% circles12.m -------------------------------------------------------------------
%
% This is a two part script. The first part creates 3 figures
% each showing a different solution to the following problem:
%
% *****************************************************
% *****************************************************
% ** **
% ** The 12 circle problem: **
% ** Draw 12 circles in a plane so that every **
% ** circle is tangent to exactly 5 others. **
% ** **
% *****************************************************
% *****************************************************
%
% - Demonstrates the utility of using complex numbers to
% hold the x and y positions of the plotted points.
%
% - Demonstrates 2 ways of creating the Trace IDs (reshape & repmat).
%
% - Demonstrates how to make circles look true by using a zero in the
% 'Position' argument (width or height).
%
% - Note that even though the calls to plt for solutions 2&3 specify
% same screen location ('Position' parameter) plt doesn't actually
% plot them on top of each other. Instead a small offset is added
% in this situation, a feature that makes it easier to create
% many plt windows so that any of them can be easily selected
% with the mouse.
%
% - The last figure shows the use of the Nocursor and -All options
% to make the cursor objects and menu box items invisible.
%
% ----- Author: ----- Paul Mennen
% ----- Email: ----- paul@mennen.org
N = 600; % number of points per circle
cN = exp((1:N)*pi*2i/(N-1))'; % N point unit circle
% Solution #3 ------------------------------------------------------
v = 1:4; r = (12+5*v)./(12-v); % radii of vertical set of 4
c = [v-1 9-v complex(4,7-r)]; % center location of all 12 circles
r = [v v r]; % radii of all 12 circles
sz = get(0,'screensize');
plt(cN*r + repmat(c,N,1),'FigName','12 Circles (solution 3)',...
'LineWidth',2,'Options','Ticks','Position',[10 sz(4)-715 620 648],...
'Xlim',[-1.6 9.5],'Ylim',[-4.8 7.5],...
'AxisPos',[1 1 1 1 1.2],'LabelY','','LabelX','',...
'TraceID',reshape(sprintf('circle%2d',1:12),8,12)');
% Solutions #1 and 2 ----------------------------------------------
% Note: there are 2 more solutions that are essentially combinations of
% solutions #1 and #2. If you find yet more solutions, please let me know!
ID = [{'middle'; 'outer'}; repmat({'group1'},5,1); repmat({'group2'},5,1);];
pi5 = pi/5; s = 1/sin(pi5); u5 = ones(1,5);
c5 = exp((0:4)*pi5*2i); % 5 point unit circle
rg1 = 1/(s-1); % radius of group 1 circles
t = 1 + 2*rg1;
rg2 = roots([s^2-1, -2*(s*sqrt(t)+rg1), t]); % radius of group 2 circles
b = c5*s*rg1*exp(pi5*1i);
for k = [2 1]
c = [0 0 b c5*s*rg2(k)]; % center location of all 12 circles
r = [1 rg2(k)*(s+1) rg1*u5 rg2(k)*u5]; % radii of all 12 circles
plt(cN*r + repmat(c,N,1),'LabelY','','LabelX','',...
'FigName',sprintf('12 Circles (solution %d)',k),'Options','Ticks',...
'TraceID',ID,'LineWidth',2,'Position',[10 50 600 0]);
end;
% The second part of this script (below) draws the solution to the
% following problem:
%
% ********************************************************
% ********************************************************
% ** **
% ** Divide a circle into n congruent parts so that **
% ** at least one part does not touch the center. **
% ** (Hint: the only known solution uses n = 12) **
% ** **
% ********************************************************
% ********************************************************
%
% A slider is also added below the plot which lets you rotate the
% image and control the rotation speed
%
N6 = N/6; f = 1:6; na = NaN+f; cN = cN';
a = repmat([3 4 5 6 1 2]*N6,N6,1); % index points for arc centers
d = cN + cN(a(:)); % N point circle with translated centers
e = [reshape(transpose(d),N6,6); na]; % insert NaNs to eliminate stray lines
f = [na; cN(N6*f); d(N6*(f-.5))]; % draw the lines bisecting the 6 congruent parts
e = [e(:)' f(:)']; % append the bisecting lines to the 6 arcs
h = plt(e,cN,'Position',[645 sz(4)-470 0 400],...
'Figname','12 circles (part 2)','Options','Ticks Nocur -All','TraceID',['div ';'circ'],...
'LabelX','','LabelY','','Xlim',[-1.1 1.1],'Ylim',[-1.1 1.1]);
set(gca,'xtick',[],'ytick',[]); % remove tick marks
hs = plt('slider',[.02 .1 .3],[1 -15 15],... % create rotation speed slider
'Rotation speed','',2,['%3d';'%3d';'%2d']);
bt = uicontrol('str','Stop','units','norm',... % create close button
'position',[.35 .02 .08 .05],'callback',...
'delete(findobj(gcf,''str'',''Stop''))'); % the callback causes the button to delete itself
while ishandle(bt) % forever loop (until you click on the stop button)
s = plt('slider',hs,'get');
if s if s<0 s=s+N-1; end; % rotate CCW/CW for Positive/Negative slider position
e = e .* cN(s);
set(h(1),'x',real(e),'y',imag(e));
end;
pause(.02);
end;
if ishandle(h(1)) plt('slider',hs,'set','visOFF'); end; % hide the slider
