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The sometimes 'Beauty' of mathematics???

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I don't know how many people have seen my 'Tag' (?) thing, at the bottom of my posts/replies?   :P
   Diagonal of 1x1 square = Root-2. Ok.
   Diagonal of 1x1x1 cube = Root-3 !!!  Beautiful !!

Doesn't that not strike people as mathematically amazing! that the internal diagonal of a 'Cube' (1x1x1) = Root-3 !!!
I love the natural relationships with certain numbers, that can easily be proven/shown, and be so simple!   8)
GREAT!!  Good-ol basic maths can show/prove it all !!!...

Then I found out recently, that there is NO KNOWN correct formula, to calculate the Circumference of an 'Ellipse' !!   :palm:
You don't believe it... then look it up!!  Sigh...   Just lost faith in Maths again...   :box:

The "elliptic integrals" that appear when calculating the circumference of an ellipse are not "elementary functions" such as combinations of sines and cosines, but are tabulated functions that can be found in math-table books, entered into the books from numerical computations.  There are other non-elementary functions that occur in science and engineering, such as Bessel functions, that are solutions of practical math problems and are also tabulated in such books.
That being said, in school where a problem could be reduced to elliptic integrals, we left it at that.  One such problem I remember (senior undergraduate classical mechanics) was how long does it take for a car door to slam shut if it is open at the start of the car's acceleration.

Euler's identity shows how five fundamental numbers are related in a single equation: \(  e^{i \pi} + 1 = 0 \)

If you like this sort of thing, then consider a string wrapped around the earth's equator (assuming the earth as a perfect sphere) and tied off at the ends. Now raise the string above the surface by some small amount all the way around, say 1 meter. How much longer is the string when raised and tied at the ends? Do same for the Moon, Sun, Jupiter, the solar system and the entire universe assuming all are spheres.  Then redo for a beachball, basketball, baseball, pingpong ball, solder ball and an atom (assume all are spheres). Now compare the results, amazing indeed :-+

If you like mathematics in electronics, look into analog filters and how the transfer functions of Butterworth, Bessel, Cauer, Chebyshev, Gauss,  & Legendre interact in the frequency and time domains. Fun stuff indeed for those interested :-+


Edit: Signal processing, both analog and digital, is another fascinating mathematical area :)

For delightful visualisations of mathematical concepts see:


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