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Beautiful Equations

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  • #91
    Sitting at home waiting for the world to re-open and/or to be called back to work, I was re-visiting some old threads I enjoyed. This is one of them. I think the first two posts are a good summary of thread - Nick the Noodle's post on the Euler identity, and Colonel Sennef's post on The Pythagorean theorem. I still feel the Pythagorean theorem is the best and most interesting (post #38), as it clearly relates geometry and (some) basic arithmetic operations. When I taught middle school math, I would at one or two points during the year put up the visual proof of the Pythagorean Theorem. The thing that makes it great is the simplicity of it, and the fact the proofs and postulates don't require a higher-level knowledge than what is being proved. This involves knowing about areas, multiplication, and raising a number to a power. In fact, when I discussed it, I was trying to teach geometry, and this is a good way to work in dimensions and exponents. A few students would understand it right away, but most got something out of it. I'm sure I didn't really get the proof the first time I saw it.

    Whereas the Euler identity, it requires a knowledge of trigonometry, complex numbers, limits, angle conventions, etc. I can see that a real mathematician would find it interesting, but as a former engineer I find the simpler Pythagorean Theorem more elegant - and easier to understand.

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    • #92
      A link to an article I found interesting - 10 visual mathematical proofs.

      https://www.zmescience.com/science/m...nd-simplicity/

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      • #93
        Maybe not a beautiful equation, but a relevant equation in the age of the virus: The Logistic Funtion. Essentially it starts from zero, and goes up/approaches 1, where it cannot go higher. The epidemiological curve is related - starts off exponential-like, then linear-like, then "flattening". The actual logistic can have negative values, which obviously doesn't apply with the number of people with diseases, but the curve can be modified to start at zero, and of course can have coefficients on the variables. Also of course, in real life people are born, die, etc. The generalized logistic function is known as Richards' Curve.

        https://en.wikipedia.org/wiki/Logistic_function

        https://en.wikipedia.org/wiki/Genera...istic_function

        ================================================== ========

        The logistic function



        One example of Richards' Curve (With a particular set of parameters

        )

        ================================================== ===

        excerpt from "The logistic function" article

        In medicine: modeling of a pandemic[edit]


        A novel infectious pathogen to which a population has no immunity may spread exponentially in the early stages, while the supply of susceptibles is plentiful. The SARS-CoV-2 virus that causes COVID-19 exhibited exponential growth early in the course of infection in several countries in spring 2020.[18] Many factors, ranging from lack of susceptibles (either through the continued spread of infection until it passes the threshold for herd immunity or reduction in the accessibility of susceptibles through physical distancing measures), exponential-looking epidemic curves may first linearize (replicating the "logarithmic" to "logistic" transition first noted by Pierre-François Verhulst, as noted above) and then reach a maximal limit.
        Last edited by lakechampainer; 15 May 20, 11:47.

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