rational polygons

[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.

[mkirkpatrick]

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.

[AntonioLao]

snowflakes are all different in shape,although there does appear a certain symmetry with them.

You are correct. It is the hexagonal symmetry I'm interested in.

[mkirkpatrick]

You are correct. It is the hexagonal symmetry I'm interested in.

What will this tell you?



regards michael.

[AntonioLao]

What will this tell you

the existence of local as well as global gauge symmetry independent of length.

[mkirkpatrick]

the existence of local as well as global gauge symmetry independent of length.

Surely we are surrounded by perfect symmetry,albeit mostly invisible,local and otherwise.


regards michael.

[AntonioLao]

mostly invisible

In mathematics, if a transformation does not exist between two domains then they are conversely invisible to each other.

[mkirkpatrick]

In mathematics, if a transformation does not exist between two domains then they are conversely invisible to each other.

Can they then still effect on another?



regards michael.

[AntonioLao]

Can they then still effect on another

That could be the quantum nonlocality that many are talking about. I' afraid I dont understand it. My idea is based on the existence of unit circle of gauge invariance, in fact two circles linked together into a Hopf ring. Transformations exist and they commute between zero and infinity but not taking the exact values of zero and infinity.

[mkirkpatrick]

That could be the quantum nonlocality that many are talking about. I' afraid I dont understand it. My idea is based on the existence of unit circle of gauge invariance, in fact two circles linked together into a Hopf ring. Transformations exist and they commute between zero and infinity but not taking the exact values of zero and infinity.

I don't fully understand it either,but it "feels" right!



regards michael.