Topological Examples in the Real World

In this section, a list is displayed of a number of physical effects which - to first order - do not depend explicity on size or shape. Therefore, these phenomena must have a topological basis for their explanation. Key topological properties are the number of parts (connectivity), the dimension, and orientability. Concepts of size and shape, metric, or connection can not enter into the explanation of such phenomena, except in an auxiliary way..

  • Planck's Radiation Formula.
    The distribution law is independent from the size and shape of the hot body.

  • Coaxial Wave-Guide Propagation.
    A hollow wave guide is a high-pass filter, but a non-simply connected co-axial cable can pass DC.

  • Chaos does not occur in Dimension 2.
    A system of ODE's has a PDE equivalent. If the PDE satisfies the Frobenius integrability theorem, according to Darboux there is always a representation in terms of 2 functions.

  • Bohm-Aharanov effect.
    Flux quanta in superconductors come in integer multiples of h/2e.

  • Gauss Law.
    The integral of the D over a bounding surface only depends upon the number, e, of
    electrons in the interior of the closed surface, no matter what the size or shape of the surface may be.

  • Topological Torsion and Spin.
    The Bohm-Aharonov effect is a 1-D period integral. Gauss Law is a 2-D period integral. There are two 3-D period integrals that have been little studied in science, but appear to have significance in hydrodynamics and plasmas as robust coherent vortex-like helical structures - insensitive to deformations in space time.

  • Quantum Transition Probability.
    Fermi's Golden Rule demonstrates that the
    transition probability is a cross-ratio projective invariant, independent from scales.

  • Thermodynamic Irreversibility.
    The idea that thermodynamic irreversibility is defined
    by the failure of the Frobenius Theorem for the 1 form of work implies that the topological dimension of the Action 1-form must be 4. In other words, thermodynamic irreversibility is an artifact of four dimensions.

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