The Ridgeway, nr Avebury, Wiltshire, UK. Reported 6th July. Map Ref:  This Page has been accessed Updated Saturday 27th  July  2013

 AERIAL SHOTS GROUND SHOTS DIAGRAMS FIELD REPORTS COMMENTS ARTICLES 07/07/13 07/07/13 11/07/13 07/07/13 20/07/13 27/07/13

 Circles as a Result of Solving Laplace's Equation in a Cylindrical Coordinate System  We believe the crop formation reported on 6th July 2013 resembles a solution to Laplace's equation in a cylindrical coordinate system. The Laplace's equation describes many physical phenomena, for example, an electric field, heat field or gravitational field. The Laplace's equation is a partial differential equation written as    On the left side of the equation we can see the Laplace operator (delta) applied to a scalar function (phi). The Laplace operator in a cylindrical coordinate system is given by    The crop formation represents a special solution dependent on the coordinate r only. Therefore, to get the solution, only the first part of this equation is applied. We obtain an equation    An analytical solution to this equation is    The constants A and B are determined by Dirichlet boundary conditions, i.e. by the known values of a scalar function on the boundaries of the solved area.  The following example shows the distribution of the electric potential inside a cylindrical capacitor (Fig. 1). The inner electrode of the cylindrical capacitor has the electric potential 8 volts and the outer electrode is grounded, i.e. the electric potential is zero volts. The radius of the inner and outer electrode is set to be equal to the crop formation. The information about how to set the ratio of these radii can be found in the centre of the crop formation (see Fig. 2). The distribution of the electric potential is solved by Laplace's equation.  Fig. 1: The cross-section of a cylindrical capacitor  Fig. 2: The information about how to set the ratio of the radii  Fig. 3 and Fig. 4 show the distribution of the electric potential in a cross-section of the cylindrical capacitor in two versions: surface and contour. Contours separate the values ​​of the electric potential by 1 volt. The computations were done by COMSOL Multiphysics.  Fig. 3: The electric potential in a cross-section of the cylindrical capacitor  Fig. 4: The electric potential in a cross-section of the cylindrical capacitor – in contours  Fig. 5: The comparison of the crop formation with the obtained results  As you can see in Fig. 5, the distances between the individual circles of the crop formation are the same as in the solution to Laplace's equation in a cylindrical coordinate system depicted in Fig. 4. As the aerial shot was not taken perpendicularly to the crop formation, the circles of the crop formation do not correspond to the circles of the solution to Laplace's equation in the upper and lower part of Fig. 5.  It seems that the creators of this crop formation try to remind us of the beauty of mathematics again. They have already done so through pictograms about Ludolph's number, golden ratio and fractals.  Jitka & Roman Hamar, The Czech Republic, 20/07/2013

Discuss this circle on our Facebook

 A precise mathematical description of the nine outer rings at Ridgeway, in terms of a linear geometric series and “collapsed space”  There has been considerable discussion as to what the Ridgeway crop picture might signify. Some workers have suggested “music” from a diatonic scale, others have suggested the “electrical potential of a cylinder” (see below), while others have suggested a “spacetime wormhole”. If we study this problem in a quantitative sense, we can see right away that the Ridgeway crop picture does not represent “music”, because the numbers for that theory do not match the actual numbers seen in crops:  Still we have not provided a precise mathematical description of those nine outer rings at Ridgeway, by which to test any possible theory. In their direct unaltered form, those radii do not even fit well to a linear geometric series, as noted in the slide above!   It turns out that we can fit those nine rings to a linear geometric series, if we “expand” the centre of the crop picture by 3 x Rc, where Rc is the radius of its small central circle. Now the measured numbers fit well to a geometric series, which keeps the same base value of 1.13 when going in to out:   How might we interpret this result in visual terms? If Rc = Rs for the radius of a Schwarzschild wormhole, then 3 x Rs would be the radius of its smallest stable orbit. Hence space seems to have “collapsed inward” at the centre of Ridgeway by that amount, just like “falling down into a wormhole”. Such space needs to be “expanded outward” again, in order for its outer region of nine rings to return to normal. This was the mathematical basis for a very clever visual illusion, designed somewhere by unknown crop artists. Many thanks to L.H. and C.M. for help.   Red Collie (Dr. Horace R. Drew)

FOR VISITING THE CROP CIRCLES.

JOIN THE CROP CIRCLE CONNECTOR MEMBERSHIP NOW ONLINE

BACK

Mark Fussell & Stuart Dike