File:Pythagoras tree 1 1 13 Summer.svg

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Description
English: Pythagoras tree
Français : Arbre de Pythagore
Русский: Дерево Пифагора
Date
Source Own work
Author Guillaume Jacquenot Gjacquenot
SVG development
Matlab Logo.png
This diagram was created with MATLAB.
MATLAB code
function M = Pythagor_tree(m,n,Colormap)
% function M = Pythagor_tree(m,n,Colormap)
% Compute Pythagoras_tree
% The Pythagoras Tree is a plane fractal constructed from squares.
% It is named after Pythagoras  because each triple of touching squares 
% encloses a right triangle, in a configuration traditionally used to
% depict the Pythagorean theorem.
% http://en.wikipedia.org/wiki/Pythagoras_tree
%
% Input : 
%       - m ( double m> 0) is the relative length of one of the side
%         right-angled triangle. The second side of the right-angle is 
%         taken to be one.
%         To have a symmetric tree, m has to be 1.
%       - n ( integer ) is the level of recursion.
%         The number of elements of tree is equal to 2^(n+1)-1.
%         A reasonnable number for n is 10.
%       - Colormap: String used to generate color of the different levels
%         of the tree.
%       All these arguments are optional: the function can run with
%       argument.
% Output : 
%       - Matrix M: Pyhagoras tree is stored in a matrix M.
%         This matrix has 5 columns.
%         Each row corresponds to the coordinate of each square of the tree
%         The two first columns give the bottom-left position of each
%         square. The third column corresponds to the orientation angle of
%         each square. The fourth column gives the size of each square. The
%         fifth column specifies the level of recursion of each square.
%         The first row corresponds to the root of the tree. It is always
%         the same
%         M(1,:) = [0 -1 0 1 1];
%         The leaf located at row i will give 2 leaves located at 2*i and
%         2*i+1.
%       - A svg file giving a vectorial display of the tree. The name of
%         file is generated from the parameter m,n,Colormap. The file is
%         stored in the current folder.
%
% 2010 02 29
% Guillaume Jacquenot
% guillaume dot jacquenot at gmail dot com

%% Check inputs
narg = nargin;
if narg <= 2
    % Colormap = 'jet';
    Colormap = 'summer';
    if narg <= 1
        n = 12; % Recursion level    
        if nargin == 0
            m = 0.8;
        end
    end
end
if m <= 0
	error([mfilename ':e0'],'Length of m has to be greater than zero');
end
if rem(n,1)~=0
	error([mfilename ':e0'],'The number of level has to be integer');
end
if ~iscolormap(Colormap)
	error([mfilename ':e1'],'Input colormap is not valid');
end
%% Compute constants
d      = sqrt(1+m^2);                  % 
c1     = 1/d;                          % Normalized length 1
c2     = m/d;                          % Normalized length 2
T      = [0 1/(1+m^2);1 1+m/(1+m^2)];  % Translation pattern  
alpha1 = atan2(m,1);                   % Defines the first rotation angle
alpha2 = alpha1-pi/2;                  % Defines the second rotation angle
pi2    = 2*pi;                         % Defines pi2
nEle   = 2^(n+1)-1;                    % Number of elements (square)
M      = zeros(nEle,5);                % Matrice containing the tree
M(1,:) = [0 -1 0 1 1];                 % Initialization of the tree

%% Compute the level of each square contained in the resulting matrix
Offset = 0;
for i = 0:n
    tmp = 2^i;
    M(Offset+(1:tmp),5) = i;
    Offset = Offset + tmp;
end
%% Compute the position and size of each square wrt its parent
for i = 2:2:(nEle-1)
    j          = i/2;
    mT         = M(j,4) * mat_rot(M(j,3)) * T;
    Tx         = mT(1,:) + M(j,1);
    Ty         = mT(2,:) + M(j,2);    
    theta1     = rem(M(j,3)+alpha1,pi2);
    theta2     = rem(M(j,3)+alpha2,pi2);
    M(i  ,1:4) = [Tx(1) Ty(1) theta1 M(j,4)*c1];
    M(i+1,1:4) = [Tx(2) Ty(2) theta2 M(j,4)*c2];
end
%% Display the tree
Pythagor_tree_plot(M,n);

%% Write results to an SVG file
Pythagor_tree_write2svg(m,n,Colormap,M);

function Pythagor_tree_write2svg(m,n,Colormap,M)
% Determine the bounding box of the tree with an offset
% Display_metadata = false;
Display_metadata = true;

nEle    = size(M,1);
r2      = sqrt(2);
LOffset = M(nEle,4) + 0.1;
min_x   = min(M(:,1)-r2*M(:,4)) - LOffset;
max_x   = max(M(:,1)+r2*M(:,4)) + LOffset;
min_y   = min(M(:,2)          ) - LOffset;  % -r2*M(:,4)
max_y   = max(M(:,2)+r2*M(:,4)) + LOffset;

% Compute the color of tree
ColorM = zeros(n+1,3);
eval(['ColorM = flipud(' Colormap '(n+1));']);
co   = 100;
Wfig = ceil(co*(max_x-min_x));
Hfig = ceil(co*(max_y-min_y));
filename = ['Pythagoras_tree_1_' strrep(num2str(m),'.','_') '_'...
             num2str(n) '_' Colormap '.svg'];
fid  = fopen(filename, 'wt');
fprintf(fid,'<?xml version="1.0" encoding="UTF-8" standalone="no"?>\n');
if ~Display_metadata
    fprintf(fid,'<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN"\n'); 
    fprintf(fid,'  "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">\n');
end
fprintf(fid,'<svg width="%d" height="%d" version="1.1"\n',Wfig,Hfig); % 
% fprintf(fid,['<svg width="12cm" height="4cm" version="1.1"\n']); % Wfig,

% fprintf(fid,['<svg width="15cm" height="10cm" '...
%              'viewBox="0 0 %d %d" version="1.1"\n'],...
%              Wfig,Hfig);
if Display_metadata
    fprintf(fid,'\txmlns:dc="http://purl.org/dc/elements/1.1/"\n');
    fprintf(fid,'\txmlns:cc="http://creativecommons.org/ns#"\n');
    fprintf(fid,['\txmlns:rdf="http://www.w3.org/1999/02/22'...
                 '-rdf-syntax-ns#"\n']);
end
fprintf(fid,'\txmlns:svg="http://www.w3.org/2000/svg"\n');
fprintf(fid,'\txmlns="http://www.w3.org/2000/svg"\n');
fprintf(fid,'\txmlns:xlink="http://www.w3.org/1999/xlink">\n');

if Display_metadata
    fprintf(fid,'\t<title>Pythagoras tree</title>\n');
    fprintf(fid,'\t<metadata>\n');
    fprintf(fid,'\t\t<rdf:RDF>\n');
    fprintf(fid,'\t\t\t<cc:Work\n');
    fprintf(fid,'\t\t\t\trdf:about="">\n');
    fprintf(fid,'\t\t\t\t<dc:format>image/svg+xml</dc:format>\n');
    fprintf(fid,'\t\t\t\t<dc:type\n');
    fprintf(fid,'\t\t\t\t\trdf:resource="http://purl.org/dc/dcmitype/StillImage" />\n');
    fprintf(fid,'\t\t\t\t<dc:title>Pythagoras tree</dc:title>\n');
    fprintf(fid,'\t\t\t\t<dc:creator>\n');
    fprintf(fid,'\t\t\t\t\t<cc:Agent>\n');
    fprintf(fid,'\t\t\t\t\t\t<dc:title>Guillaume Jacquenot</dc:title>\n');
    fprintf(fid,'\t\t\t\t\t</cc:Agent>\n');
    fprintf(fid,'\t\t\t\t</dc:creator>\n');
    fprintf(fid,'\t\t\t\t<cc:license\n');
    fprintf(fid,'\t\t\t\t\t\trdf:resource="http://creativecommons.org/licenses/by-nc-sa/3.0/" />\n');
    fprintf(fid,'\t\t\t</cc:Work>\n');
    fprintf(fid,'\t\t\t<cc:License\n');
    fprintf(fid,'\t\t\t\trdf:about="http://creativecommons.org/licenses/by-nc-sa/3.0/">\n');
    fprintf(fid,'\t\t\t\t<cc:permits\n');
    fprintf(fid,'\t\t\t\t\trdf:resource="http://creativecommons.org/ns#Reproduction" />\n');
    fprintf(fid,'\t\t\t\t<cc:permits\n');
    fprintf(fid,'\t\t\t\t\trdf:resource="http://creativecommons.org/ns#Reproduction" />\n');
    fprintf(fid,'\t\t\t\t<cc:permits\n');
    fprintf(fid,'\t\t\t\t\trdf:resource="http://creativecommons.org/ns#Distribution" />\n');
    fprintf(fid,'\t\t\t\t<cc:requires\n');
    fprintf(fid,'\t\t\t\t\trdf:resource="http://creativecommons.org/ns#Notice" />\n');
    fprintf(fid,'\t\t\t\t<cc:requires\n');
    fprintf(fid,'\t\t\t\t\trdf:resource="http://creativecommons.org/ns#Attribution" />\n');
    fprintf(fid,'\t\t\t\t<cc:prohibits\n');
    fprintf(fid,'\t\t\t\t\trdf:resource="http://creativecommons.org/ns#CommercialUse" />\n');
    fprintf(fid,'\t\t\t\t<cc:permits\n');
    fprintf(fid,'\t\t\t\t\trdf:resource="http://creativecommons.org/ns#DerivativeWorks" />\n');
    fprintf(fid,'\t\t\t\t<cc:requires\n');
    fprintf(fid,'\t\t\t\t\trdf:resource="http://creativecommons.org/ns#ShareAlike" />\n');
    fprintf(fid,'\t\t\t</cc:License>\n');
    fprintf(fid,'\t\t</rdf:RDF>\n');
    fprintf(fid,'\t</metadata>\n'); 
end
fprintf(fid,'\t<defs>\n');
fprintf(fid,'\t\t<rect width="%d" height="%d" \n',co,co);
fprintf(fid,'\t\t\tx="0" y="0"\n');
fprintf(fid,'\t\t\tstyle="fill-opacity:1;stroke:#00d900;stroke-opacity:1"\n');
fprintf(fid,'\t\t\tid="squa"\n');
fprintf(fid,'\t\t/>	\n');
fprintf(fid,'\t</defs>\n');
fprintf(fid,'\t<g transform="translate(%d %d) rotate(180) " >\n',...
                round(co*max_x),round(co*max_y));
for i = 0:n
    fprintf(fid,'\t\t<g style="fill:#%s;" >\n',...
                generate_color_hexadecimal(ColorM(i+1,:)));            
    Offset = 2^i-1;
    for j = 1:2^i
        k = j + Offset;
        fprintf(fid,['\t\t\t<use xlink:href="#squa" ',...
                     'transform="translate(%+010.5f %+010.5f)'...
                     ' rotate(%+010.5f) scale(%8.6f)" />\n'],...
                    co*M(k,1),co*M(k,2),M(k,3)*180/pi,M(k,4));   
    end
    fprintf(fid,'\t\t</g>\n');
end
fprintf(fid,'\t</g>\n');
fprintf(fid,'</svg>\n');
fclose(fid);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function M = mat_rot(x)
c = cos(x);
s = sin(x);
M=[c -s; s c];

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function H = Pythagor_tree_plot(D,ColorM)
if numel(ColorM) == 1
    ColorM = flipud(summer(ColorM+1));
end
H = figure('color','w');
hold on
axis equal
axis off
for i=1:size(D,1)
    cx    = D(i,1);
    cy    = D(i,2);
    theta = D(i,3);
    si    = D(i,4);    
    M     = mat_rot(theta);
    x     = si*[0 1 1 0 0];
    y     = si*[0 0 1 1 0];
    pts   = M*[x;y];
    fill(cx+pts(1,:),cy+pts(2,:),ColorM(D(i,5)+1,:));
    % plot(cx+pts(1,1:2),cy+pts(2,1:2),'r');
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function Scolor = generate_color_hexadecimal(color)
Scolor = '000000';
for i=1:3
    c = dec2hex(round(255*color(i)));
    if numel(c)==1
        Scolor(2*(i-1)+1) = c;
    else
        Scolor(2*(i-1)+(1:2)) = c;
    end
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function  res = iscolormap(cmap)
% This function returns true if 'cmap' is a valid colormap
LCmap = {...
    'autumn'
    'bone'
    'colorcube'
    'cool'
    'copper'
    'flag'
    'gray'
    'hot'
    'hsv'
    'jet'
    'lines'
    'pink'
    'prism'
    'spring'
    'summer'
    'white'
    'winter'
};

res = ~isempty(strmatch(cmap,LCmap,'exact'));

Licensing[edit]

I, the copyright holder of this work, hereby publish it under the following licenses:
GNU head Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.
w:en:Creative Commons

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current23:16, 28 February 2010Thumbnail for version as of 23:16, 28 February 2010618 × 420 (2.05 MB)Gjacquenot (talk | contribs){{Information |Description={{en|1=Pythagoras tree}} {{fr|1=Arbre de Pythagore}} |Source={{own}} |Author=Gjacquenot |Date=2010-03-01 |Permission= |other_versions= }} Category:Pythagoras trees
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