MATH089 Project 2 - Patterns, Tilings, Crystals

Posted: 09/15/21

Due: 10/04/21, 11:55PM

1Introduction

2Models and methods

2.1Two-dimensional tilings by regular polygons

2.1.1Tiling with squares

Julia (1.6.1) session in GNU TeXmacs 

∴ 
function square(x0,y0)
  x=[x0,x0+1,x0+1,x0,x0];
  y=[y0,y0,y0+1,y0+1,y0];
  plot(x,y,"-k");
end;
 
∴ 
figure(1); clf(); square(0,0); axis("equal");
 
∴ 
function xtile(m,y)
  square.(0:m-1,y)
end;
 
∴ 
clf(); xtile(10,0); axis("equal");
 
∴ 
function tile(m,n)
  xtile.(m,0:n-1)
end;
 
∴ 
clf(); tile(8,8); axis("equal");
 
∴ 
savefig("/home/student/courses/MATH089/images/chessboard.eps");
 
∴ 

          

        

      

    

    

      

        

          
        

          
        

          

            

              Figure 1. Tiling of the plane through squares
            

          

        

      

    

    
2.1.2Tiling with triangles by translation

    

      

        

          

            
∴ 

            
function triangle(x0,y0)
  x=[x0,x0+1,x0+0.5,x0];
  y=[y0,y0,y0+sqrt(3.0)/2,y0];
  plot(x,y,"-k");
end;

          

        

      

    

    

      

        

          

            
∴ 

            
clf(); triangle(0,0); axis("equal");

          

        

      

    

    

      

        

          

            
∴ 

            
function xtile(m,x,y)
  triangle.(x:x+m-1,y)
end;

          

        

      

    

    

      

        

          

            
∴ 

            
clf(); xtile(10,0,0); axis("equal");

          

        

      

    

    

      

        

          

            
∴ 

            
function tile(m,n)
  xtile.(m,(0:n-1)*0.5,(0:n-1)*sqrt(3.0)/2)
end;

          

        

      

    

    

      

        

          

            
∴ 

            
clf(); tile(8,8); axis("equal");

          

        

      

    

    

      

        

          

            
∴ 

            
savefig("/home/student/courses/MATH089/images/goboard.eps");

          

        

      

    

    

      
    

    

      

        

          
        

          
        

          

            

              Figure 2. Tiling of the plane through
              triangles
            

          

        

      

    

    
2.1.3Tiling with triangles by translation
    along normals and rotation or reflection

    

      

        

          

            
∴ 

            
clf(); triangle(0,0); axis("equal");

          

        

      

    

    

      

        

          

            

              UndefVarError(:triangle)
            

          

        

      

    

    

      

        

          

            
∴ 

            
function PlotTri(X)
  x=[X[1,:]; X[1,1]]
  y=[X[2,:]; X[2,1]]
  plot(x,y,"-k");
end;

          

        

      

    

    

      

        

          

            
∴ 

            
X=[0 1 0.5; 0 0 sqrt(3)/2];

          

        

      

    

    

      

        

          

            
∴ 

            
clf(); PlotTri(X); axis("equal");

          

        

      

    

    

      

        

          

            
∴ 

            
function TileTri(X)
  PlotTri(X)
  Xm = 0.5*(X[:,2]+X[:,3]); R=Xm-X[:,1]
  Xr = Xm + R; Y=copy(X); Y[:,1]=Xr; PlotTri(Y)
  Xm = 0.5*(X[:,3]+X[:,1]); R=Xm-X[:,2]
  Xr = Xm + R; Y=copy(X); Y[:,2]=Xr; PlotTri(Y)
  Xm = 0.5*(X[:,1]+X[:,2]); R=Xm-X[:,3]
  Xr = Xm + R; Y=copy(X); Y[:,3]=Xr; PlotTri(Y)
end;

          

        

      

    

    

      

        

          

            
∴ 

            
clf(); TileTri(X);

          

        

      

    

    

      

        

          

            
∴ 

            
function TileTri(X,n)
  if (n==0) return end
  PlotTri(X)
  Xm = 0.5*(X[:,2]+X[:,3]); R=Xm-X[:,1]
  Xr = Xm + R; Y=copy(X); Y[:,1]=Xr; PlotTri(Y); TileTri(Y,n-1)
  Xm = 0.5*(X[:,3]+X[:,1]); R=Xm-X[:,2]
  Xr = Xm + R; Y=copy(X); Y[:,2]=Xr; PlotTri(Y); TileTri(Y,n-1)
  Xm = 0.5*(X[:,1]+X[:,2]); R=Xm-X[:,3]
  Xr = Xm + R; Y=copy(X); Y[:,3]=Xr; PlotTri(Y); TileTri(Y,n-1)
end;

          

        

      

    

    

      

        

          

            
∴ 

            
clf(); TileTri(X,6);

          

        

      

    

    

      

        

          

            
∴ 

            
function TileTri(X,n,s)
  if (n==0) return end
  PlotTri(X)
  Xm = 0.5*(X[:,2]+X[:,3]); R=Xm-X[:,1]
  Xr = Xm + R; Y=copy(X); Y[:,1]=Xr;
  Yc=(Y[:,1]+Y[:,2]+Y[:,3])/3; Y=s*(Y-Xm)+Xm;
  PlotTri(Y); TileTri(Y,n-1)
  Xm = 0.5*(X[:,3]+X[:,1]); R=Xm-X[:,2]
  Xr = Xm + R; Y=copy(X); Y[:,2]=Xr;
  Yc=(Y[:,1]+Y[:,2]+Y[:,3])/3; Y=s*(Y-Xm)+Xm;
  PlotTri(Y); TileTri(Y,n-1)
  Xm = 0.5*(X[:,1]+X[:,2]); R=Xm-X[:,3]
  Xr = Xm + R; Y=copy(X); Y[:,3]=Xr;
  Yc=(Y[:,1]+Y[:,2]+Y[:,3])/3; Y=s*(Y-Xm)+Xm;
  PlotTri(Y); TileTri(Y,n-1)
end;

          

        

      

    

    

      

        

          

            
∴ 

            
clf(); TileTri(X,3,1.0/3.0)

          

        

      

    

    

      

        

          

            

              DimensionMismatch("dimensions must match: a has dims
              (Base.OneTo(2), Base.OneTo(3)), must have singleton at dim
              2")
            

          

        

      

    

    

      

        

          

            
∴ 

            

          

        

      

    

    
2.2Tiling with concave polygons

    
2.2.1Maltese cross

    

      
    

    
2.3Three-dimensional tilings by regular
    polyhedra

    
2.3.1Tilings by cubes

    

      
    

    

      

        

          

            
∴ 

            
function cube(x0,y0,z0,c)
  h=0.025;
  x=[x0+h; x0+1-h; x0+1-h; x0+h; x0+h]
  y=[y0+h; y0+h; y0+1-h; y0+1-h; y0+h]
  z=ones(5)*(z0+h)
  plot3D(x,y,z,c); plot3D(x,y,z.+(1-h),c);
  x=ones(5)*(x0+h)
  y=[y0+h; y0+h; y0+1-h; y0+1-h; y0+h]
  z=[z0+h; z0+1-h; z0+1-h; z0+h; z0+h]
  plot3D(x,y,z,c); plot3D(x.+(1-h),y,z,c); 
end;

          

        

      

    

    

      

        

          

            
∴ 

            
clf(); cube(0,0,0,"g"); cube(1,0,0,"r");

          

        

      

    

    

      

        

          

            
∴ 

            
function xtile(x0,y0,z0,m,c)
  cube.((0:m-1).+x0,y0,z0,c)
end;

          

        

      

    

    

      

        

          

            
∴ 

            
clf(); xtile(0,0,0,5,"r");

          

        

      

    

    

      

        

          

            
∴ 

            
function xytile(x0,y0,z0,m,n,c)
  xtile.(x0,(0:n-1).+y0,z0,m,c)
end;

          

        

      

    

    

      

        

          

            
∴ 

            
clf(); xytile(0,0,0,5,5,"r");

          

        

      

    

    

      

        

          

            
∴ 

            
function xyztile(x0,y0,z0,m,n,p,c)
  xytile.(x0,y0,(0:p-1).+z0,m,n,c)
end;

          

        

      

    

    

      

        

          

            
∴ 

            
clf(); xyztile(0,0,0,5,5,5,"g");

          

        

      

    

    

      

        

          

            
∴ 

            

          

        

      

    

    
2.3.2Tetrahedral tiling

    

      Define a function to draw a tetrahedron given the coordinates of its
      four vertices
    

    

      

        

          

            
∴ 

            
function PlotTet(X)
   for k=0:3
     PlotTri3D( circshift(X,(0,k))[:,1:3] )
   end
end;

          

        

      

    

    

      
    

    

      Define a function to draw each face triangle in 3D
    

    

      

        

          

            
∴ 

            
function PlotTri3D(X)
  for k=0:2
    Y=circshift(X,(0,k))[:,1:2]
    x=Y[1,:]; y=Y[2,:]; z=Y[3,:]
    plot3D(x,y,z,"-k")
  end
end;

          

        

      

    

    

      
    

    

      Test the trianle drawing function
    

    

      

        

          

            
∴ 

            
X=[1 0 0; 0 1 0; 0 0 1];

          

        

      

    

    

      

        

          

            
∴ 

            
clf(); PlotTri3D(X);

          

        

      

    

    

      

        

          

            
∴ 

            
BaseDir="/home/student/courses/MATH089/images/";

          

        

      

    

    

      

        

          

            
∴ 

            
savefig(BaseDir*"P02FigTri.eps");

          

        

      

    

    

      

        

          

            

              
            

          

        

      

    

    

      

        

          

            
∴ 

            

          

        

      

    

    

      Test the tetrahedron drawing function
    

    

      

        

          

            
∴ 

            
s3=sqrt(3); s23=sqrt(2/3); s13=sqrt(1/3);

          

        

      

    

    

      

        

          

            
∴ 

            
T=[0 0 -0.5 0.5; 0 s3/3 -s3/6 -s3/6; s23 -s13 -s13 -s13];

          

        

      

    

    

      

        

          

            
∴ 

            
clf(); PlotTet(T);

          

        

      

    

    

      

        

          

            
∴ 

            
savefig(BaseDir*"P02FigTet.eps");

          

        

      

    

    

      

        

          

            

              
            

          

        

      

    

    

      

        

          

            
∴ 

            

          

        

      

    

    

      Define a function t otile 3D-space by tetrahedra
    

    

      

        

          

            
∴ 

            
function TileTet(X,n)
end;

          

        

      

    

    

      

        

          

            
∴ 

            

          

        

      

    

    

      

        

          

            

              

                
              

              

                
              

              

                
              

            

          

        

          
        

          

            

              Figure 3. 
            

          

        

      

    

    
2.4Polygons with fractal boundaries

    

      Repeated subdivision of polygon edges combined with geometric
      transformations can lead to non-intuitive results. Consider the process
      depicted in Fig. 4, described mathematically by taking the
      coordinates
    

    

      X=[
        
          x1
          x2
        
        
          y1
          y2
        
      ],
    

    

      forming the end-to-end vector
    

    

      R=[
        
          x2
        
        
          y2
        
      ]-[
        
          x1
        
        
          y1
        
      ],
    

    

      and drawing as output four line segments, that are modifications of
      R by scaling and rotation
    

    

      Xa=X1+13⁡M(0)R,Xb=Xa+13M(π3)R,Xc=Xb+13M(-π3)R
    

    

      M(θ)=[
        
          cos⁡θ
          -sin⁡θ
        
        
          sin⁡θ
          cos⁡θ
        
      ]
    

    

      The above is implemented in the ScRot function:
    

    

      

        

          

            

              Julia (1.6.1) session in GNU TeXmacs
            

          

        

      

    

    

      

        

          

            
∴ 

            
function ScRot(X0, s, θ, R)
   X0 + s*[cos(θ) -sin(θ); sin(θ) cos(θ)]*R
end;

          

        

      

    

    

      

        

          

            
∴ 

            
ScRot([1; 0], 1/3, π/3, [1; 0])

          

        

      

    

    

      

        

          

            

              

                [
                  
                    1.1666666666666667
                  
                  
                    0.28867513459481287
                  
                ]
                (1)
              

            

          

        

      

    

    

      

        

          

            
∴ 

            

          

        

      

    

    

      

        

          
        

          
        

          

            

              Figure 4. 
            

          

        

      

    

    

      

        

          

            
∴ 

            
function PlotSeg(X)
end function

          

        

      

    

    

      
    

    

      
    

    
2.4.1Koch snowflake

    

      

        

          

            
∴ 

            
function star(x0,y0,c)
   b0=0.5; z=0.0
   c0=sqrt(3.0); c1=c0/3; c2=c0/6
   x = [z,b0,-b0,z]
   y = [c1,-c2,-c2,c1]
   plot(x,y,c,x,-y,c);
end;

          

        

      

    

    

      

        

          

            
∴ 

            
clf(); star(0,0,"r-"); axis("equal");

          

        

      

    

    

      

        

          

            
∴ 

            
star(0,0,"w-");

          

        

      

    

    

      

        

          

            
∴ 

            

          

        

      

    

    

      
    

    

      

        

          

            
∴ 

            
function Koch(m)
 
end;

          

        

      

    

    

      

        

          

            
∴ 

            

          

        

      

    

    
3Results

    
4Discussion and conclusion