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