MATLAB code to automate the search for a relation between the diagonals of regular polygons and the metallic ratios.

metallicratiosearch.m

sideLength = 1;
differenceThreshold = 0; % show results that differ at most this much.
%differenceThreshold = 0.0000000001;

metallicCount = 1000000; % to be safe, make sure this is greater than sideLength * (the maximum value of n)
metallicRatios = zeros(1, metallicCount);
for ratioNumber = 1:metallicCount
    metallicRatios(1, ratioNumber) = (ratioNumber + sqrt(4 + ratioNumber^2))/2;
end

tic
parfor n = 1:4000 % determines which polygons we examine.
    
    if (mod(n, 400) == 0)
        disp(["Checking Polygon with ", num2str(n), " sides."]);
        drawnow;
    end
    
   internalAngleSum = (n-2) * 180;
   singleAngle = internalAngleSum / n; % unused atm.
   exteriorAngle = 360/n;
   exteriorAngleRad = exteriorAngle*pi/180 ;
   
   for diagCount = 1:floor(n/2) - 1 %diagCount describes which diagonal this is (1st, 2nd etc.)
       
        diagStart = [0, 0];
        diagEnd = [0, 0];
        
        %trace the edges of the polygon up to the end of the diagonal, then measure distance:
        for rotCount = 1:(diagCount+1)
            %rotate each line segment by the correct amount and stick it onto the end of the last one
            theta = rotCount * exteriorAngleRad;
            rotMtx = [cos(theta) -sin(theta) ;sin(theta) cos(theta)] ;
            rotatedLineSegmentEnd = [0, sideLength] * rotMtx;
            diagEnd = diagEnd + rotatedLineSegmentEnd;
        end
        
        distance = norm(diagEnd - diagStart); % this is the length of the diagonal
        ratio = distance / sideLength;
        
        for ratioNumber = floor(ratio):ceil(ratio)  % don't need to check past these limits
            if (abs(ratio - metallicRatios(ratioNumber)) <= differenceThreshold)
                text = ["Diagonal #", num2str(diagCount), "of a",num2str(n),"sided polygon is related to ratio #", num2str(ratioNumber)];
                disp(text);
                format long
                disp(ratio);
                disp(metallicRatios(ratioNumber));
            end
        end

   end
end
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