The sometimes 'Beauty' of mathematics???

[GlennSprigg]

I don't know how many people have seen my 'Tag' (?) thing, at the bottom of my posts/replies?   
   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!   
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' !!   
You don't believe it... then look it up!!  Sigh...   Just lost faith in Maths again...   

[TimFox]

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.

[tggzzz]

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

[mawyatt]

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

Best,

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

[daqq]

For delightful visualisations of mathematical concepts see:
https://www.youtube.com/c/3blue1brown/videos
https://www.youtube.com/c/PrimerLearning/videos

[Brumby]

If you like this sort of thing, then consider a string wrapped around the earth's equator (assuming the earth as a perfect sphere)

That constraint is more limiting than the problem requires.  You simply need to assume the earth's equator is a perfect circle.  The same applies to the other bodies.


The question, however, is a simple one - and some very basic algebra will yield the answer.   

[mawyatt]

If you like this sort of thing, then consider a string wrapped around the earth's equator (assuming the earth as a perfect sphere)

That constraint is more limiting than the problem requires.  You simply need to assume the earth's equator is a perfect circle.  The same applies to the other bodies.


The question, however, is a simple one - and some very basic algebra will yield the answer.   

Yes, but 3D bodies seem more useful in visualization 

And yes the answer is very simple, but astounding 

Best,

[magic]

I have heard a different version:

Imagine a circle lying on the Earth's equator.
Then increase its circumference by 1 meter.
Is it enough for a mouse to pass under it?

[TimFox]

Another story from my school days, roughly 50 years ago.  The senior (and very distinguished) professor in the mathematics department “crossed the aisle” to the physics department colloquium to tell us about recent work in mathematics that might be of interest to us, including work on generalizing the concept of “function”.
He started with an anecdote about his technical work during WW II, where his security clearance allowed him to read classified technical reports.  One report computed a function relevant to aircraft design and tabulated it as y = f(x).  The tabulated values were all close to 1.  He and a buddy went back to the original equations and found they could be reduced to f(x) = sin²(x) + cos²(x).
This is an example of mathematical beauty:  even though you need recourse to tabulated values for trig functions of arbitrary values, it is straightforward to prove relations between them (what we call “trig identities”) with (in this case) Euclidean geometry.  Draw a circle with radius equal to the hypotenuse of a right triangle and apply the Pythagorean theorem to the other two sides that represent sine and cosine.

[armandine2]

sometimes the numbers seem to be well chosen - almost like someone was trying to get you to remember the formula!

[rstofer]

Let's say a person has an BS is Applied Mathematics and an MS in Mathematics.  Where do they find a job?  What kind of industries hire theoretical mathematicians?

[TimFox]

sometimes the numbers seem to be well chosen - almost like someone was trying to get you to remember the formula!

In primary-grade arithmetic (say, fourth grade), the numbers in the homework problems always "came out even", so that the clever students knew that it was probably wrong when the process did not come out even.

[mawyatt]

Let's say a person has an BS is Applied Mathematics and an MS in Mathematics.  Where do they find a job?  What kind of industries hire theoretical mathematicians?

We had a whole staff of mathematicians including my colleagues wife a Fulbright Scholar, she studied at a French university (I can't remember the details . They were mostly working on Kalman filters for SOTA Inertial Nav Systems based upon ESG and RLG Gyros, but this was ~45 years ago. Also had a colleague professor when I was an adjunct that studied under Nash at Princeton, he was brilliant and authored a grad level book on Complex Variables!! However, today I'm not sure where one would look outside a university for work specifically in advanced mathematics.

Best,

[mawyatt]

sometimes the numbers seem to be well chosen - almost like someone was trying to get you to remember the formula!

In primary-grade arithmetic (say, fourth grade), the numbers in the homework problems always "came out even", so that the clever students knew that it was probably wrong when the process did not come out even.

Recall from undergrad calculus a test problem with integration by parts, you kept coming up with the same integral over and over that you started with, and another term. The clever student, not me  realized the answer was the other term divided by 2

Best,

[TimFox]

I’m retired, and not an authority on the current job market, but at my employer we had high-ranking employees with advanced degrees in mathematics working on image processing and similar topics.

[daqq]

Let's say a person has an BS is Applied Mathematics and an MS in Mathematics.  Where do they find a job?  What kind of industries hire theoretical mathematicians?

It depends - if you are actually good there's any number of serious engineering companies, whether they are software, hardware, civil engineering... There are real world applications even (or maybe especially) for the really advanced stuff.

If you are not particularly good, no idea.

[rstofer]

I’m retired, and not an authority on the current job market, but at my employer we had high-ranking employees with advanced degrees in mathematics working on image processing and similar topics.

I would think the hot topics are cryptology or machine learning and other branches of AI.  I know Google is into AI bigtime and so is Microsoft plus, perhaps, government agencies.  In addition to math skills, I suspect programming skills are absolutely required.  MATLAB, Octave, wxMaxima, Python (and all of the libraries, especially AI libraries) and so on.  I'm really old so I would throw in a dash of Fortran as well.

But I'm just guessing...

[TimFox]

By the time I retired, the younger employees doing the analytical work mainly used Matlab, but I found it easier to refer the relevant problems to them rather than learn it myself.
The nice thing about Matlab is that my employer could purchase a license (for a quite reasonable price) that allowed our guys to develop "stand-alone" applications that could be used internally by the non-Matlab folks and even sold to our customers.

[EPAIII]

You would be surprised.

Start with the folks who drill for oil and gas.

Let's say a person has an BS is Applied Mathematics and an MS in Mathematics.  Where do they find a job?  What kind of industries hire theoretical mathematicians?

[Brumby]

I have heard a different version:

Imagine a circle lying on the Earth's equator.
Then increase its circumference by 1 meter.
Is it enough for a mouse to pass under it?

I like that variation.   

[rstofer]

I like "Gabriel's Horn" better.  Consider the curve 1/x for x between 1 and infinity - it's a really long horn...

Compute the volume of revolution and you will find that it takes PI units of paint to fill it completely.

Then compute the surface area and you will find that it takes an infinite amount of paint to cover the interior surface.

That's just weird!



Hints:

Volume = integral from 1 to infinity of pi * radius squared dx but r = 1/x so integral of pi/x² dx = pi

Surface Area = integral from 1 to infinity of 2 * pi * r which is the integral of 2 * pi / x dx.  Remember, integral of 1/x = ln(x) and the result diverges (runs off to infinity)

[magic]

You may run out of paint on the exterior, but the interior cannot be painted with a layer of any thickness because most of it is too narrow

If you want to see some seriously fucked up consequences of playing with all that "infinity" stuff, look up the Banach-Tarski paradox
(not going to pretend I know the proof of that)

[TimFox]

Another similar paradox applied to natural objects is to compute the coastline of Norway.  Obviously, the area of Norway (with mountains projected back onto the globe) is finite, since you can draw a closed curve around it, of finite area, that encloses it.  If you do the actual experiment of computing the physical coastline as you go to finer and finer map scales, the value (in meters) increases without bound.  (Please don't quibble about atomic dimensions.  This is math.)
An ideal geometric figure that does the same:  the Koch snowflake, with finite area but infinite perimeter.
Start with an equilateral triangle, of side 1.  Total perimeter = 3.
Iteration 1, on each of the three sides of that triangle erect another equilateral triangle whose side is 1/3 of the first.  This gives you a Star of David hexagram.  Total perimeter = 4
Iteration 2, on each line segment (12 for this step) of that figure erect another triangle of length 1/3.
etc.
There exists a circle that encloses all of the resulting figures, but the perimeter increases without bound.  The length after N iterations is 3 x (4/3)^N
The asymptotic area (after an infinite number of iterations) is (8/5) x the area of the original triangle.
(I read this discussion many years ago in Mandelbrot's popular introduction to fractal geometry.)

[rstofer]

You may run out of paint on the exterior, but the interior cannot be painted with a layer of any thickness because most of it is too narrow

If I can have quantifiably small internal diameters, I should be able to have really tiny paint molecules.

My math skills are too weak for either of the last two paradoxes.  I just try to balance my checkbook and that isn't easy as it sometimes requires imaginary numbers.

[TimFox]

The interior surface area might be infinite, but the volume of paint required to paint the interior is finite due to decreasing clearance, even if the thickness goes to subatomic dimensions.  The volume of paint must be less than the volume of the interior (equal to pi according to the post, I haven't checked), which must be less than an external volume that contains the horn.
Paint at Home Depot is sold by volume.