Reconstruction Wrotham 2000

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 cross-lines, such that the woven-like 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 = r₁ cos φ_i

(1)

y_i = r₁ sin φ_i

(2)

u_i = x_i + r₂ cos –a·φ_i

(3)

v_i = y_i + r₂ sin –a·φ_i

(4)

p_i = x_i + c·r₂ cos –a·φ_i

(5)

q_i = y_i + c·r₂ sin –a·φ_i

(6)

With the following parameter setting:

a = 2

b = 2.6

c = 0.55

r₂ = b·r₁

φ_i = 0,...,2π

the set of formula's will generate this pattern:

mathematical representation of Celtic knot

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 r₁. Each position of point A is determined by the value of φ; φ runs counter-clockwise.

12.

Another circle with radius r₂ (= 2.6 times r₁) 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 r₂) from point A on the same ray as point B.

Since 1-February-2005