Reconstruction of the 2000 Wrotham formation 

1. 

Start with constructing a equilateral triangle. 

2. 

Draw three circles centered at each corner of the triangle
and tangent to the opposite sides. 

3. 

Remove the parts of the circles, that lie outside the triangle. 

4. 

Draw circles centered at the midpoint of each side of the triangle,
and tangent to the larger circles. 

5. 

From the same midpoints, draw smaller circles,
touching the other two circles. 

6. 

As with step 2, draw circles from each corner of the triangle,
that touch the latter smaller circles at the far side. 

7. 

Again, remove from the last drawn circles all parts outside
the triangle. 

8. 

Remove from all circles introduced in steps 4 and 5,
all parts inside the triangle; then, remove the triangle. 

9. 

Finally, remove repeatedly two of the four crosslines,
such that the wovenlike knot pattern appears. 

10. 
courtesy
The Crop Circle Connector
original image by:
Andrew King 
The final reconstruction fits neatly. 


I tried to find mathematical expressions with a graphical
representation similar to the pattern of the Wrotham formation.
I found the following quite complicated formulas.


x_{i} =
r_{1}
cos φ_{i}

(1) 

y_{i} =
r_{1}
sin φ_{i}  (2) 

u_{i} = x_{i} +
r_{2}
cos –a·φ_{i}  (3) 

v_{i} = y_{i} +
r_{2}
sin –a·φ_{i}  (4) 

p_{i} = x_{i} +
c·r_{2}
cos –a·φ_{i}  (5) 

q_{i} = y_{i} +
c·r_{2}
sin –a·φ_{i}  (6) 

With the following parameter setting:


a = 2
 

b = 2.6
 

c = 0.55
 

r_{2} = b·r_{1}
 

φ_{i} = 0,...,2π
 
the set of formula's will generate this pattern:



To understand what happens, look at the following pictures:


11. 

Formula's (1) and (2) describe a point A
(x_{i},y_{i}) "walking"
around a circle with radius r_{1}.
Each position of point A is determined by the value of φ;
φ runs counterclockwise. 

12. 

Another circle with radius r_{2}
(= 2.6 times r_{1})
has its center in point A. An other point B
(u_{i},v_{i}) is walking
around this second circle, in the opposite direction,
and two times as fast (formula's (3) and (4)). 

13. 

Following the path of point B results in the outer
loop of the Celtic knot pattern.
The last two formula's (5) and (6) generate the inner loop.
It is the path of a point C
(p_{i},q_{i}),
that is just some distance (0.55 times r_{2})
from point A on the same ray as point B. 



Copyright © 2000, Zef Damen, The Netherlands Personal use only, commercial use prohibited.

Since 1February2005

