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
 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

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');
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);
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,'\t<title>Pythagoras tree</title>\n');
fprintf(fid,'\t\t<rdf:RDF>\n');
fprintf(fid,'\t\t\t<cc:Work\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</cc:Work>\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<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</rdf:RDF>\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',...
Offset = 2^i-1;
for j = 1:2^i
k = j + Offset;
'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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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

I, the copyright holder of this work, hereby publish it under the following licenses:
 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.
 This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported, 2.5 Generic, 2.0 Generic and 1.0 Generic license. You are free: to share – to copy, distribute and transmit the work to remix – to adapt the work Under the following conditions: attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. share alike – If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one.

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current23:16, 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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