# Networks .. and those who made them possible

### The history of communication on Earth is extensive. Every living thing has means of communication. Trees, insects, molluscs, the entirety of the plant and animal kingdoms all communicating between and within individuals and communities of organisms, as they, and the cells which compose them, organise to feed, reproduce and defend themselves.

#### In an Euler Cycle, the Path ends where the Path began. In a Hamilton Cycle, the Path, likewise, ends where it began, such that the initial and final vertices are identical (the only allowable repeated vertices in a Hamilton Cycle or Path)

##### Notice that in Euler Paths we are looking at crossing (following) edges. In Hamilton Paths it’s vertices we are looking at.  #### Related to the Bridges of Kӧnigsberg problem in network theory, and often attributed to William Rowan Hamilton: Hamiltonian paths and cycles are named after the man who invented the icosian game, now also known as Hamilton’s puzzle, which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Hamilton). This solution does not generalize to arbitrary graphs.

##### Light Waves, Radio Waves, Photons, Black Bodies, Planck’s Experiment, Einstein’s ‘thought experiments’, Electrons, Mass, Energy, Bosons and Gravitation
###### E & H are examples of  dependent “Vector Fields”. At every point in a vector field there exists a force on a body with magnitude and direction ie a vector. Fields of temperature, mass or energy values are examples, on the other hand, of scalar fields. Multi-dimensional Tensor fields exist in physics (eg Mechanical Stress in solids and viscous fluids, and space-time curvature Tensors in General Relativity).

.. Just for clarity, Special Relativity (1905) is concerned with reconciling physics to the space-time transformation ‘metric’ revealed in Hertz’s Experiment’s verification of the existence of radio waves, but predictable from Maxwell’s equations before him. A ‘metric’ here refers to finding a formulaic way we may consistently model relative velocities, and other physical properties, between 2 observers traveling separately in space-time, in “light” of the fact that there is really nowhere to be taken as a zero velocity point and that we may only consistently measure velocities relative to the local velocity of light (radio waves equally). Einstein discovered the metric inherent in Maxwell’s equations, relating to electromagnetism, and argued that there can be only one metric in this universe, and that therefore Newton, a man who apparently walked out of his first opera, was wrong (in 1687) to assume a Euclidean ‘orthogonal’ space for physical reality, which naturally seemed to separate time from space. The metric of Maxwell’s equations revealed skewed axes in 4 dimensions where the upper limit to universal velocities is the speed of light. By progressing from this point, Einstein was able to unite our concepts of Electricity and Magnetism as well as uniting Energy and Mass.  Our concepts of Energy and Mass were unified by Einstein applying the Conservation of Energy Principle followed by the Conservation of Momentum Principle (in special relativistic formats) to a collision (in a thought-experiment) where a so-called Black Body absorbs a colliding photon fully. By equating the total Energy and Momentum prior to the collision with the total Energy and Momentum after, and utilising the Planck Relation for Energy of a photon E = hf, Albert was able to show with simple algebra and calculus that the total Energy of that system post collision E = mc2, where m is the mass of the black body (the photon has zero mass itself) and c is the velocity of light.

Planck’s experiment some few years after Hertz’s experiment, revealed to Einstein an interpretation of results stating light (and thus radio waves) is actually composed of light “particles” or quanta called photons. Albert deduced this publicly in a paper in 1904, giving birth to a field of research still continuing today, called quantum mechanics. Due to the incompatibility of the Classical Wave based theory of light with a particle based theory at the time, the effort to find the linking equations between quantum mechanics and the classical theory (adequate until Planck’s Experiment) led scientists to a probabilistic theory which Einstein always disowned. Incidentally Erwin Schrödinger also believed that there exists a deterministic underlying continuous theory possible in physics. The possibility that many events could happen simultaneously is required for the theory to work.

The answer in any case seems to lie in the success Schrӧdinger had in 1926, (and Werner Heisenberg at the same time), with an approach that replaced the value for total field Energy (E) with hf in a single frequency (laser) light field’s theoretical “Work Function”. The basic experimental relation E = hf (Energy of a photon = its frequency f multiplied by Plank’s constant h) is reliable and verifiable by anyone who wants to repeat Planck’s Experiment. The result of Schrӧdinger’s substitution in the field equation had to be integrated to uncover the general “Quantum Mechanical” version of the Wave Equation, which as required, approaches the behaviour of the Classical Wave Equation, as the value of Planck’s Constant is made to approach zero. This requirement is a way of mimicking the idea that Energy in a light field be taken as independent of frequency, in the classical limit. The Classical Wave Equation also applies to Sound waves in the limit, and thus there exist “Phonons” or stress/pressure particles, since the Quantum Mechanical Wave Equation also applies more accurately to sound/pressure/stress waves, including both types of seismic wave the shear waveform or aftershock, and the direct wave or primary shock, in similarity with the sonic boom and aftershock with supersonic objects (the shear wave arrives last). These waves are all carried in the final result by phonons in rays spreading out from the source(s).

The theoretical Schrӧdinger and Heisenberg treatments also matched predictions from practical absorption spectra experiments with Hydrogen, in a model of the Hydrogen Atom, with its single electron as a wave-particle, obeying the new Wave Equation, absorbing energy in stable quantum stages as predicted. The wavelike nature of electron beams themselves was experimentally established when Electron diffraction, in fact, was observed (1927) by C.J. Davisson and L.H. Germer in New York and by G.P. Thomson in Aberdeen, Scot. , thus supporting an underlying principle of quantum mechanics, “Wave Particle Duality”.

The Theory of General Relativity (1915) went further than Einstein’s Special Theory, as Special Relativity still rested upon a ‘flat’ space-time cosmology, whereas the General Theory concerned itself with further revelations, now about Gravitation; specifically, that it is caused by the local magnitude and direction of curvature of space-time in a “hyper-volume” (possibly a multiverse) of a larger number of dimensions (larger than 4, but otherwise unspecified).  This actual curvature of space-time is caused by the presence of matter (such as the earth or the sun or a pencil), and Einstein gave equations which accurately predict the behaviour of our solar system as well as real galaxies, contrary to Newton’s inconsistent predictions.  (Although, Einstein was never good with pencils .. they never weigh enough and they move too slowly – moreover it was Newton’s Mechanics that enabled the development of the fundamental Impulse/Momentum Equation of Rocketry and the Finite Element Method of Stress and Strain analysis, for the Apollo rocketships, that took men to the moon successfully, and any navigator since well before Newton would have been happy to plot the course – as it was done in Euclidean space).

However, much has happened in physics since publication of the General Theory of Relativity in 1915 ..  starting with quantum mechanics, Schrӧdinger, Heisenberg, Dirac and Stephen Hawking’s life devoted to reaching past Einstein into Black Holes .. go search .. and remember that although God may not play dice, people may be required to, in physics, because the human intellect needs a way to comprehend wave-particle duality, and many other probabilistic phenomena, such as the question of how is the particle, the Higgs boson, which confers “mass” on some objects (although not all), related to the universal curvature-of-space-time tensor in a generally-relativistic quantum mechanics, or, how would a Higgs boson be related to Gravitons, as emitted by pulsars?

###### The following link is also a wikipedia article. History Of The Internet. (wiki) 