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The law of the prolonged sine was observed when measuring strength of the reaction of the plant stems and roots in response to turning from their usual vertical orientation. Such organisms maintained their usual vertical growth, and, if turned, start bending back toward the vertical. The prolonged sine law was observed when measuring the dependence of the bending speed from the angle of reorientation.
The observed law
It was observed that deviation form the desired growth direction by more than the 90 degrees causes further increase of the bending speed. After turning the 135 degrees the reoriented plant or fungi understands that it is places "head down" and bends faster than turned by just 45 degrees.
Following the popular hypothesis of the mechanism of the plant spatial orientation, the bending from the horizontal position is caused by some small heavy particles that after turning put the pressure on the side cell wall, irritating some system and activating the bending process. The pressure of such particle to the cell was would be proportional to the sine of the reorientation angle and would be equal for the reorientations by both 45 and 135 degrees.