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Three points always makes a plane and a fixed plane. So, the end points of the three legs make a plane. This is a result of the postulate that the minimum number of points to determine a plane is three, Or the postulate that two intersecting straight lines are sufficient to determine a plane.To fix the horizontal plane for the legs is easy with three legs. The monopad may be shaky but more easily manageble. More than three legs is too redundant and still may not provide the directional precision , due to slight differences in the legs and any force may distort the directional precision. So why take more risk with unnecessary additional legs.
It is not clear which postulates you are talking about. But the reason for using a three legged stand rather than any other number by photographer is that a three legged stand provides a stable support for the camera. In terms of geometry this can be explained by the fact (or postulate) that any three points define a plane completely and uniquely.
A stand with only two legs will be unstable. And a stand with four legs is likely to have only three legs touching the ground, while the fourth leg slightly raised above the ground. When such stand is tilted to the side of the leg not touching the ground, that leg will touch the ground but some other leg will be raised. Thus the stand with four legs will tend to rock or move as the photographer is trying to work. This of course will make the stand unfit to provide a steady, unmoving support to the camera. At best a four legged stand will provide stable support with only three of its legs with the fourth leg providing no support.
It is worthwhile noting that a five legged stand will also provide a stable support. This will be because any three legs of the five leg stand will be touching the ground. This way only three of the five legs will be used. There for a five legged stand will only increase the weight and cost of the stand without providing any additional benefit. Also for best support the three legs supporting the stand should form a near equilateral triangle. This may not always be the case with the three supporting legs of stand with more than three legs.
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