Cross Checked Rows
California grows extensive row crops (grapes, peaches, almonds) on large plots of level land. The rows and columns of fruit and nut plantings are planted and pruned to very even tolerances to enable mechanized harvesting, trellising, and even irrigation (notice the drip hose also suspended by the trellis). With cross-checked rows one notices many different diagonals. If the space between rows and columns are equal, the most obvious diagonal is 45 degrees - the pathway leading away from the camera in the right center. The diagonals are even given names for their application in crystallography - the three-dimensional structure of atoms arranged in a regular lattice. This main 45 degree diagonal is called the (1,1) direction because the diagonal is formed by "sideways one unit, depthways one unit". The diagonal in the center of the image is the (2,1) diagonal (sideways two, depthways one). The next diagonals in the series are (3,1), (4,1), etc. In between some of these (n,1) series of diagonals are higher order diagonals such as (3,2) or (5, 3). If the size of the field of the row crop were infinite, and the plants or posts were perfectly straight and narrow, there would be no limit to the number of diagonals visible. One sees the same effect in empty theater seats, military grave plots, or any regular array of objects.
What is amazing about viewing the cross checked rows viewed from a moving car window (at 60 miles/hour) is that although the individual plants or posts are a blur, an infinite array of diagonals are still visible in the second image, and even more pronounced. This blurred image was taken from a long-enough time exposure such that all the individual trunks of this nut orchard are just a blur, yet many diagonal paths are visible - similar to interference of waves creating standing patterns as the waves are moving. No matter where the camera is positioned, the camera will always see the (3,2) diagonal in the same direction. One series of diagonals has been labeled beginning with (2,1) on the left and progressing with (n, n+1) series which eventually becomes the (1,1) diagonal - the convergence of the series for very large n. The identity of the diagonals was verified by carefully measuring the position angles of these diagonals within the digital image with geometry. Of course many more series are visible in the blurred image.
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