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Some Formulae

Scans from the papers of Lawrence Edwards

Examples of the formulae in use

(Images © G. Calderwood; the mathematical content is public domain.)



An Artificial Example

Plot points A, B, C and D on a line

  1. Calculate the positions of invariant points X and Y
  2. Plot these positions on the line
  3. Draw a line through X, and a line through Y
  4. Select a point on the line through Y
  5. Project plotted point B through the selected point on to the line through X
  6. Project plotted point C through the selected point on to the line through X
  7. These projected points are the directrices controlling the transformation
  8. Use the directrices, and points B and C, to locate points A and D of the transformation

If the positions of points X and Y have been correctly calculated and plotted, Points A and D should fall, by pure construction, on their plotted positions.


Given four points A, B, C & D on the line, and their spacings, we want the invariants, X & Y

a = AB = 2,
c = BC = 4,
d = CD = 3,
x = XB, y = BY.

We need the cross ratio, R, of the four points A, B, C & D to find the multiplier, m.









These values are measured outwards from B, and tested by construction, above.  The agreement of construction with calculation is excellent.

A Natural Example


We use John's arrows and ruler to obtain three sets of three intervals (My markings)


Set           a              c              d              R                 m                    Type
 s1           4.8           5.3           7.0         3.697        0.85+i0.51           circling
 s2           5.3           7.0         10.2         3.914        0.96+i0.28           circling
 s3           7.0         10.2         12.8         4.415           1.883                growth

We see that only the third set, s3, gives real invariants: the X invariant lies 16.86 mm (by John's ruler) left of the fourth arrow from the left.

Using the position of this invariant for the check-by-construction, we see that the positions of the nodes corresponding to s3 are accurately traced, but that other node-positions are not.

This suggests that we may have a measure that varies smoothly in "amplitude" along the length of the bamboo.


With thanks to John Blackwood.
His book, “Geometry in Nature”, is published by
Floris Books,
15, Harrison Gardens,