Spiral

t=0:sqrt(pi)/pi^2:5*pi;

r=sqrt(t);

lent=length(t);

x=zeros(lent,8);

y=zeros(lent,8);

x2=zeros(lent,8);

y2=zeros(lent,8);

pd=-0.75:0.25:1;

for n=1:8

x(:,n)=r.*cos(t+pd(n)*pi);

y(:,n)=r.*sin(t+pd(n)*pi);

x2(:,n)=-r.*cos(t+pd(n)*pi);

y2(:,n)=r.*sin(t+pd(n)*pi);

end

h_s=scatter(x(:)',y(:)',50+10*((x(:)').^2+(y(:)').^2),...

((x(:)').^2+(y(:)').^2),'filled');

colormap( fliplr(hsv) );

t=linspace(0.1,14*pi,500);

r=0.2./sqrt(t);

x=r.*cos(t);

y=r.*sin(t);

plot(x,y);

%% Baravelle Spiral Square

% This spiral is made from four square.On each step

% all squares are rotated at pi/4 and scaled down

% by 1/sqrt(2).

%% Four Squares

% Length is taken as 1.All the vertices and faces of the squares,

% are kept together.Coordinates are rotated in polar coordinates

% and drawn in cartesian.This makes the rotation easier.

l = 1;

vertices = [0 0;l 0;l l;0 l;-l l;-l 0;-l -l;0 -l;l -l];

faces = [1 2 3 4;

1 4 5 6;

1 6 7 8;

1 8 9 2];

vert = vertices;

[theta,r] = cart2pol(vertices(:,1),vertices(:,2));

%% Draw the Spiral

% On each step ,all the squares are rotated by pi/4

for k = 0:pi/4:4*pi

[vert(:,1),vert(:,2)] = pol2cart(theta-k,r);

patch('Vertices',vert,'Faces',faces,...

'facecolor','flat','edgecolor','none',...

'FaceVertexCData',[1 2 3 4]');

r = r/sqrt(2);

end

axis equal off

%% Baravelle Spiral triangle

% This spiral is made from three triangle.On each step

% all triangles are rotated at an angle pi/3 and scaled down

% by 1/2.

%% Three equilateral Traingles

% Length is taken as 1.Vertices represent an equilateral triangle.

% Vertices consist of 7 points. O is the centre of the triangle,

% A,B,C are the vertex and D,E,F are the centre of edges.

% D,E,F are connected to the O.Coordinates are rotated in polar

% coordinates and converted to cartesian before they are drawn.

l = 1;

vertices = [0 0; %O

0 l/(2*sin(pi/3)); %A

-l/2 -l/2*tan(pi/6); %B

l/2 -l/2*tan(pi/6); %C

-l/2*cos(pi/3) l/(2*sin(pi/3))*cos(pi/3)^2; %D

0 -l/2*tan(pi/6); %E

l/2*cos(pi/3) l/(2*sin(pi/3))*cos(pi/3)^2];%F

faces = [1 5 2 7;1 6 3 5;1 6 4 7];

vert = vertices;

[theta,r] = cart2pol(vertices(:,1),vertices(:,2));

%% Draw the Spiral

% On each step , rotate each triangle by pi/3

for k = 0:pi/3:3*pi

[vert(:,1),vert(:,2)] = pol2cart(theta-k,r);

patch('Vertices',vert,'Faces',faces,...

'facecolor','flat','edgecolor','none',...

'FaceVertexCData',[1 2 3]');

r = r/2;

end

axis equal off

%% Circle Spiral

% Create a Spiral of Circles.The spiral is made of many

% cicles with increasing radius and placed at some angular offset.

%% Make the first circle

% Make a cicle in polar coordinates and convert

% to cartesian.Place the circle on y-axis at some offset

t = linspace(0,2*pi,200);

r = 1;

[x,y] = pol2cart(t,r);

y = y-min(y); %place the cicle on +ve side of y-axis

[t,r] = cart2pol(x,y);

%% Draw the spiral

% Draw many cicles starting from the first one and gradually

% increase the radius.Also add a small angle to rotate circle

% about the origin.

for k = 1:1/(4*pi):2*pi

r = 1.025*r; %increase the radius of circle

[x,y] = pol2cart(t+k,r);

patch(x,y,'r','facecolor','none','edgecolor',[1/k 1-1/k 1-1/k]);

end

axis equal

box on