[mkirkpatrick]

Originally Posted by AntonioLao

The area of a rectangle is the product of its length (L) and its width (W). However, the largest area of a quadrilateral inscribed in a circle is a square or the same as a rectangle with length equal to width, L=W. If the circumscribed circle has unit radius then the area of the inscribed square is 2 square units and the sides are irrationals square root of 2. if an inscribed rectangle has unit width then the length is again an irrational square root of 3 whose area is also square root of 3 square units. Without further proofs, it is logical to assert that no rational quadrilaterals can be inscribed in a circle. However, for general polygons, it can be demonstrated that a regular rational hexagon of unit sides can be inscribed in a unit circle.

In nature, this distinctive pattern can be observed in the structure of snowflakes or snow crystals. If it can be experimentally measured that the sizes of these hexagonal structures are consistently of the same size then it proves the existence of global symmetry for earthbound unit circles. On the other hand, if these structures are of different sizes then global invariant symmetry does not exist even though local invariant symmetry does.

If my memory serves me well I believe snowflakes are all different in shape,although
there does appear a certain symmetry with them.

regards michael.