In the book My Painted Elephant by Keren Sutcliffe, Brenda makes the word count of the chapters proportional to the derivative of the lengths of the hypotenuse of each successive square root triangle in the Theodorus Spiral.

Brenda was getting frustrated that the ‘Struggling to get it right’ chapter was now too long and she would need to rearrange the story again or maybe she thought she could add two hundred words to every chapter to make the second derivative spiral right. She had already times the change in hypotenuse by ten thousand and then divided it by two. It was becoming difficult to manage. She pondered the idea of investigating the word lengths if she calculated the change in the change of the hypotenuse of each triangle. It would then become a story with chapter lengths determined by the third derivative of the Theodorus Spiral. She was madly cutting and pasting a bit from here and into there. She was frantic because she realised she was chopping and changing the only copy. She should have made a new copy and started altering that. She only had a mini laptop with no mouse, just a worn out touch pad to work with. This was insane.     Except from My Painted Elephant

(Change in hypotenuse X 1000 / 2) + 200 = Brenda’s target word count for each chapter.

1 1.00 The beginning: the longest part (1968/1966)

2 1.41 Max : the information trader in 2021 (1644/1596)

3 1.73 Brenda’s first night (1450/1307)

4 2.00 Struggling to get it right (1320/1322)

5 2.24 It’s a shabobble (1220/1202)

6 2.45 Max’s friends (1145/1059)

7 2.65 The adventures of Kesbooks Skytower (1083/1080)

8 2.82 Charlotte’s full house (1033/1044)

9 3.00 Max’s party (991/939)

10 3.16 Who is Claude? (954/856)

11 3.32 What is in this world for Brenda? (921/737)

12 3.46 Kembla the blacksmith (893/676)

13 3.60 Theodorus has something to say (868/677)

14 3.74 Phone a friend (845/687)

15 3.87 It’s all gone (825/454)

16 4.00 Time to go home (806/454)

17 4.12 The end: the shortest part (789/490)

%Created by Keren Sutcliffe 2011

clear all

total_angle=0;
winding=0;
finish=16; %finish = 16 creates the spiral that Theodorus completed
           %change the value of finish to any number

fprintf(' x       hyp    change in hyp  chngeinchngehyp   angle     chngeangle     X2           Y2          winding    circum   chngeincircum\n');

for x=1:finish

    hyp=sqrt(x+1);
    chngeinhyp=1/(2*sqrt(x+1));
    chngechngehyp=1/(-4*sqrt(x+1));
    angle=atan(1/sqrt(x));
    degangle=angle*180/pi;
    chngeangle=(1/(1+x))*(1/(2*sqrt(x)));
    total_angle=total_angle+angle;
    degtotal_angle=total_angle*180/pi;
    X=hyp*cos(total_angle);
    Y=hyp*sin(total_angle);
    winding=degtotal_angle/360;
    circum=2*pi*hyp;
    chngeincircum=1/(2*circum);
    fprintf('%3i  %10f   %10f   %10f   %10f    %10f    %10f    %10f   %5f  %5f %5f \n',x ,hyp,chngeinhyp,chngechngehyp,degangle,chngeangle,X,Y,winding,circum, chngeincircum);

    figure(1)
    title('Theodorus Spiral')
    axis equal;
    xlabel ('x');
    ylabel ('y');

    hold on
    plot ([0 1], [0 0])   %First spoke
    plot ([0 X], [0 Y])   %spokes

    figure (2)
    title ('Theodorus Spiral First Derivative of the Hypotenuse(change in the hypotenuse)')
    xlabel ('x');
    ylabel ('y');

    hold on
    plot ([x x], [0 chngeinhyp]) %first derivative
    plot (x, chngeinhyp)

    figure (3)
    title ('Theodorus Spiral Second Derivative of the Hypontenuse (change in the change of the hypotenuse)')
    xlabel ('x');
    ylabel ('y');

    hold on
    plot ([x x], [0 chngechngehyp])   %second derivative

    figure (4)
    title ('Theodorus Spiral Derivative of the Circumferance (change in the circumferance)')
    xlabel ('x');
    ylabel ('y');

    hold on
    plot ([x x], [0 chngeincircum])  %derivative of circumferance
end

 x       hyp    change in hyp  chngeinchngehyp   angle     chngeangle     X2           Y2          winding    circum   chngeincircum
  1    1.414214     0.353553    -0.176777    45.000000      0.250000      1.000000      1.000000   0.125000  8.885766 0.056270 
  2    1.732051     0.288675    -0.144338    35.264390      0.117851      0.292893      1.707107   0.222957  10.882796 0.045944 
  3    2.000000     0.250000    -0.125000    30.000000      0.072169     -0.692705      1.876209   0.306290  12.566371 0.039789 
  4    2.236068     0.223607    -0.111803    26.565051      0.050000     -1.630810      1.529856   0.380082  14.049629 0.035588 
  5    2.449490     0.204124    -0.102062    24.094843      0.037268     -2.314982      0.800536   0.447012  15.390598 0.032487 
  6    2.645751     0.188982    -0.094491    22.207654      0.029161     -2.641800     -0.144552   0.508700  16.623746 0.030077 
  7    2.828427     0.176777    -0.088388    20.704811      0.023623     -2.587164     -1.143058   0.566213  17.771532 0.028135 
  8    3.000000     0.166667    -0.083333    19.471221      0.019642     -2.183032     -2.057759   0.620300  18.849556 0.026526 
  9    3.162278     0.158114    -0.079057    18.434949      0.016667     -1.497113     -2.785436   0.671508  19.869177 0.025165 
 10    3.316625     0.150756    -0.075378    17.548401      0.014374     -0.616280     -3.258865   0.720254  20.838968 0.023994 
 11    3.464102     0.144338    -0.072169    16.778655      0.012563      0.366304     -3.444680   0.766861  21.765592 0.022972 
 12    3.605551     0.138675    -0.069338    16.102114      0.011103      1.360698     -3.338937   0.811589  22.654347 0.022071 
 13    3.741657     0.133631    -0.066815    15.501360      0.009905      2.286752     -2.961547   0.854648  23.509527 0.021268 
 14    3.872983     0.129099    -0.064550    14.963217      0.008909      3.078259     -2.350387   0.896213  24.334672 0.020547 
 15    4.000000     0.125000    -0.062500    14.477512      0.008069      3.685127     -1.555584   0.936428  25.132741 0.019894 
 16    4.123106     0.121268    -0.060634    14.036243      0.007353      4.074023     -0.634302   0.975418  25.906237 0.019300