Geometry Analysis of the Rockville Crop Formation

The late mathematician and astronomer Gerald Hawkins made this discovery back in the early 90's after looking into the complex designs that were beginning to come out in crop circles. What he discovered were 4 new theorems that existed beyond Euclidean Geometry!

Amazingly enough these significant aspects have been found to exist within the design of R1. We have had the privilege to have a highly qualified individual work with us on this. His detailed report offers graphics and the scientific breakdown of how he arrived at his conclusions. As well it offers a statistical analysis of what the chances are of these ratios occurring randomly. The odds are an astounding 1/45 million!

This talented individual's name is Dr. Jean-Noel Aubrun. We would like to extend our thanks to Dr. Aubrun for his hard work and important contribution. He has become a team member with us through this process offering support and consultations along the way. With a background like Dr. Aubrun's there is no doubt our work will carry a serious tone for scientific consideration. If our hoaxers would only come forward now, within the next week before this report is made public, perhaps they could take a small quiz that would show their knowledge of these advanced theorems and how they applied them in their designs.

Jean-Noel Aubrun holds a PhD in physics from the Sorbonne and an engineering diploma from Ecole de Physique et Chimie de Paris. He has been working for NASA and major US aerospace industries for over 30 years and directed several million-dollar research projects. Under his Marjan Research company, he acts as a scientific consultant for various aerospace and astronomy projects. His contributions to a wide variety of technological areas, especially in controls, electro-optical systems, mathematical modeling and simulation, have been nationally recognized and earned him several awards. Dr. Aubrun is the author of over 70 publications in scientific journals. His interest in the crop circle phenomenon started in the early 90's after participating in fieldwork in England under the direction of Mr. Colin Andrews.

The geometric properties of the July 2003 Rockville, California crop formation were derived from aerial photographs and ground measurements. Since neither photos nor ground measurements are accurate enough to assemble a perfect geometry model, these inputs were used to build a computer model representing some best fit of the data. Professional graphics/drafting software was used to obtain an idealized image of the formation.

At first, the size of each circle was reproduced as the best possible match, and then the whole picture was assembled. Certain assumptions, born out of the direct visual study of the formation as well as of historical data, were made concerning symmetry and other special relationships between the various elements.

The first striking feature on this formation is the definite pattern governing the ratios between the diameters of the various circles. As observed several times by the late Professor Gerald Hawkins in the crop formations of England, all the circles of the Rockville, CA formation are following the musical scale ratios. Specifically, if the diameter of the smallest of the circles is taken to represent the frequency of an F, all the other circles can be represented by a precise note of the diatonic scale (white keys on the piano).

Calling F_o the frequency (or note) representing the four smallest circles, one finds that the others circles can be defined by the notes C₁, F₁, A₂, C₂, G₂, C₃, and D₄. Table I lists the classical diatonic ratios covering three octaves where C₀ is chosen to have the value of 1. The bold numbers are the ones corresponding to the circles appearing in the formation.

Name	Ratio	Name	Ratio	Name	Ratio	Name	Ratio	Name	Ratio
		A₁	5/3	A₂	10/3	A₃	20/3	A₄	40/3
		B₁	15/8	B₂	15/4	B₃	15/2	B₄	15
C₀	1	C₁	2	C₂	4	C₃	8	C₄	16
D₀	9/8	D₁	9/4	D₂	9/2	D₃	9	D₄	18
E₀	5/4	E₁	5/2	E₂	5	E₃	10	E₄	20
F₀	4/3	F₁	8/3	F₂	16/3	F₃	32/3	F₄	64/3
G₀	3/2	G₁	3	G₂	6	G₃	12	G₄	24

The present analysis uses the aerial photography shown in Figure 1. With the help of the computer graphics program, one circle is first created to match as well as possible the dimensions of the smallest circles, and then all the others are created according to the ratios mentioned above. Then common scaling, projection and rotation are applied to these circles to reflect the aerial photograph particular view.

Each computer-generated circle is then laid down on top of the corresponding photographed circle as shown in Figure 2. Because of the distance distortion introduced by the camera, the size of circle D₄ must be adjusted based on ground measurements, but all the others can be used directly and generally exhibit a very good match with the ground formation. There are of course discrepancies in shape and size of various degrees that will be discussed in later sections.

In the next phase of this study we evaluate the spatial relationships between the circles. From the aerial photograph one can determine that the three main circles (C₃�s) have their centers roughly on the vertices of an equilateral triangle. In addition they seem to present an interesting tangency property, that is, one can draw three lines that are each (with some degree of approximation) tangent to all three circles. This pattern, shown in Figure 3 is quite interesting because it is precisely the First Euclidian Theorem discovered by Pr. G. Hawkins in the 1988 Cheesefoot Head formation in Southern England. This theorem states that the circle passing through the tangency points (and also through the center of the three main circles) is in the exact ration 4/3 with respect to these main circles. Moreover, it can be shown that the ratio of the diameters of the three main circles to the small circle inscribed in the triangle formed by the three tangents is exactly 3/1. These properties can be matched by allocating the note F₁ to the small inscribed circle, C₃ to the main circles, and F₃ to the circle passing through the three centers.

In this next step, we are examining the position of the three F₁ ground circles with respect to the central pattern of Figure 3. From the distance measured on the photograph (and correlated by ground measurements as well), it appears that these circles are tangent to the construction circle shown in Figure 3. Now, if one constructs yet another circle tangent to the three F₁ circles and on the outside of them, one finds again a remarkable property: the outside circle can be exactly represented by the note C₄. In addition, the circle passing through the centers of these F₁ circles can also be exactly represented by the note A₄, as shown in Figure 4.

Finally the position of the small circles F₀ is found to be tangent to the construction circle C₄. As the photograph seems also to indicate, the center of F₀ aligns with the corresponding centers of F₁ and C₃ as shown in Figure 5. These two conditions completely determine the position of these small circles.

Figure 4: Positioning The Three F₁ Circles Figure 5: Positioning The Four Small Circles

To complete this process, the two circles inside the top circle C₃ of the pattern are identified as C₂ and G₂, again exact diatonic ratios. Although the rest of the formation above this top circle is obviously distorted on the ground and is loosing its geometric accuracy, it contains four other circles that appear to be placed also according to the diatonic ratio rule. Using ground measurements to determine the length of the path between circles, one can nicely fit two more construction circles (i.e. circles that are not imprinted in the crop but contain the geometry for properly spacing the actual circles). This is shown in Figure 6 where four more tones appear, namely C₁, A₂,A₃ and D₄. The pattern on the figure starts at the top circle C₃ of the main pattern discussed previously and is rotated 90 degrees to better fit the page.

The final step of this analysis is to match the �theoretical� geometry pattern established in the previous section, with the actual pattern on the ground. The full pattern is assembled as shown in Figure 7 (the top circle D₄ is not shown). The filled colored circles represent the pattern imprinted on the ground, the lighter circles and lines are geometrical constructs that mathematically link those circles. Once the full pattern is constructed it is then scaled, rotated and positioned to best match the pattern on the ground. The aerial photograph is used for that purpose. The superimposition of computer model and actual formation is shown in Figure 8. It should be noted that the top of the pattern, while conserving the respective sizes of the circles, has only been matched topologically, since its alignment to the bottom formation is obviously considerably distorted.

While the main core of the formation (containing the circles F₀, F₁, C₂ , G₂ and C₃) is at first sight pretty well matched on the ground, there are also some significant discrepancies. The major ones are found in the gross misalignment of some of the F₀circles and the lack of tangency of one of the lines connecting the C₃ circles. The less obvious ones have to do with the regularity of the circular pattern themselves, especially for circle G₂ . Ground measurements have also revealed that the diameter of some of the circles varies with the direction of measurement.

One may wonder at the intricate geometry revealed in this seemingly simple combination of circles. Several remarkable properties and theorems are expressed that combine diatonic ratios and new Euclidian theorems in a very sophisticated manner. The preliminary analysis of this geometry presented in this report probably merely scratches the surface, and there could be more hidden properties and meanings.

From the analysis of the geometric design itself, even as preliminary as it is, one may definitively conclude that these constructions are not the result of mere coincidences, and that their level of sophistication is much beyond an ordinary high school level education.

However, the �execution� on the ground of the intended geometry appears to have been poorly carried out, a feature that does not sit well with the idea of some advanced technology being involved.

It is not the purpose of this report to conjecture or elaborate on what or who is responsible for such paradoxical construction. This part of the analysis is at this time left to the reader to ponder upon and to other investigators and researchers to look at all the facts and data available and come to their own conclusions. All that can be said at this point is that the Rockville formation is a mystery waiting to be solved.