# faces.py  data for various n-cubes, n-spheres
#      various n dimensions for n-dimensional object volume, area, edge length
#      the number of m-dimensional sub objects
#            execute    python  faces.py > faces_py.out

def main():
    face = [0 for i in range(12)]
    print("faces.py running, data for various n-cubes, n-spheres, n dimensions")
    print("edge length L for cubes, radius R for spheres")
    print(" ")
    for n in range(0, 11):
      if n>3 :
         print("n=",n,"-cube")

      if n==3 :
         print("n=3-cube cube")

      if n==2 :
         print("n=2-cube square")

      if n==1 :
         print("n=1-cube line")

      if n==0 :
        print("0-cube point")
        print("vertices  = 1")
        print(" ")
        continue

      for m in range (n, -1, -1):
        # print("loop m=",m) # debug
        face[m] = pwr2(n-m)*comb(n,m)
        # print("face[",m,"]=",face[m]) # debug
        if m>3 :
          if face[m]==1 :
            print(m,"-cubes = ", face[m],"      volume=L^",m)
          else:
            print(m,"-cubes = ", face[m],"      volume=",face[m],"*L^",m)


        if m==3 :
           print("cubes     = ", face[m],"      volume=",face[m],"L^3")

        if m==2 :
           print("2D faces  = ", face[m],"      area=",face[m],"*L^2")

        if m==1 :
           print("edges     = ", face[m],"      length=",face[m],"*L")

        if m==0 :
           print("vertices  = ", face[m])


      # end m
      print(" ")

    # end n
    print(" ")
    print(" ")
    print(" cubes of n dimensions") 
    print(" with edge length E, n>=3,  volume = E^n, area=(2D faces)*E*E") 
    print(" n-cube has 2^n vertex, with n edges at each vertex") 
    print(" n>=m, there are pwr2(n-m)*comb(n,m)  n-m sub cubes")
    print(" ")
    print("Spheres up to 10 dimensions ")
    print(" ")
    print("              D-1 surface         D volume")
    print("2D circle     2      Pi   R       1      Pi   R^2")
    print("3D sphere     4      Pi   R^2     4/3    Pi   R^3")
    print("4D 4-sphere   2      Pi^2 R^3     1/2    Pi^2 R^4")
    print("5D 5-sphere   8/3    Pi^2 R^4     8/15   Pi^2 R^5")
    print("6D 6-sphere   1      Pi^3 R^5     1/6    Pi^3 R^6")
    print("7D 7-sphere   16/15  Pi^3 R^6     16/105 Pi^3 R^7")
    print("8D 8-sphere   1/3    Pi^4 R^7     1/24   Pi^4 R^8")
    print("9D 9-sphere   32/105 Pi^4 R^8     32/945 Pi^4 R^9")
    print("10D 10-sphere 1/12   Pi^5 R^9     1/120  Pi^5 R^10")
    print(" ")
    print("volume V_n(R)= Pi^(n/2) R^n / gamma(n/2+1)")
    print("gamma(integer) = factorial(integer-1)  gamma(5) = 24")
    print("gamma(1/2) = sqrt(Pi), gamma(n/2+1) = (2n)! sqrt(Pi)/(4^n n!)")
    print("or V_2k(R) = Pi^k R^2k/k! , V_2k+1 = 2 k! (4Pi)^k R^(2k+1)/(2k+1)!")
    print("surface area A_n(R) = d/dR V_n(R)")
    print(" ")
    print(" ")
    print("one definition of sequence of n-spheres")
    print("a1, a2, a3, a4, a5, a6, a7 are angles, typ: theta, phi, ...")
    print(" ")
    print("x1, x2, x3, x4, x5, x6, x7, x8 are orthogonal coordinates")
    print("x1^2 + x2^2 + x3^2 + x4^2 + x5^2 + x6^2 + x7^2 +x8^2 = R^2")
    print(" Radius R = sqrt(R^2)")
    print(" ")
    
    print("2D circle")
    print("x1 = R cos(a1)    typ: x  theta")
    print("x2 = R sin(a1)    typ: y  theta")
    print(" ")
    print("a1 = arctan(x2/x1) or a1 = acos(x1/R)")
    print(" ")
    print("3D sphere")
    print("x1 = R cos(a1)            typ: z  phi")
    print("x2 = R sin(a1) cos(a2)    typ: x  phi  theta")
    print("x3 = R sin(a1) sin(a2)    typ: y  phi  theta")
    print(" ")
    print("a1 = arctan(sqrt(x2^2+x3^2)/x1) or a1 = acos(x1/R)")
    print("a2 = arctan(x3/x2) or a2 = acos(x2/sqrt(x2^2+x3^2))")
    print(" ")
    
    print("4D 4-sphere continuing systematic notation, notice pattern")
    print("x1 = R cos(a1)")
    print("x2 = R sin(a1) cos(a2)")
    print("x3 = R sin(a1) sin(a2) cos(a3)")
    print("x4 = R sin(a1) sin(a2) sin(a3)")
    print(" ")
    print("a1 = acos(x1/sqrt(x1^2+x2^2+x3^2+x4^2))")
    print("a2 = acos(x2/sqrt(x2^2+x3^2+x4^2))")
    print("a3 = acos(x3/sqrt(x3^2+x4^2)) if x4>=0")
    print("a3 = 2 Pi - acos(x3/sqrt(x3^2+x4^2)) if x4<0")
    print(" ")
    
    print("5D 5-sphere")
    print("x1 = R cos(a1)")
    print("x2 = R sin(a1) cos(a2)")
    print("x3 = R sin(a1) sin(a2) cos(a3)")
    print("x4 = R sin(a1) sin(a2) sin(a3) cos(a4)")
    print("x5 = R sin(a1) sin(a2) sin(a3) sin(a4)")
    print(" ")
    print("a1 = acos(x1/sqrt(x1^2+x2^2+x3^2+x4^2+x5^2))")
    print("a2 = acos(x2/sqrt(x2^2+x3^2+x4^2+x5^2))")
    print("a3 = acos(x3/sqrt(x3^2+x4^2+x5^2))")
    print("a4 = acos(x4/sqrt(x4^2+x5^2)) if x5>=0")
    print("a4 = 2 Pi - acos(x4/sqrt(x4^2+x5^2)) if x5<0")
    print(" ")
    
    print("6D 6-sphere")
    print("x1 = R cos(a1)")
    print("x2 = R sin(a1) cos(a2)")
    print("x3 = R sin(a1) sin(a2) cos(a3)")
    print("x4 = R sin(a1) sin(a2) sin(a3) cos(a4)")
    print("x5 = R sin(a1) sin(a2) sin(a3) sin(a4) cos(a5)")
    print("x6 = R sin(a1) sin(a2) sin(a3) sin(a4) sin(a5)")
    print(" ")
    print("a1 = acos(x1/sqrt(x1^2+x2^2+x3^2+x4^2+x5^2+x6^2))")
    print("a2 = acos(x2/sqrt(x2^2+x3^2+x4^2+x5^2+x6^2))")
    print("a3 = acos(x3/sqrt(x3^2+x4^2+x5^2+x6^2))")
    print("a4 = acos(x4/sqrt(x4^2+x5^2+x6^2))")
    print("a5 = acos(x5/sqrt(x5^2+x6^2)) if x6>=0")
    print("a5 = 2 Pi - acos(x5/sqrt(x5^2+x6^2)) if x6<0")
    print(" ")
    
    print("7D 7-sphere")
    print("x1 = R cos(a1)")
    print("x2 = R sin(a1) cos(a2)")
    print("x3 = R sin(a1) sin(a2) cos(a3)")
    print("x4 = R sin(a1) sin(a2) sin(a3) cos(a4)")
    print("x5 = R sin(a1) sin(a2) sin(a3) sin(a4) cos(a5)")
    print("x6 = R sin(a1) sin(a2) sin(a3) sin(a4) sin(a5) cos(a6)")
    print("x7 = R sin(a1) sin(a2) sin(a3) sin(a4) sin(a5) sin(a6)")
    print(" ")
    print("a1 = acos(x1/sqrt(x1^2+x2^2+x3^2+x4^2+x5^2+x6^2+x7^2))")
    print("a2 = acos(x2/sqrt(x2^2+x3^2+x4^2+x5^2+x6^2+x7^2))")
    print("a3 = acos(x3/sqrt(x3^2+x4^2+x5^2+x6^2+x7^2))")
    print("a4 = acos(x4/sqrt(x4^2+x5^2+x6^2+x7^2))")
    print("a5 = acos(x5/sqrt(x5^2+x6^2+x7^2))")
    print("a6 = acos(x6/sqrt(x6^2+x6^2)) if x7>=0")
    print("a6 = 2 Pi - acos(x6/sqrt(x6^2+x7^2)) if x7<0")
    print(" ")
    
    print("8D 8-sphere")
    print("x1 = R cos(a1)")
    print("x2 = R sin(a1) cos(a2)")
    print("x3 = R sin(a1) sin(a2) cos(a3)")
    print("x4 = R sin(a1) sin(a2) sin(a3) cos(a4)")
    print("x5 = R sin(a1) sin(a2) sin(a3) sin(a4) cos(a5)")
    print("x6 = R sin(a1) sin(a2) sin(a3) sin(a4) sin(a5) cos(a6)")
    print("x7 = R sin(a1) sin(a2) sin(a3) sin(a4) sin(a5) sin(a6) cos(a7)")
    print("x8 = R sin(a1) sin(a2) sin(a3) sin(a4) sin(a5) sin(a6) sin(a7)")
    print(" ")
    print("a1 = acos(x1/sqrt(x1^2+x2^2+x3^2+x4^2+x5^2+x6^2+x7^2+x8^2))")
    print("a2 = acos(x2/sqrt(x2^2+x3^2+x4^2+x5^2+x6^2+x7^2+x8^2))")
    print("a3 = acos(x3/sqrt(x3^2+x4^2+x5^2+x6^2+x7^2+x8^2))")
    print("a4 = acos(x4/sqrt(x4^2+x5^2+x6^2+x7^2+x8^2))")
    print("a5 = acos(x5/sqrt(x5^2+x6^2+x7^2+x8^2))")
    print("a6 = acos(x6/sqrt(x5^2+x6^2+x7^2+x8^2))")
    print("a7 = acos(x7/sqrt(x7^2+x8^2)) if x8>=0")
    print("a7 = 2 Pi - acos(x7/sqrt(x7^2+x8^2)) if x8<0")
    print(" ")
    print("faces.py finished")
  # end main 

def pwr2(n):
  pwr = 1
  if n<=0 :
     return 1

  for i in range(0, n) :
    pwr = pwr*2

  return pwr

# end pwr2

def fct(n):
  nfct = 1
  if n<=0 :
    return 1

  for i in range(2, n+1):
    nfct = nfct*i

  return nfct

# end fct

def comb(n, m): # n things taken m at a time
  if m>n :
    return 0

  return fct(n)/(fct(m)*fct(n-m))

# end comb

if __name__ == '__main__':
   main()
  
# end faces.py