a, -0.5], [0.5, 0.5, -a, -0.5], [-0.5, -0.5, a, -0.5], [0.5, -0.5, a, -0.5], [-0.5, 0.5, a, -0.5], [0.5, 0.5, a, -0.5], [-0.5, -0.5, -a, 0.5], [0.5, -0.5, -a, 0.5], [-0.5, 0.5, -a, 0.5], [0.5, 0.5, -a, 0.5], [-0.5, -0.5, a, 0.5], [0.5, -0.5, a, 0.5], [-0.5, 0.5, a, 0.5], [0.5, 0.5, a, 0.5] ] tutcup :: [[Double]] tutcup = [ [1, -1/sqrt 3, 5/sqrt 6, 1/sqrt 2] , [-1, -1/sqrt 3, 5/sqrt 6, 1/sqrt 2] , [0, 2/sqrt 3, 5/sqrt 6, 1/sqrt 2] , [2, -2/sqrt 3, 1/sqrt 6, 1/sqrt 2] , [-2, -2/sqrt 3, 1/sqrt 6, 1/sqrt 2] , [0, 4/sqrt 3, 1/sqrt 6, 1/sqrt 2] , [1, 3/sqrt 3, -3/sqrt 6, 1/sqrt 2] , [1, -3/sqrt 3, -3/sqrt 6, 1/sqrt 2] , [-1, 3/sqrt 3, -3/sqrt 6, 1/sqrt 2] , [-1, -3/sqrt 3, -3/sqrt 6, 1/sqrt 2] , [2, 0, -3/sqrt 6, 1/sqrt 2] , [-2, 0, -3/sqrt 6, 1/sqrt 2] , [1, 3/sqrt 3, 3/sqrt 6, -1/sqrt 2] , [1, -3/sqrt 3, 3/sqrt 6, -1/sqrt 2] , [-1, 3/sqrt 3, 3/sqrt 6, -1/sqrt 2] , [-1, -3/sqrt 3, 3/sqrt 6, -1/sqrt 2] , [2, 0, 3/sqrt 6, -1/sqrt 2] , [-2, 0, 3/sqrt 6, -1/sqrt 2] , [2, 2/sqrt 3, -1/sqrt 6, -1/sqrt 2] , [-2, 2/sqrt 3, -1/sqrt 6, -1/sqrt 2] , [0, -4/sqrt 3, -1/sqrt 6, -1/sqrt 2] , [1, 1/sqrt 3, -5/sqrt 6, -1/sqrt 2] , [-1, 1/sqrt 3, -5/sqrt 6, -1/sqrt 2] , [0, -2/sqrt 3, -5/sqrt 6, -1/sqrt 2] ] runcinatedTesseract :: [[Double]] runcinatedTesseract = [[i*a, j*b, k*c, l*d] | i <- pm, j <- pm, k <- pm, l <- pm, (a,b,c,d) <- s] where pm = [-1,1] x = 1 + sqrt 2 s = [(1,1,1,x),(1,1,x,1),(1,x,1,1),(x,1,1,1)] runcinated5cells :: [[Double]] runcinated5cells = [ [sqrt(5/2), 1/sqrt 6, 1/sqrt 3, 1] ,[sqrt(5/2), 1/sqrt 6, 1/sqrt 3, -1] ,[-sqrt(5/2), -1/sqrt 6, -1/sqrt 3, -1] ,[-sqrt(5/2), -1/sqrt 6, -1/sqrt 3, 1] ,[sqrt(5/2), 1/sqrt 6, -2/sqrt 3, 0] ,[-sqrt(5/2), -1/sqrt 6, 2/sqrt 3, 0] ,[sqrt(5/2), -sqrt(3/2), 0, 0] ,[-sqrt(5/2), sqrt(3/2), 0, 0] ,[0, 2*sqrt(2/3), 1/sqrt 3, 1] ,[0, 2*sqrt(2/3), 1/sqrt 3, -1] ,[0, -2*sqrt(2/3), -1/sqrt 3, -1] ,[0, -2*sqrt(2/3), -1/sqrt 3, 1] ,[0, 2*sqrt(2/3), -2/sqrt 3, 0] ,[0, -2*sqrt(2/3), 2/sqrt 3, 0] ,[0, 0, sqrt 3, 1] ,[0, 0, sqrt 3, -1] ,[0, 0, -sqrt 3, 1] ,[0, 0, -sqrt 3, -1] ,[0, 0, 0, 2] ,[0, 0, 0, -2] ] truncated5cells :: [[Double]] truncated5cells = [ [3/sqrt 10, -1/sqrt 6, 2/sqrt 3, 2] ,[3/sqrt 10, -1/sqrt 6, 2/sqrt 3, -2] ,[3/sqrt 10, -1/sqrt 6, -4/sqrt 3, 0] ,[3/sqrt 10, 3/sqrt 6, 0, 2] ,[3/sqrt 10, 3/sqrt 6, 0, -2] ,[3/sqrt 10, 3/sqrt 6, sqrt 3, 1] ,[3/sqrt 10, 3/sqrt 6, sqrt 3, -1] ,[3/sqrt 10, 3/sqrt 6, -sqrt 3, 1] ,[3/sqrt 10, 3/sqrt 6, -sqrt 3, -1] ,[3/sqrt 10, -5/sqrt 6, 1/sqrt 3, 1] ,[3/sqrt 10, -5/sqrt 6, 1/sqrt 3, -1] ,[3/sqrt 10, -5/sqrt 6, -2/sqrt 3, 0] ,[-2/sqrt 10, 2/sqrt 6, 2/sqrt 3, 2] ,[-2/sqrt 10, 2/sqrt 6, 2/sqrt 3, -2] ,[-2/sqrt 10, 2/sqrt 6, -4/sqrt 3, 0] ,[-2/sqrt 10, -sqrt 6, 0, 0] ,[-7/sqrt 10, 1/sqrt 6, 1/sqrt 3, 1] ,[-7/sqrt 10, 1/sqrt 6, 1/sqrt 3, -1] ,[-7/sqrt 10, 1/sqrt 6, -2/sqrt 3, 0] ,[-7/sqrt 10, -3/sqrt 6, 0, 0] ] truncated24cells :: [[Double]] truncated24cells = [permuteList p [0,i,j*2,k*3] | p <- permutations 4, i <- pm, j <- pm , k <- pm] where pm = [-1,1] thex :: [[Double]] thex = nub [ permuteList p [0,0,i,j*2] | p <- permutations 4, i <- pm, j <- pm] where pm = [-1,1] truncatedTetrahedron :: [[Double]] truncatedTetrahedron = [ [1/sqrt 6, -2/sqrt 3, 2] , [1/sqrt 6, -2/sqrt 3, -2] , [1/sqrt 6, 4/ sqrt 3, 0] , [-3/sqrt 6, 0, 2] , [-3/sqrt 6, 0, -2] , [-3/sqrt 6, sqrt 3, 1] , [-3/sqrt 6, -sqrt 3, 1] , [-3/sqrt 6, sqrt 3, -1] , [-3/sqrt 6, -sqrt 3, -1] , [5/sqrt 6, -1/sqrt 3, 1] , [5/sqrt 6, -1/sqrt 3, -1] , [5/sqrt 6, 2/sqrt 3, 0] ] daVinci :: [[Double]] daVinci = [ [1.61352, -0.43234, 1.1862], [1.18118, -1.18118, 1.1862], [0.43234, -1.61352, 1.1862], [-0.43234, -1.61352, 1.1862], [-1.18118, -1.18118, 1.1862], [-1.61352, -0.43234, 1.1862], [-1.61352, 0.43234, 1.1862], [-1.18118, 1.18118, 1.1862], [-0.43234, 1.61352, 1.1862], [0.43234, 1.61352, 1.1862], [1.18118, 1.18118, 1.1862], [1.61352, 0.43234, 1.1862], [1.61352, -0.43234, -1.1862], [1.61352, 0.43234, -1.1862], [1.18118, 1.18118, -1.1862], [0.43234, 1.61352, -1.1862], [-0.43234, 1.61352, -1.1862], [-1.18118, 1.18118, -1.1862], [-1.61352, 0.43234, -1.1862], [-1.61352, -0.43234, -1.1862], [-1.18118, -1.18118, -1.1862], [-0.43234, -1.61352, -1.1862], [0.43234, -1.61352, -1.1862], [1.18118, -1.18118, -1.1862], [2.0102, 0.53863, 0], [1.47157, 1.47157, 0], [0.53863, 2.0102, 0], [-0.53863, 2.0102, 0], [-1.47157, 1.47157, 0], [-2.0102, 0.53863, 0], [-2.0102, -0.53863, 0], [-1.47157, -1.47157, 0], [-0.53863, -2.0102, 0], [0.53863, -2.0102, 0], [1.47157, -1.47157, 0], [2.0102, -0.53863, 0], [0.89068, 0.23866, 1.77777], [0.89068, -0.23866, 1.77777], [0.65202, -0.65202, 1.77777], [0.23866, -0.89068, 1.77777], [-0.23866, -0.89068, 1.77777], [-0.65202, -0.65202, 1.77777], [-0.89068, -0.23866, 1.77777], [-0.89068, 0.23866, 1.77777], [-0.65202, 0.65202, 1.77777], [-0.23866, 0.89068, 1.77777], [0.23866, 0.89068, 1.77777], [0.65202, 0.65202, 1.77777], [0.65202, -0.65202, -1.77777], [0.89068, -0.23866, -1.77777], [0.89068, 0.23866, -1.77777], [0.65202, 0.65202, -1.77777], [0.23866, 0.89068, -1.77777], [-0.23866, 0.89068, -1.77777], [-0.65202, 0.65202, -1.77777], [-0.89068, 0.23866, -1.77777], [-0.89068, -0.23866, -1.77777], [-0.65202, -0.65202, -1.77777], [-0.23866, -0.89068, -1.77777], [0.23866, -0.89068, -1.77777], [0, 0, 2.04922], [0, 0, -2.04922]] -- Cuboctohadron4D - does not work cuboctahedron4d :: [[Double]] cuboctahedron4d = nub [ permuteList p [0,1,1,2] | p <- permutations 4] -- OLOID -- twocircles :: [[Double]] twocircles = circle1 `union` circle2 where circle1 = [[cos (realToFrac i * 2*pi/30), sin (realToFrac i * 2*pi/30), 0] | i <- [0 .. 29]] circle2 = [[0, cos (realToFrac i * 2*pi/30) - 1, sin (realToFrac i * 2*pi/30)] | i <- [0 .. 29]] translate3 :: [[Double]] -> [Double] -> [[Double]] translate3 points u = map (zipWith (+) u) points duplicate3 :: [[Double]] -> [Double] -> [[Double]] duplicate3 points u = nub $ points ++ translate3 points u rgg :: [[Double]] rgg = [[-5,-5, 16], [-5, 8, 3 ], [ 4,-1, 3 ], [ 4,-5, 7], [ 4,-1,-10], [ 4,-5,-10], [-5, 8,-10], [-5,-5,-10]] centricCube :: [[Double]] centricCube = [[-1,-1,-1],[-1,-1, 1],[-1, 1,-1],[-1, 1, 1],[ 1,-1,-1], [ 1,-1, 1],[ 1, 1,-1],[ 1, 1, 1],[ 0, 0, 0]] squareLattice :: [[Double]] squareLattice = [[0,0],[0,1],[0,2] ,[1,0],[1,1],[1,2] ,[2,0],[2,1],[2,2]] centricSquare :: [[Double]] centricSquare = [[0,0],[0,2],[2,0],[2,2],[1,1]] cuboctahedron :: [[Double]] cuboctahedron = [[i,j,0] | i <- [-1,1], j <- [-1,1]] ++ [[i,0,j] | i <- [-1,1], j <- [-1,1]] ++ [[0,i,j] | i <- [-1,1], j <- [-1,1]] truncatedCuboctahedron :: [[Double]] truncatedCuboctahedron = [[i, j * (1 + sqrt 2), k * (1 + 2*sqrt 2)] | i <- pm, j <- pm, k <- pm] ++ [[j * (1 + sqrt 2), i, k * (1 + 2*sqrt 2)] | i <- pm, j <- pm, k <- pm] ++ [[j * (1 + sqrt 2), k * (1 + 2*sqrt 2), i] | i <- pm, j <- pm, k <- pm] ++ [[i, k * (1 + 2*sqrt 2), j * (1 + sqrt 2)] | i <- pm, j <- pm, k <- pm] ++ [[k * (1 + 2*sqrt 2), i, j * (1 + sqrt 2)] | i <- pm, j <- pm, k <- pm] ++ [[k * (1 + 2*sqrt 2), j * (1 + sqrt 2), i] | i <- pm, j <- pm, k <- pm] where pm = [-1,1] rhombicDodecahedron :: [[Double]] rhombicDodecahedron = [[-1.0, 0.0, 0.0], [-0.5,-0.5,-0.5], [-0.5,-0.5, 0.5], [ 0.0,-1.0, 0.0], [-0.5, 0.5,-0.5], [-0.5, 0.5, 0.5], [ 0.0, 1.0, 0.0], [ 1.0, 0.0, 0.0], [ 0.5,-0.5,-0.5], [ 0.5,-0.5, 0.5], [ 0.5, 0.5,-0.5], [ 0.5, 0.5, 0.5], [ 0.0, 0.0,-1.0], [ 0.0, 0.0, 1.0]] faceCenteredCubic :: [[Double]] faceCenteredCubic = [[-1,-1,-1],[-1,-1,1],[-1,1,-1],[-1,1,1] ,[1,-1,-1],[1,-1,1],[1,1,-1],[1,1,1] ,[1,0,0],[-1,0,0] ,[0,1,0],[0,-1,0] ,[0,0,1],[0,0,-1]] waves :: [[Double]] waves = [f u v | u <- seq_u, v <- seq_v] where f u v = [u, 0.05*(sin(v*4*pi)+sin(u*4*pi)), v] frac p q = realToFrac p/ realToFrac q n = 10 seq_u = [8 * frac i n - 4 | i <- [0 .. n]] seq_v = [4 * frac i n - 2 | i <- [0 .. n]] waves2D :: [[Double]] waves2D = concatMap (\y' -> map (\[x,y] -> [x, y+y']) sinusoid) seq_y where seq_x = [realToFrac i * pi/2 | i <- [0 .. 20]] sinusoid = map (\(x,y) -> [x,y]) (zip seq_x (map sin seq_x)) seq_y = [realToFrac 2*i | i <- [0 .. 20]] type Vertex = [Double] ncube :: Int -> [Vertex] ncube n = concatMap (mapM (\x -> nub [x,-x])) [replicate n 1] cube3 :: [[Double]] cube3 = [[i,j,k] | i <- [-1,1], j <- [-1,1], k <- [-1,1]] cube4 :: [[Double]] cube4 = [[i,j,k,l] | i <- [-1,1], j <- [-1,1], k <- [-1,1], l <- [-1,1]] cube5 :: [[Double]] cube5 = [[i,j,k,l,m] | i <- [-1,1], j <- [-1,1], k <- [-1,1], l <- [-1,1], m <- [-1,1]] cube6 :: [[Double]] cube6 = [[i,j,k,l,m,n] | i <- [-1,1], j <- [-1,1], k <- [-1,1], l <- [-1,1], m <- [-1,1], n <- [-1,1]] mobiusStrip :: [[Double]] mobiusStrip = map (\(u,v) -> [ cos u * (1 + v/2 * cos(u/2)) , sin u * (1 + v/2 * cos(u/2)) , v/2 * sin(u/2) ]) uv where uv = [(u,v) | u <- u_, v <- v_] u_ = [i/50 * 2 *pi | i <- [0 .. 50]] v_ = [-1,1] -- [i/50 | i <- [-50 .. 50]] irregularPolyhedron :: [[Double]] irregularPolyhedron = [ [ -0.586233 , 0.192482 , -1.9732e-2 ] , [ -0.344233 , 0.301871 , 0.358301 ] , [ 0.344233 , -0.301871 , 0.358301 ] , [ 0.179112 , -0.514398 , -0.290554 ] , [ 1.1612e-2 , -0.564879 , 0.137037 ] , [ -1.1612e-2 , 0.564879 , 0.137037 ] , [ -0.179112 , 0.514398 , -0.290554 ] , [ -0.151548 , -0.185001 , -0.529915 ] , [ -5.5686e-2 , -0.430284 , -0.411612 ] , [ -0.364663 , -0.382652 , -0.277601 ] , [ -0.394163 , -0.151218 , -0.422176 ] , [ 5.5686e-2 , 0.430284 , -0.411612 ] , [ 0.364663 , 0.382652 , -0.277601 ] , [ 0.394163 , 0.151218 , -0.422176 ] , [ 0.151548 , 0.185001 , -0.529915 ] , [ 0.361372 , 0.268614 , 0.393607 ] , [ 0.246149 , 0.511456 , 0.185169 ] , [ 0.417404 , 0.426728 , -1.1876e-2 ] , [ 0.510787 , 0.234322 , 0.148969 ] , [ -0.361372 , -0.268614 , 0.393608 ] , [ -0.510787 , -0.234322 , 0.148969 ] , [ -0.417404 , -0.426728 , -1.1876e-2 ] , [ -0.246149 , -0.511456 , 0.185169 ] , [ 0.430396 , -0.334808 , -0.249032 ] , [ 0.586233 , -0.192482 , -1.9732e-2 ] , [ 0.587567 , -0.188189 , -2.1408e-2 ] , [ 0.509201 , -0.136702 , -0.282307 ] , [ -0.430396 , 0.334808 , -0.249032 ] , [ -0.509201 , 0.136702 , -0.282307 ] , [ -0.587567 , 0.188189 , -2.1408e-2 ] , [ -0.272269 , -0.118712 , 0.520703 ] , [ -0.248355 , 9.5223e-2 , 0.535651 ] , [ 0.272269 , 0.118712 , 0.520703 ] , [ -2.22e-4 , 2.388e-3 , 0.617335 ] , [ 2.22e-4 , -2.388e-3 , 0.617335 ] , [ 0.248355 , -9.5223e-2 , 0.535651 ] ] truncatedCube :: [[Double]] truncatedCube = let x = 1 + sqrt 2 in [[i, j*x, k*x] | i <- [-1,1], j <- [-1,1], k <- [-1,1]] ++ [[i*x, j, k*x] | i <- [-1,1], j <- [-1,1], k <- [-1,1]] ++ [[i*x, j*x, k] | i <- [-1,1], j <- [-1,1], k <- [-1,1]] truncatedTesseract :: [[Double]] truncatedTesseract = let x = 1 + sqrt 2 in [[i, j*x, k*x, l*x] | i <- [-1,1], j <- [-1,1], k <- [-1,1], l <- [-1,1]] ++ [[i*x, j, k*x, l*x] | i <- [-1,1], j <- [-1,1], k <- [-1,1], l <- [-1,1]] ++ [[i*x, j*x, k, l*x] | i <- [-1,1], j <- [-1,1], k <- [-1,1], l <- [-1,1]] ++ [[i*x, j*x, k*x, l] | i <- [-1,1], j <- [-1,1], k <- [-1,1], l <- [-1,1]] projectedTruncatedTesseract :: [[Double]] projectedTruncatedTesseract = [ [-x1,-x2,-x2], [-x4,-x3,-x3], [-x1,-x2, x2], [-x4,-x3, x3], [-x1, x2,-x2], [-x4, x3,-x3], [-x1, x2, x2], [-x4, x3, x3], [ x1,-x2,-x2], [ x4,-x3,-x3], [ x1,-x2, x2], [ x4,-x3, x3], [ x1, x2,-x2], [ x4, x3,-x3], [ x1, x2, x2], [ x4, x3, x3], [-x2,-x1,-x2], [-x3,-x4,-x3], [-x2,-x1, x2], [-x3,-x4, x3], [-x2, x1,-x2], [-x3, x4,-x3], [-x2, x1, x2], [-x3, x4, x3], [ x2,-x1,-x2], [ x3,-x4,-x3], [ x2,-x1, x2], [ x3,-x4, x3], [ x2, x1,-x2], [ x3, x4,-x3], [ x2, x1, x2], [ x3, x4, x3], [-x2,-x2,-x1], [-x3,-x3,-x4], [-x2,-x2, x1], [-x3,-x3, x4], [-x2, x2,-x1], [-x3, x3,-x4], [-x2, x2, x1], [-x3, x3, x4], [ x2,-x2,-x1], [ x3,-x3,-x4], [ x2,-x2, x1], [ x3,-x3, x4], [ x2, x2,-x1], [ x3, x3,-x4], [ x2, x2, x1], [ x3, x3, x4], [-x6,-x6,-x6], [-x5,-x5,-x5], [-x6,-x6, x6], [-x5,-x5, x5], [-x6, x6,-x6], [-x5, x5,-x5], [-x6, x6, x6], [-x5, x5, x5], [ x6,-x6,-x6], [ x5,-x5,-x5], [ x6,-x6, x6], [ x5,-x5, x5], [ x6, x6,-x6], [ x5, x5,-x5], [ x6, x6, x6], [ x5, x5, x5]] where a = 1 + sqrt 2 x1 = 2 / (sqrt(1 + 3*a*a) + a) -- 0.29 x2 = 2*a / (sqrt(1 + 3*a*a) + a) -- 0.71 x3 = 2*a / (sqrt(1 + 3*a*a) - a) -- 2.56 x4 = 2 / (sqrt(1 + 3*a*a) - a) -- 1.06 x5 = 2*a / (sqrt(1 + 3*a*a) - 1) -- 1.46 x6 = 2*a / (sqrt(1 + 3*a*a) + 1) -- 0.91 rectifiedTesseract :: [[Double]] rectifiedTesseract = let x = sqrt 2 in [[0, j*x, k*x, l*x] | j <- [-1,1], k <- [-1,1], l <- [-1,1]] ++ [[i*x, 0, k*x, l*x] | i <- [-1,1], k <- [-1,1], l <- [-1,1]] ++ [[i*x, j*x, 0, l*x] | i <- [-1,1], j <- [-1,1], l <- [-1,1]] ++ [[i*x, j*x, k*x, 0] | i <- [-1,1], j <- [-1,1], k <- [-1,1]] cantellatedTesseract :: [[Double]] cantellatedTesseract = let x = 1 + sqrt 2 in map (map (/ sqrt(2 + 2*x*x))) $ [[i, j, k*x, l*x] | i <- [-1,1], j <- [-1,1], k <- [-1,1], l <- [-1,1]] ++ [[i*x, j, k, l*x] | i <- [-1,1], j <- [-1,1], k <- [-1,1], l <- [-1,1]] ++ [[i*x, j*x, k, l] | i <- [-1,1], j <- [-1,1], k <- [-1,1], l <- [-1,1]] ++ [[i, j*x, k, l*x] | i <- [-1,1], j <- [-1,1], k <- [-1,1], l <- [-1,1]] ++ [[i, j*x, k*x, l] | i <- [-1,1], j <- [-1,1], k <- [-1,1], l <- [-1,1]] ++ [[i*x, j, k*x, l] | i <- [-1,1], j <- [-1,1], k <- [-1,1], l <- [-1,1]] projectedCantellatedTesseract :: [[Double]] projectedCantellatedTesseract = map stereographic cantellatedTesseract where stereographic x = map (/(1-x!!3)) [2 * x!!0, 2 * x!!1, 2 * x!!2] octaplex :: [[Double]] octaplex = map (map (/ sqrt 2)) $ [[i,j,0,0] | i <- pm, j <- pm] ++ [[i,0,j,0] | i <- pm, j <- pm] ++ [[i,0,0,j] | i <- pm, j <- pm] ++ [[0,0,i,j] | i <- pm, j <- pm] ++ [[0,i,0,j] | i <- pm, j <- pm] ++ [[0,i,j,0] | i <- pm, j <- pm] where pm = [-1,1] nonConvexPolyhedron :: [[Double]] nonConvexPolyhedron = [[i*x, j*x, k*x] | i <- [-1,1], j <- [-1,1], k <- [-1,1]] ++ [[i*y, 0, 0] | i <- [-1,1]] ++ [[0, j*y, 0] | j <- [-1,1]] ++ [[0, 0, k*y] | k <- [-1,1]] where x = 2.1806973249 y = 3.5617820682 -- tetrahedra: (i 2.18, j 2.18, k 2.18), (i 3.56, 0, 0), (0, j 3.56, 0), (0, 0, k 3.56) dodecahedron :: [[Double]] -- it's icosahedron ! dodecahedron = let phi = (1 + sqrt 5)/2 in [[0,i,j] | i <- [-1,1], j <- [-phi, phi]] ++ [[j,0,i] | i <- [-1,1], j <- [-phi, phi]] ++ [[i,j,0] | i <- [-1,1], j <- [-phi, phi]] icosahedron :: [[Double]] icosahedron = dodecahedron duoprism330 :: [[Double]] duoprism330 = [a ++ b | a <- triangle, b <- p30] where triangle = [[cos (realToFrac i * 2*pi/3), sin (realToFrac i * 2*pi/3)] | i <- [0 .. 2]] p30 = [[cos (realToFrac i * 2*pi/30), sin (realToFrac i * 2*pi/30)] | i <- [0 .. 29]] duoprism35 :: [[Double]] duoprism35 = [a ++ b | a <- triangle, b <- pentagon] where triangle = [ [sqrt 3 / 2 , 0.5] , [-sqrt 3 / 2, 0.5] , [0 , -1 ] ] pentagon = [ [1 , 0 ] , [cos(2*pi/5), sin(2*pi/5)] , [cos(4*pi/5), sin(4*pi/5)] , [cos(6*pi/5), sin(6*pi/5)] , [cos(8*pi/5), sin(8*pi/5)] ] duoprism34 :: [[Double]] duoprism34 = [a ++ b | a <- triangle, b <- square] where triangle = [ [sqrt 3 / 2 , 0.5] , [-sqrt 3 / 2, 0.5] , [0 , -1 ] ] square = [[i * sqrt 2 / 2, j * sqrt 2 / 2] | i <- [-1,1], j <- [-1,1]] triangularDuoprism :: [[Double]] triangularDuoprism = [a ++ b | a <- triangle, b <- triangle] where triangle = [ [sqrt 3 / 2 , 0.5] , [-sqrt 3 / 2, 0.5] , [0 , -1 ] ] hexagonalDuoprism :: [[Double]] hexagonalDuoprism = [a ++ b | a <- hexagon, b <- hexagon] where hexagon = [ [sqrt 3 / 2, 0.5] , [0 , 1 ] , [-sqrt 3 / 2, 0.5] , [-sqrt 3 / 2, -0.5] , [0 , -1 ] , [sqrt 3 / 2 , -0.5] ] projectedHexagonalDuoprims :: [[Double]] projectedHexagonalDuoprims = [ [ 1.8945800837529239 , 1.0938363213560542 , 1.8945800837529239 ] , [ 4.181540550352056 , 2.414213562373096 , 0.0 ] , [ 1.8945800837529239 , 1.0938363213560542 , -1.8945800837529239 ] , [ 0.904836765142137 , 0.522407749927483 , -0.904836765142137 ] , [ 0.7174389352143009 , 0.4142135623730951 , 0.0 ] , [ 0.904836765142137 , 0.522407749927483 , 0.904836765142137 ] , [ 0.0 , 2.1876726427121085 , 1.8945800837529239 ] , [ 0.0 , 4.828427124746192 , 0.0 ] , [ 0.0 , 2.1876726427121085 , -1.8945800837529239 ] , [ 0.0 , 1.044815499854966 , -0.904836765142137 ] , [ 0.0 , 0.8284271247461902 , 0.0 ] , [ 0.0 , 1.044815499854966 , 0.904836765142137 ] , [ -1.8945800837529239 , 1.0938363213560542 , 1.8945800837529239 ] , [ -4.181540550352056 , 2.414213562373096 , 0.0 ] , [ -1.8945800837529239 , 1.0938363213560542 , -1.8945800837529239 ] , [ -0.904836765142137 , 0.522407749927483 , -0.904836765142137 ] , [ -0.7174389352143009 , 0.4142135623730951 , 0.0 ] , [ -0.904836765142137 , 0.522407749927483 , 0.904836765142137 ] , [ -1.8945800837529239 , -1.0938363213560542 , 1.8945800837529239 ] , [ -4.181540550352056 , -2.414213562373096 , 0.0 ] , [ -1.8945800837529239 , -1.0938363213560542 , -1.8945800837529239 ] , [ -0.904836765142137 , -0.522407749927483 , -0.904836765142137 ] , [ -0.7174389352143009 , -0.4142135623730951 , 0.0 ] , [ -0.904836765142137 , -0.522407749927483 , 0.904836765142137 ] , [ 0.0 , -2.1876726427121085 , 1.8945800837529239 ] , [ 0.0 , -4.828427124746192 , 0.0 ] , [ 0.0 , -2.1876726427121085 , -1.8945800837529239 ] , [ 0.0 , -1.044815499854966 , -0.904836765142137 ] , [ 0.0 , -0.8284271247461902 , 0.0 ] , [ 0.0 , -1.044815499854966 , 0.904836765142137 ] , [ 1.8945800837529239 , -1.0938363213560542 , 1.8945800837529239 ] , [ 4.181540550352056 , -2.414213562373096 , 0.0 ] , [ 1.8945800837529239 , -1.0938363213560542 , -1.8945800837529239 ] , [ 0.904836765142137 , -0.522407749927483 , -0.904836765142137 ] , [ 0.7174389352143009 , -0.4142135623730951 , 0.0 ] , [ 0.904836765142137 , -0.522407749927483 , 0.904836765142137 ] ] hexaSquare :: [[Double]] hexaSquare = [a ++ b | a <- hexagon, b <- square] where hexagon = [ [sqrt 3 / 2, 0.5] , [0 , 1 ] , [-sqrt 3 / 2, 0.5] , [-sqrt 3 / 2, -0.5] , [0 , -1 ] , [sqrt 3 / 2 , -0.5] ] square = [[i * sqrt 2 / 2, j * sqrt 2 / 2] | i <- [-1,1], j <- [-1,1]] cubinder :: [[Double]] cubinder = [a ++ b | a <- circle, b <- square] where circle = [[cos (realToFrac i * 2*pi/30), sin (realToFrac i * 2*pi/30)] | i <- [0 .. 29]] square = [[i * sqrt 2 / 2, j * sqrt 2 / 2] | i <- [-1,1], j <- [-1,1]] duoprism1616 :: [[Double]] duoprism1616 = [a ++ b | a <- p16, b <- p16] where p16 = [[cos (realToFrac i * 2*pi/16), sin (realToFrac i * 2*pi/16)] | i <- [0 .. 15]] duocylinder :: [[Double]] duocylinder = [a ++ b | a <- circle, b <- circle] where circle = [[cos (realToFrac i * 2*pi/30), sin (realToFrac i * 2*pi/30)] | i <- [0 .. 29]] reuleuxTetrahedron :: [[Double]] reuleuxTetrahedron = [ [0.5 / sqrt 3, -0.5, 0.5 / sqrt 6] , [sqrt 3 / 3, 0, -0.5 / sqrt 6] , [0.5 / sqrt 3, 0.5, -0.5 / sqrt 6] , [0, 0, 0.5 * sqrt 3 / sqrt 2] ] hexacosichoron :: [[Double]] hexacosichoron = [[i*0.5, j*0.5, k*0.5, l*0.5] | i <- pm, j <- pm, k <- pm, l <-pm] ++ [[0, 0, 0, i] | i <- pm] ++ [[0, 0, i, 0] | i <- pm] ++ [[0, i, 0, 0] | i <- pm] ++ [[i, 0, 0, 0] | i <- pm] ++ [ permuteList p [phi/2, 1/2, 1/2/phi, 0] | p <- permutations4, isEvenPermutation p] ++ [ permuteList p [phi/2, 1/2, -1/2/phi, 0] | p <- permutations4, isEvenPermutation p] ++ [ permuteList p [phi/2, -1/2, 1/2/phi, 0] | p <- permutations4, isEvenPermutation p] ++ [ permuteList p [phi/2, -1/2, -1/2/phi, 0] | p <- permutations4, isEvenPermutation p] ++ [ permuteList p [-phi/2, 1/2, 1/2/phi, 0] | p <- permutations4, isEvenPermutation p] ++ [ permuteList p [-phi/2, 1/2, -1/2/phi, 0] | p <- permutations4, isEvenPermutation p] ++ [ permuteList p [-phi/2, -1/2, 1/2/phi, 0] | p <- permutations4, isEvenPermutation p] ++ [ permuteList p [-phi/2, -1/2, -1/2/phi, 0] | p <- permutations4, isEvenPermutation p] where permutations4 = permutations 4 phi = (1 + sqrt 5) / 2 pm = [-1,1] snub24cell :: [[Double]] snub24cell = [ permuteList p [0, i, i'*phi, i''*phi*phi] | i <- pm, i' <- pm, i'' <- pm, p <- permutations4, isEvenPermutation p] where permutations4 = permutations 4 phi = (1 + sqrt 5) / 2 pm = [-1,1] dodecaplex :: [[Double]] dodecaplex = nub $ [permuteList p [0.0, 0.0, i, j] | i <- pm2 , j <- pm2, p <- perms4] ++ [permuteList p [i, j, k, l] | i <- pm, j <- pm, k <-pm, l <- pmsqrt5, p <- perms4] ++ [permuteList p [i ,j ,k ,l] | i <- pmphipowminus2, j <- pmphi, k <- pmphi, l <- pmphi, p <- perms4] ++ [permuteList p [i ,j ,k ,l] | i <- pmphipowminus1, j <- pmphipowminus1, k <- pmphipowminus1, l <- pmphipow2, p <- perms4] ++ [permuteList p [0, i, j, k] | i <- pmphipowminus2, j <- pm, k <- pmphipow2, p <- perms4, isEvenPermutation p] ++ [permuteList p [0, i, j, k] | i <- pmphipowminus1, j <- pmphi, k <- pmsqrt5, p <- perms4, isEvenPermutation p] ++ [permuteList p [i, j, k, l] | i <- pmphipowminus1, j <- pm, k <- pmphi, l <- pm2, p <- perms4, isEvenPermutation p] where pm = [-1.0,1.0] pm2 = [-2, 2] perms4 = permutations 4 pmsqrt5 = [-sqrt 5, sqrt 5] pmphipowminus2 = [-1/phi/phi, 1/phi/phi] pmphipow2 = [-phi*phi, phi*phi] phi = (1 + sqrt 5) / 2 -- pmphiminus1 = phi-1 pmphi = [phi, -phi] pmphipowminus1 = [-1/phi, 1/phi] spheresPack :: [[Double]] spheresPack = [ [2,1,1], [4,1,1] , [1, 1 + sqrt 3, 1], [3, 1 + sqrt 3, 1], [5, 1 + sqrt 3, 1] , [2, 1 + 2*sqrt 3, 1], [4, 1 + 2*sqrt 3, 1] , [3, 1 + sqrt 3 / 3, 1 + 2*sqrt 6 / 3] , [2, 1 + 4*sqrt 3 / 3, 1 + 2*sqrt 6 / 3], [4, 1 + 4*sqrt 3 / 3, 1 + 2*sqrt 6 / 3] , [3, 1 + sqrt 3, 1 + 4*sqrt 6 /3] , [3, 1 + sqrt 3 / 3, 1 - 2*sqrt 6 / 3] , [2, 1 + 4*sqrt 3 / 3, 1 - 2*sqrt 6 / 3], [4, 1 + 4*sqrt 3 / 3, 1 - 2*sqrt 6 / 3] , [3, 1 + sqrt 3, 1 - 4*sqrt 6 /3] ] randomInCircle :: Int -> IO [[Double]] randomInCircle n = do g1 <- newStdGen let theta = map (*(2*pi)) (take n (randoms g1 :: [Double])) g2 <- newStdGen let rho = map (/2) (take n (randoms g2 :: [Double])) return $ zipWith (\r a -> [(r+0.5) * cos a, (r+0.5) * sin a]) rho theta randomInSquare :: Int -> IO [[Double]] randomInSquare n = do g <- newStdGen return $ chunksOf 2 (take (2*n) (randoms g :: [Double])) randomInSphere :: Int -> IO [[Double]] randomInSphere n = do g1 <- newStdGen let theta = map (*(2*pi)) (take n (randoms g1 :: [Double])) g2 <- newStdGen let phi = map (*pi) (take n (randoms g2 :: [Double])) g3 <- newStdGen let rho = take n (randoms g3 :: [Double]) return $ zipWith3 (\r a b -> [r * cos a * sin b, r * sin a * sin b, r * cos b ]) rho theta phi regularSphere :: Int -> [[Double]] regularSphere n = concatMap (\a -> map (s2c a) phi) theta where theta = map (*(2*pi)) [frac i n | i <- [0 .. n-1]] phi = map (*pi) [frac i n | i <- [1 .. n-1]] frac :: Int -> Int -> Double frac p q = realToFrac p / realToFrac q s2c :: Double -> Double -> [Double] s2c th ph = [cos th * sin ph, sin th * sin ph, cos ph] randomOnSphere :: Int -> Double -> IO [[Double]] randomOnSphere n r = do g <- newStdGen let x = take (2*n) (randoms g :: [Double]) let u_ = map (*(2*pi)) (take n x) let v_ = drop n x return $ zipWith (\u v -> [r * cos u * sin (acos (2*v-1)), r * sin u * sin (acos (2*v-1)), r * (2*v-1) ]) u_ v_ randomInCube :: Int -> IO [[Double]] randomInCube n = do g <- newStdGen return $ chunksOf 3 (take (3*n) (randoms g :: [Double])) randomOnTorus :: Int -> Double -> Double -> IO [[Double]] randomOnTorus n c a = do g <- newStdGen let x = take (2*n) (randoms g :: [Double]) let u_ = map (*(2*pi)) (take n x) let v_ = map (*(2*pi)) (drop n x) return $ zipWith (\u v -> [cos u * (c + a * cos v), sin u * (c + a * cos v), a * sin v ]) u_ v_ qcube1,qcube2,qcube3,qcube4,qcube5 :: [[Double]] qcube1 = [ [ 0.0 , 0.6180339887498949 , 1.618033988749895 ] , [ -0.6180339887498949 , 1.618033988749895 , 0.0 ] , [ 1.618033988749895 , 0.0 , 0.6180339887498949 ] , [ 1.0 , 1.0 , -1.0 ] , [ 0.6180339887498949 , -1.618033988749895 , 0.0 ] , [ -1.0 , -1.0 , 1.0 ] , [ -1.618033988749895 , 0.0 , -0.6180339887498949 ] , [ 0.0 , -0.6180339887498949 , -1.618033988749895 ] ] qcube2 = [ [ -1.0 , 1.0 , 1.0 ] , [ 0.6180339887498949 , 1.618033988749895 , 0.0 ] , [ 0.0 , 0.6180339887498949 , -1.618033988749895 ] , [ 0.0 , -0.6180339887498949 , 1.618033988749895 ] , [ 1.618033988749895 , 0.0 , 0.6180339887498949 ] , [ 1.0 , -1.0 , -1.0 ] , [ -1.618033988749895 , 0.0 , -0.6180339887498949 ] , [ -0.6180339887498949 , -1.618033988749895 , 0.0 ] ] qcube3 = [ [ -0.6180339887498949 , 1.618033988749895 , 0.0 ] , [ 0.0 , 0.6180339887498949 , -1.618033988749895 ] , [ 0.0 , -0.6180339887498949 , 1.618033988749895 ] , [ 1.0 , 1.0 , 1.0 ] , [ 1.618033988749895 , 0.0 , -0.6180339887498949 ] , [ 0.6180339887498949 , -1.618033988749895 , 0.0 ] , [ -1.618033988749895 , 0.0 , 0.6180339887498949 ] , [ -1.0 , -1.0 , -1.0 ] ] qcube4 = [ [ -1.0 , 1.0 , 1.0 ] , [ 1.0 , -1.0 , 1.0 ] , [ 1.0 , 1.0 , 1.0 ] , [ 1.0 , 1.0 , -1.0 ] , [ 1.0 , -1.0 , -1.0 ] , [ -1.0 , -1.0 , 1.0 ] , [ -1.0 , 1.0 , -1.0 ] , [ -1.0 , -1.0 , -1.0 ] ] qcube5 = [ [ 0.0 , 0.6180339887498949 , 1.618033988749895 ] , [ 0.6180339887498949 , 1.618033988749895 , 0.0 ] , [ 1.0 , -1.0 , 1.0 ] , [ 1.618033988749895 , 0.0 , -0.6180339887498949 ] , [ -1.618033988749895 , 0.0 , 0.6180339887498949 ] , [ -1.0 , 1.0 , -1.0 ] , [ -0.6180339887498949 , -1.618033988749895 , 0.0 ] , [ 0.0 , -0.6180339887498949 , -1.618033988749895 ] ] teapot :: [[Double]] teapot = [ [-3, 1.64999997615814, 0] , [-2.98710989952087, 1.64999997615814, -0.0984380021691322] , [-2.98710989952087, 1.64999997615814, 0.0984380021691322] , [-2.98537993431091, 1.56731998920441, -0.0492190010845661] , [-2.98537993431091, 1.56731998920441, 0.0492190010845661] , [-2.9835000038147, 1.48308002948761, 0] , [-2.98188996315002, 1.72346997261047, -0.0492190010845661] , [-2.98188996315002, 1.72346997261047, 0.0492190010845661] , [-2.97656011581421, 1.79852998256683, 0] , [-2.97090005874634, 1.48620998859406, -0.0984380021691322] , [-2.97090005874634, 1.48620998859406, 0.0984380021691322] , [-2.96388006210327, 1.79533994197845, -0.0984380021691322] , [-2.96388006210327, 1.79533994197845, 0.0984380021691322] , [-2.96220993995667, 1.57017004489899, -0.133594006299973] , [-2.96220993995667, 1.57017004489899, 0.133594006299973] , [-2.95864009857178, 1.72056996822357, -0.133594006299973] , [-2.95864009857178, 1.72056996822357, 0.133594006299973] , [-2.95313000679016, 1.64999997615814, -0.168750002980232] , [-2.95313000679016, 1.64999997615814, 0.168750002980232] , [-2.95247006416321, 1.40374004840851, -0.0492190010845661] , [-2.95247006416321, 1.40374004840851, 0.0492190010845661] , [-2.93770003318787, 1.49447000026703, -0.168750002980232] , [-2.93770003318787, 1.49447000026703, 0.168750002980232] , [-2.93523001670837, 1.85214996337891, -0.0492190010845661] , [-2.93523001670837, 1.85214996337891, 0.0492190010845661] , [-2.93358993530273, 1.32011997699738, 0] , [-2.93044996261597, 1.78692996501923, -0.168750002980232] , [-2.93044996261597, 1.78692996501923, 0.168750002980232] , [-2.93037009239197, 1.41149997711182, -0.133594006299973] , [-2.93037009239197, 1.41149997711182, 0.133594006299973] , [-2.92188000679016, 1.32553005218506, -0.0984380021691322] , [-2.92188000679016, 1.32553005218506, 0.0984380021691322] , [-2.91278004646301, 1.84416997432709, -0.133594006299973] , [-2.91278004646301, 1.84416997432709, 0.133594006299973] , [-2.90625, 1.91015994548798, 0] , [-2.89422988891602, 1.90456998348236, -0.0984380021691322] , [-2.89422988891602, 1.90456998348236, 0.0984380021691322] , [-2.89138007164001, 1.57910001277924, -0.196875005960464] , [-2.89138007164001, 1.57910001277924, 0.196875005960464] , [-2.8909900188446, 1.33980000019073, -0.168750002980232] , [-2.8909900188446, 1.33980000019073, 0.168750002980232] , [-2.89065003395081, 1.71208000183105, -0.196875005960464] , [-2.89065003395081, 1.71208000183105, 0.196875005960464] , [-2.88346004486084, 1.24579000473022, -0.0483429990708828] , [-2.88346004486084, 1.24579000473022, 0.0483429990708828] , [-2.86346006393433, 1.25713002681732, -0.132717996835709] , [-2.86346006393433, 1.25713002681732, 0.132717996835709] , [-2.86265993118286, 1.43482995033264, -0.196875005960464] , [-2.86265993118286, 1.43482995033264, 0.196875005960464] , [-2.8625500202179, 1.88982999324799, -0.168750002980232] , [-2.8625500202179, 1.88982999324799, 0.168750002980232] , [-2.84999990463257, 1.64999997615814, -0.224999994039536] , [-2.84999990463257, 1.64999997615814, 0.224999994039536] , [-2.84970998764038, 1.16155004501343, 0] , [-2.84710001945496, 1.82081997394562, -0.196875005960464] , [-2.84710001945496, 1.82081997394562, 0.196875005960464] , [-2.84193992614746, 1.94692003726959, -0.0492190010845661] , [-2.84193992614746, 1.94692003726959, 0.0492190010845661] , [-2.8289999961853, 1.76139998435974, -0.224999994039536] , [-2.8289999961853, 1.76139998435974, 0.224999994039536] , [-2.82867002487183, 1.17597997188568, -0.0949330031871796] , [-2.82867002487183, 1.17597997188568, 0.0949330031871796] , [-2.82470011711121, 1.52193999290466, -0.224999994039536] , [-2.82470011711121, 1.52193999290466, 0.224999994039536] , [-2.82115006446838, 1.93519997596741, -0.133594006299973] , [-2.82115006446838, 1.93519997596741, 0.133594006299973] , [-2.81230998039246, 1.18719005584717, -0.168750002980232] , [-2.81230998039246, 1.18719005584717, 0.168750002980232] , [-2.80501008033752, 1.28997004032135, -0.196875005960464] , [-2.80501008033752, 1.28997004032135, 0.196875005960464] , [-2.79727005958557, 1.38311004638672, -0.224999994039536] , [-2.79727005958557, 1.38311004638672, 0.224999994039536] , [-2.78906011581421, 1.99013996124268, 0] , [-2.78836011886597, 1.69931995868683, -0.196875005960464] , [-2.78836011886597, 1.69931995868683, 0.196875005960464] , [-2.77820992469788, 1.98283004760742, -0.0984380021691322] , [-2.77820992469788, 1.98283004760742, 0.0984380021691322] , [-2.77442002296448, 1.52737998962402, -0.196875005960464] , [-2.77442002296448, 1.52737998962402, 0.196875005960464] , [-2.77356004714966, 1.09860002994537, -0.0843750014901161] , [-2.77356004714966, 1.09860002994537, 0.0843750014901161] , [-2.76641011238098, 1.84511995315552, -0.224999994039536] , [-2.76641011238098, 1.84511995315552, 0.224999994039536] , [-2.76033997535706, 1.90090000629425, -0.196875005960464] , [-2.76033997535706, 1.90090000629425, 0.196875005960464] , [-2.74959993362427, 1.96355998516083, -0.168750002980232] , [-2.74959993362427, 1.96355998516083, 0.168750002980232] , [-2.74831008911133, 1.78569996356964, -0.196875005960464] , [-2.74831008911133, 1.78569996356964, 0.196875005960464] , [-2.74688005447388, 1.64999997615814, -0.168750002980232] , [-2.74688005447388, 1.64999997615814, 0.168750002980232] , [-2.73125004768372, 1.00780999660492, 0] , [-2.72756004333496, 1.73587000370026, -0.168750002980232] , [-2.72756004333496, 1.73587000370026, 0.168750002980232] , [-2.72036004066467, 1.69082999229431, -0.133594006299973] , [-2.72036004066467, 1.69082999229431, 0.133594006299973] , [-2.71948003768921, 1.24977004528046, -0.224999994039536] , [-2.71948003768921, 1.24977004528046, 0.224999994039536] , [-2.71677994728088, 1.14468002319336, -0.196875005960464] , [-2.71677994728088, 1.14468002319336, 0.196875005960464] , [-2.71288990974426, 1.64999997615814, -0.0984380021691322] , [-2.71288990974426, 1.64999997615814, 0.0984380021691322] , [-2.7089900970459, 1.54176998138428, -0.133594006299973] , [-2.7089900970459, 1.54176998138428, 0.133594006299973] , [-2.70354008674622, 1.42640995979309, -0.168750002980232] , [-2.70354008674622, 1.42640995979309, 0.168750002980232] , [-2.70097994804382, 1.03784000873566, -0.168750002980232] , [-2.70097994804382, 1.03784000873566, 0.168750002980232] , [-2.70000004768372, 1.64999997615814, 0] , [-2.69965004920959, 2.0107901096344, -0.0483460016548634] , [-2.69965004920959, 2.0107901096344, 0.0483460016548634] , [-2.69711995124817, 1.68792998790741, -0.0492190010845661] , [-2.69711995124817, 1.68792998790741, 0.0492190010845661] , [-2.69412994384766, 1.72746002674103, -0.0984380021691322] , [-2.69412994384766, 1.72746002674103, 0.0984380021691322] , [-2.68661999702454, 1.54668998718262, -0.0492190010845661] , [-2.68661999702454, 1.54668998718262, 0.0492190010845661] , [-2.68263006210327, 1.76234996318817, -0.133594006299973] , [-2.68263006210327, 1.76234996318817, 0.133594006299973] , [-2.68147993087769, 1.9964599609375, -0.13272100687027] , [-2.68147993087769, 1.9964599609375, 0.13272100687027] , [-2.68144011497498, 1.72426998615265, 0] , [-2.67574000358582, 1.27084994316101, -0.196875005960464] , [-2.67574000358582, 1.27084994316101, 0.196875005960464] , [-2.67265009880066, 1.44068002700806, -0.0984380021691322] , [-2.67265009880066, 1.44068002700806, 0.0984380021691322] , [-2.67025995254517, 1.80040001869202, -0.168750002980232] , [-2.67025995254517, 1.80040001869202, 0.168750002980232] , [-2.667799949646, 1.84623003005981, -0.196875005960464] , [-2.667799949646, 1.84623003005981, 0.196875005960464] , [-2.66279006004333, 1.9050999879837, -0.224999994039536] , [-2.66279006004333, 1.9050999879837, 0.224999994039536] , [-2.66093993186951, 1.44608998298645, 0] , [-2.66018009185791, 1.75436997413635, -0.0492190010845661] , [-2.66018009185791, 1.75436997413635, 0.0492190010845661] , [-2.63858008384705, 1.78567004203796, -0.0984380021691322] , [-2.63858008384705, 1.78567004203796, 0.0984380021691322] , [-2.63438010215759, 1.10390996932983, -0.224999994039536] , [-2.63438010215759, 1.10390996932983, 0.224999994039536] , [-2.63073992729187, 1.95674002170563, -0.196875005960464] , [-2.63073992729187, 1.95674002170563, 0.196875005960464] , [-2.62655997276306, 1.78007996082306, 0] , [-2.625, 2.04375004768372, 0] , [-2.62463998794556, 1.30501997470856, -0.132813006639481] , [-2.62463998794556, 1.30501997470856, 0.132813006639481] , [-2.60642004013062, 1.31745004653931, -0.0484380014240742] , [-2.60642004013062, 1.31745004653931, 0.0484380014240742] , [-2.60631990432739, 2.02643990516663, -0.0949449986219406] , [-2.60631990432739, 2.02643990516663, 0.0949449986219406] , [-2.59179997444153, 2.01298999786377, -0.168750002980232] , [-2.59179997444153, 2.01298999786377, 0.168750002980232] , [-2.57172989845276, 1.83429002761841, -0.168750002980232] , [-2.57172989845276, 1.83429002761841, 0.168750002980232] , [-2.56777000427246, 1.16997003555298, -0.168750002980232] , [-2.56777000427246, 1.16997003555298, 0.168750002980232] , [-2.55460000038147, 1.18304002285004, -0.0953150019049644] , [-2.55460000038147, 1.18304002285004, 0.0953150019049644] , [-2.54975008964539, 1.89058995246887, -0.196875005960464] , [-2.54975008964539, 1.89058995246887, 0.196875005960464] , [-2.5495400428772, 0.878983974456787, -0.0843750014901161] , [-2.5495400428772, 0.878983974456787, 0.0843750014901161] , [-2.5464301109314, 1.83196997642517, -0.13272100687027] , [-2.5464301109314, 1.83196997642517, 0.13272100687027] , [-2.53749990463257, 1.20000004768372, 0] , [-2.52720999717712, 1.81920003890991, -0.0483460016548634] , [-2.52720999717712, 1.81920003890991, 0.0483460016548634] , [-2.51874995231628, 1.94530999660492, -0.224999994039536] , [-2.51874995231628, 1.94530999660492, 0.224999994039536] , [-2.51682996749878, 0.932671010494232, -0.196875005960464] , [-2.51682996749878, 0.932671010494232, 0.196875005960464] , [-2.47183990478516, 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2.27940011024475, 0.0702759996056557] , [3.26724004745483, 2.2781400680542, -0.025891000404954] , [3.26724004745483, 2.2781400680542, 0.025891000404954] , [3.27272009849548, 2.30776000022888, -0.0931639969348907] , [3.27272009849548, 2.30776000022888, 0.0931639969348907] , [3.27421998977661, 2.25, -0.0820309966802597] , [3.27421998977661, 2.25, 0.0820309966802597] , [3.29534006118774, 2.2770299911499, -0.10784900188446] , [3.29534006118774, 2.2770299911499, 0.10784900188446] , [3.29999995231628, 2.25, 0] , [3.31404995918274, 2.30330991744995, -0.131835997104645] , [3.31404995918274, 2.30330991744995, 0.131835997104645] , [3.33072996139526, 2.30984997749329, -0.0543459989130497] , [3.33072996139526, 2.30984997749329, 0.0543459989130497] , [3.33388996124268, 2.324049949646, -0.112499997019768] , [3.33388996124268, 2.324049949646, 0.112499997019768] , [3.33488988876343, 2.31701993942261, -0.0814089998602867] , [3.33488988876343, 2.31701993942261, 0.0814089998602867] , [3.34236001968384, 2.2800600528717, -0.039733998477459] , [3.34236001968384, 2.2800600528717, 0.039733998477459] , [3.35542988777161, 2.30270004272461, 0] , [3.35925006866455, 2.31465005874634, -0.0967160016298294] , [3.35925006866455, 2.31465005874634, 0.0967160016298294] , [3.37912011146545, 2.31658005714417, -0.0299929995089769] , [3.37912011146545, 2.31658005714417, 0.0299929995089769] , [3.38684010505676, 2.30481004714966, -0.0769039988517761] , [3.38684010505676, 2.30481004714966, 0.0769039988517761] , [3.40220999717712, 2.32644009590149, -0.0656249970197678] , [3.40220999717712, 2.32644009590149, 0.0656249970197678] , [3.40638995170593, 2.31850004196167, -0.035631999373436] , [3.40638995170593, 2.31850004196167, 0.035631999373436] , [3.40838003158569, 2.31542992591858, 0] , [3.42811989784241, 2.32733988761902, 0] ] voronoiCellTeapot :: [[Double]] voronoiCellTeapot = [0, 1.5, 0] : [ [ -0.18667466541973257 , 0.5968790484510281 , 0.5717139272637414 ] , [ -0.15441948465435382 , 0.7326922941987842 , 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4.366114397309236e-2 ] , [ 0.7740595934432167 , 0.862499490380288 , 9.433094929942933e-2 ] , [ 0.8122497597351315 , 1.3634301491185252 , 0.39698702598827496 ] , [ 0.7740579104368239 , 0.8624980151653314 , 9.43372496836588e-2 ] , [ 2.3194160942461885e-2 , 2.6785629054845606e-2 , 2.319416094249719e-2 ] , [ 0.4720306023200158 , 0.6074273900130303 , 0.39216297275000733 ] , [ 2.215103618775549e-2 , 2.5331770412978624e-2 , 2.2151036187755824e-2 ] , [ 0.39216297275000733 , 0.6074273900130303 , 0.4720306023200158 ] , [ -0.18893822974950683 , 0.9120771417338428 , 0.8765366554024406 ] , [ -0.15370753430700942 , 0.9074581172580759 , 0.8796369493525763 ] , [ -0.19282328992414305 , 1.0886815800201057 , 0.8962126634940768 ] , [ -0.21959663365004273 , 0.90801535760842 , 0.8664476852626979 ] , [ 0.6553927028305585 , 0.6599214690464071 , 7.68209537002328e-2 ] , [ 0.6480770038782534 , 0.6480770038782535 , 3.783067537459344e-2 ] , [ 0.6480770038782537 , 0.6480770038782535 , -3.7830675374592546e-2 ] , [ 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[ -0.1916029836825906 , 1.29931244670657 , 0.8997455590151622 ] , [ 0.0 , 1.0862351104914478 , 0.9157767877158491 ] , [ 0.0 , 1.4739386059327377 , 0.8813815581455101 ] , [ -3.497482895265056e-2 , 0.9074580789365942 , 0.8918549800719824 ] , [ -6.080525584912345e-2 , 0.8753821429287596 , 0.86310582524099 ] , [ 7.812642760551707e-3 , 7.812642760551707e-3 , 7.812642760550714e-3 ] , [ 0.6491833829319444 , 0.6692949179238175 , 0.1694761444119623 ] , [ -2.7755575615628914e-17 , 0.7373497776866849 , 0.7373497776866843 ] , [ -7.812642760551686e-3 , 7.812642760551715e-3 , 7.812642760551602e-3 ] , [ -0.3581881925466474 , 0.9124253481133149 , 0.8235344289369642 ] , [ -0.4468960748716194 , 1.1765076558719516 , 0.8091216099094026 ] , [ -0.4474984305598345 , 1.2598574705147236 , 0.8102438227139427 ] , [ -0.36541277208493106 , 1.2970587028449376 , 0.8457844093935878 ] , [ -0.5194379814088046 , 1.2957898827631247 , 0.7622224951613341 ] , [ -0.518569427022487 , 1.0903982826657406 , 0.7590404339629727 ] 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