Reconstruction of the2000 Woodborough Hill (3) formation |
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1. |
Draw a circle. | ||||

2. |
Divide the circumference into 44 equal parts and draw the rays. (Notice that this necessitates the construction of a regular 11-fold polygon, which can not be done exactly by the "ruler-and-compass" construction method (this applies equally to 7-, 9-, 13-, 14-, 18-, 19-...fold polygons). See for instance Eric Weisstein's excellent World of Mathematics for constructible polygons). | ||||

3. |
Draw the two diagonals as shown in the figure. Construct a circle centered at the center of the pattern, passing through the intersection of the two diagonals. | ||||

4. |
Draw two other diagonals as shown, and construct again a circle passing through the intersection. | ||||

5. |
Construct a small circle, centered at the intersection of the second circle and the horizontal centerline, with a radius equal to the distance between the two circles (of the previous steps). | ||||

6. |
This small circle is used to "distribute" this distance along the horizontal centerline. Therefore, construct a series of small circles, centered at every next intersection with the horizontal centerline, each with a radius equal to that of the first one, thirteen in total. | ||||

6a. |
Accurate measurement reveals the remarkable fact, that the last small circle almost exactly touches the outer circle. The difference (enlarged here) only amounts to 0.0006114114... times the radius of the outer circle, or 1 into 1635.56! Therefore, the series divides the distance between the smaller central circle and the outer circle (almost exactly) into 15 equal parts. | ||||

7. |
Construct 13 concentric circles passing through the intersections of the series of small circles and the horizontal centerline. | ||||

8. |
Starting from the intersection of the second last central circle and the horizontal centerline, draw a line to the intersection of the next ray (up) and the preceding central circle. Repeat this until the smallest central circle is reached. (These lines essentially form a , obeying the rule of a "travelling" point the distance of which to the origin is linear proportional to the angle it makes with the horizontal, spiralr = a θ).
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9. |
Repeat the same construction in the other direction, down in stead of up. | ||||

10. |
Repeat the construction of the upper spiral another 21 times (22 in total), evenly distributed around the circle. | ||||

11. |
And do this also for the lower spiral. | ||||

12. |
Making black every other "triangle" (one of the sides is circular!) in a checkerboard-like fashion reveals the reconstruction of the 2000 Woodborough Hill (3) formation (black represents standing crop). The pattern resembles very much the positioning of seeds in for instance sunflowers or the scales in a pinecone; a very good site dealing with this matter can be found at R. Knott's personal pages. | ||||

13. |
courtesy The Crop Circle Connector photo by: UKox Image | The final result, matched with the aerial image. | |||

For comparison, two more reconstructions follow: | |||||

the 1997 Alton Barnes formation. | |||||

courtesy The Crop Circle Connector images by Steve Alexander |
the 2000 Uffington White Horse formation (largest circle only). | ||||

Reconstruction of the 1997 Alton Barnes formation | |||||

1. |
Draw a circle. Draw the horizontal and vertical centerlines. | ||||

2. |
Construct an equilateral triangle with the top at the upper endpoint of the vertical centerline. | ||||

3. |
Construct three more equilateral triangles, at each of the other endpoints. | ||||

4. |
Construct a circle touching two sides of the first triangle and touching the outer circle, all at the inner sides. | ||||

5. |
Construct two more similar circles touching the other pairs of the sides of the triangle. | ||||

6. |
Repeat steps 4 and 5 for the other three triangles. | ||||

7. |
Leaving out the outer circle, the centerlines and the four triangles finishes the reconstruction of the 1997 Alton Barnes formation. | ||||

8. |
Would the Circlemakers have finished this formation in the same fashion as the Woodborough Hill (3) formation, then it should perhaps have looked like this. Through the intersections of the existing circles (except for the outermost), draw concentric circles around the main center. | ||||

9. |
Make black every other "triangle" (now all three sides are curved) in a checkerboard-like fashion. | ||||

Reconstruction of the 2000 Uffington White Horse formation (largest circle only) | |||||

1. |
Draw a circle. | ||||

2. |
Construct a hexagon (regular 6-sided polygon). | ||||

3. |
Centered at each of the corners of the hexagon, construct circles (6 in total) each passing through two other corners that form together with its center an equilateral triangle. Remove all parts of the circles that fall outside the hexagon. | ||||

4. |
The intersections of the previous circles form a new, smaller hexagon. For each of the cornerpoints of this hexagon, repeat the construction of step 3, and remove everything outside the smaller hexagon. | ||||

5. |
Repeat the previous step one more time for the newly formed still smaller hexagon.... | ||||

6. |
...and again a last time. | ||||

7. |
Making black every other "triangle" (here again with all three sides curved) in a checkerboard-like way concludes the reconstruction of the 2000 Uffington White Horse formation (largest circle only). Black again represents standing crop. | ||||

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

Since 1-February-2005 |