// faces.java  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
//            compile    javac faces.java
//            execute    java  faces > faces_java.out

public class faces
{
  public faces()
  {
    int n, m;
    int face[] = new int[11];

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

  int pwr2(int n)
  {
    int pwr = 1;
    if(n<=0) return 1;
    for(int i=0; i<n; i++) pwr = pwr*2;
    return pwr;
  } // end pwr2

  int fct(int n)
  {
    int nfct = 1;
    if(n<=0) return 1;
    for(int i=2; i<=n; i++) nfct = nfct*i;
    return nfct;
  } // end fct

  int comb(int n, int m) // n things taken m at a time
  {
    if(m>n) return 0;
    return fct(n)/(fct(m)*fct(n-m));
  } // end comb

  public static void main (String[] args)
  {
    new faces();
  }
} // end faces.java