geometric infinity

[AntonioLao]

Three acceptable geometries of space-time are Euclidean, Riemannian, and Lobachevskian. Although Einstein’s general theory of relativity is based on Riemannian geometry of positive curvature, it is still undecided whether it is true in general for both macrocosmic and microcosmic structures and superstructures. In the quantum domain of quantum mechanics and quantum field theories the descriptive effectiveness of Lobachevskian hyperbolic geometry of negative curvature dominates that of Riemannian spherical geometry of positive curvature as exemplified by the Heisenberg uncertainty principle and the Lagrangian energy function, contrasting the Hamiltonian of general relativity. The formers use differentiation while the latter uses integration.

However, it is mathematically possible to embed geometries among each other. For example, one embeds the rational Euclidean geometry within the irrational transcendental geometry whenever a regular polygon of finite number of sides is inscribed within a circle. As the number of sides of the regular polygon approaches infinity and simultaneously each side length approaches zero the measure of its perimeter approaches the measure of the circumference of the circle, which is used to define the value of �� as the ratio of the circumference over the diameter. Conversely, it is doable to inscribe a circle within a given arbitrary polygon of finite number of sides. Likewise, this implies embedding different forms of geometric structures sharing common points of contact. If these processes of inscribing and circumscribing are carried-out indefinitely then the outcome describes an approach toward geometric infinity.

[mkirkpatrick]

Three acceptable geometries of space-time are Euclidean, Riemannian, and Lobachevskian. Although Einstein’s general theory of relativity is based on Riemannian geometry of positive curvature, it is still undecided whether it is true in general for both macrocosmic and microcosmic structures and superstructures. In the quantum domain of quantum mechanics and quantum field theories the descriptive effectiveness of Lobachevskian hyperbolic geometry of negative curvature dominates that of Riemannian spherical geometry of positive curvature as exemplified by the Heisenberg uncertainty principle and the Lagrangian energy function, contrasting the Hamiltonian of general relativity. The formers use differentiation while the latter uses integration.

However, it is mathematically possible to embed geometries among each other. For example, one embeds the rational Euclidean geometry within the irrational transcendental geometry whenever a regular polygon of finite number of sides is inscribed within a circle. As the number of sides of the regular polygon approaches infinity and simultaneously each side length approaches zero the measure of its perimeter approaches the measure of the circumference of the circle, which is used to define the value of �� as the ratio of the circumference over the diameter. Conversely, it is doable to inscribe a circle within a given arbitrary polygon of finite number of sides. Likewise, this implies embedding different forms of geometric structures sharing common points of contact. If these processes of inscribing and circumscribing are carried-out indefinitely then the outcome describes an approach toward geometric infinity.

You somewhat lost me here Antonio,I need it a wee bit simpler for my tiny brain.

regards michael.

[AntonioLao]

I'm afraid I dont know how to make it much simpler except for the analogy that the concept of infinite geometries is like the infinite layers of an onion that each layer is peeled away with lot of tears of happy understanding. Unfortunately, no one can peel an infinitely layered onion forever. Moreover, realistically, an infinitely layered edible onion does not exist. But an infinitely layered space-time onion can theoretically exist as defined by the axiom of infinity.

[mkirkpatrick]

I'm afraid I dont know how to make it much simpler except for the analogy that the concept of infinite geometries is like the infinite layers of an onion that each layer is peeled away with lot of tears of happy understanding.

Ah! Now that I do understand,thank you for that.

regards michael.

[AntonioLao]

You are welcome. My question still remains whether infinity can now be defined as a number? Furthermore, I am asserting that it is an even number.

[mkirkpatrick]

You are welcome. My question still remains whether infinity can now be defined as a number? Furthermore, I am asserting that it is an even number.

Yes I think it can by the number 10 expressed with a circle containing the number 1,and ten is an even number!

regards michael.

[AntonioLao]

But 10 is not the largest even number as what infinity supposes. In the binary system 10 equals 2 of the decimal system. The way to prove it is first proving the largest twin prime such that 6N-1 is the lesser prime and 6N+1 is the larger prime and 6N equals infinity as N approaches infinity.

[mkirkpatrick]

But 10 is not the largest even number as what infinity supposes. In the binary system 10 equals 2 of the decimal system. The way to prove it is first proving the largest twin prime such that 6N-1 is the lesser prime and 6N+1 is the larger prime and 6N equals infinity as N approaches infinity.

Thanks for that answer Antonio,its just that some eastern teachings indicate that one with a circle around it represents the eternal motion of evolution,so I thought that perhaps this was it.

regards michael.

[AntonioLao]

Thanks for clarifying your idea about the power of symbols. Like written languages, we still communicate much with symbols, including numerical symbols. It is agreed by Western culture, especially ones using the English language that the symbol for infinity is an horizontal figure of sleeping eight. A vertical awaking figure eight has been used to represent the eight-fold way of Eastern mysticism as a circle above another circle. For me, these two configurations represent only two properties of directional invariance. The third property can be symbolized by a circle behind a circle or a circle in front of a circle. If they are of equal size then there is no way for us to make the distinction.

[mkirkpatrick]

Thanks for clarifying your idea about the power of symbols. Like written languages, we still communicate much with symbols, including numerical symbols. It is agreed by Western culture, especially ones using the English language that the symbol for infinity is an horizontal figure of sleeping eight. A vertical awaking figure eight has been used to represent the eight-fold way of Eastern mysticism as a circle above another circle. For me, these two configurations represent only two properties of directional invariance. The third property can be symbolized by a circle behind a circle or a circle in front of a circle. If they are of equal size then there is no way for us to make the distinction.

Thanks for clearing that up Antonio,what can embrace infinity? Consciousness perhaps?

regards michael.