Author Topic: The sometimes 'Beauty' of mathematics???  (Read 5315 times)

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Offline The Electrician

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Re: The sometimes 'Beauty' of mathematics???
« Reply #25 on: August 05, 2021, 11:17:07 pm »
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:

Of course there is a correct formula.  The relevant observation is found on this page: http://www.numericana.com/answer/ellipse.htm#elliptic

"There is no simple exact formula:  There are simple formulas but they are not exact, and there are exact formulas but they are not simple."

Also here: https://wj32.org/wp/2012/12/15/formula-for-the-circumference-of-an-ellipse/
 

Offline Circlotron

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Re: The sometimes 'Beauty' of mathematics???
« Reply #26 on: August 05, 2021, 11:21:52 pm »
The one I like is phi, 1.6180339.....
In particular
phi^2=phi+1
1/phi=phi-1
 

Offline vk6zgo

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Re: The sometimes 'Beauty' of mathematics???
« Reply #27 on: August 05, 2021, 11:56:40 pm »
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?

Q: What does a maths graduate say?

A: "You want fries with that?"
 
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Offline vk6zgo

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Re: The sometimes 'Beauty' of mathematics???
« Reply #28 on: August 06, 2021, 12:04:00 am »
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 !!!

"Is this not strange & wonderful?"

Quote


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:
 

Offline free_electron

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Re: The sometimes 'Beauty' of mathematics???
« Reply #29 on: August 06, 2021, 12:49:49 am »
3x+1
Professional Electron Wrangler.
Any comments, or points of view expressed, are my own and not endorsed , induced or compensated by my employer(s).
 

Offline RJSV

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Re: The sometimes 'Beauty' of mathematics???
« Reply #30 on: August 06, 2021, 05:06:56 am »
First year, college CALCULUS:
   That instructor...Well, he had done some 'colorfull' career stuff previous (to teaching). Yeah, he was a PRO race-car driver, in national competition.
Had an usual zest for living, and a very dynamic speaking style, presenting thinking theory behind the Calculus. The 'Infinitesimal' : what it is, and why use
it ?
   Anyway, oh and one other thing: The professional
racing had cost him a LEG so there was that. Us students never asked about that, directly.
===========================================

   After writing lots of CODE and using binary / hex number notations, I have noticed some coincidences in the number translations.
For example decimal '64' is a kind of top end, when counting using 6 bits, and if you reverse, to 64hex, that translates to decimal '100'.

For quick 'Rule of Thumb', I use a casual rule of tens;
Meaning that with 10 bits, you have 1k.
With 20 bits: you get 1 Meg,
With 30 bits, you've got 1 giga (roughly).
   Makes for quick estimating.
 

Online tggzzz

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Re: The sometimes 'Beauty' of mathematics???
« Reply #31 on: August 06, 2021, 09:09:08 am »
First year, college CALCULUS:

Everybody at my state school did calculus[1] for external exams at 16yo. Great fun.

[1] differentiation and integration of polynomials except 1/x
There are lies, damned lies, statistics - and ADC/DAC specs.
Glider pilot's aphorism: "there is no substitute for span". Retort: "There is a substitute: skill+imagination. But you can buy span".
Having fun doing more, with less
 

Offline Ed.Kloonk

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Re: The sometimes 'Beauty' of mathematics???
« Reply #32 on: August 06, 2021, 09:53:44 am »
iratus parum formica
 

Offline GlennSpriggTopic starter

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Re: The sometimes 'Beauty' of mathematics???
« Reply #33 on: August 06, 2021, 11:39:48 am »
O.P. here...    MY simplistic 'brain', at the current moment anyway...), tried to solve this by Reverse thinking!!!
Or at least from a totally different 'Angle'... (excuse the pun, but NOT actually a 'pun')   8)

Consider a perfect circle. Ok, the calculation of it's 'circumference' is OBVIOUSLY  C = 2 Pi r.
Now rotate that circle 90-deg until it 'appears' as a straight line!!  Now the APPARENT (from the viewers perspective!), is just 2 x D   :phew:
So to ME!, all the 'InBetween' apparent 'Circumferences, from one's view point!, while observing such an apparent Circumference, MUST
simply be not only between those 2 above limits, but MUST be directly related mathematically as a result of that specific angle we are viewing
it from?   'One' needs to let go of thoughts about a 'physical' Ellipse per-se, and one that is 'apparent' from the angle we are viewing it!?

Surely then, the actual 'Maths' is simple!!  Without those complex theorems/styles being applied?, or 'trying' to apply them  ??   :-+
Diagonal of 1x1 square = Root-2. Ok.
Diagonal of 1x1x1 cube = Root-3 !!!  Beautiful !!
 

Offline RJSV

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Re: The sometimes 'Beauty' of mathematics???
« Reply #34 on: August 06, 2021, 07:18:00 pm »
But but but... Hey folks, don't keep me in suspense:
   Increase circumference by 1 meter:
   Ï€ X D = C
Then what is C + 1 ?
   (Ï€ X D) + 1 = C + 1
Solve for new D value, (don't have immediate clue)...
 

Offline RJSV

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Re: The sometimes 'Beauty' of mathematics???
« Reply #35 on: August 06, 2021, 07:26:50 pm »
(continued).
... Getting schetchy here, but my inclination is to set up, maybe TWO 'simultaneous' equations. Having a diameter D1 and a later diameter D2.
You already have C1 and C2 (C2 is just 1 meter larger).
   At this point, realize that we need, one of those MATH types, here.
 

Offline mathsquid

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Re: The sometimes 'Beauty' of mathematics???
« Reply #36 on: August 08, 2021, 03:31:54 pm »
Let \$r\$ be the radius of the earth. Put a rope around the earth at the equator to get a circle. Its circumference is \$2\pi r\$. Now take another rope that is one meter longer, and use it to make a circle that is \$a\$ meters above the earth.

Since the longer rope is one meter longer, the circumference of the bigger circle is \$ 2 \pi r + 1\$.

The radius of the bigger circle is \$r + a\$, thus the circumference of the bigger circle is \$2\pi (r+a) \$

Together we have
$$2 \pi r + 1 = 2\pi (r+a)$$
$$2 \pi r + 1 = 2\pi r + 2\pi a$$
$$ 1 =  2\pi a$$
$$ a = \frac{1}{2 \pi}$$
« Last Edit: August 08, 2021, 03:33:57 pm by mathsquid »
 

Offline mawyatt

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Re: The sometimes 'Beauty' of mathematics???
« Reply #37 on: August 08, 2021, 04:22:19 pm »
Using a little elementary calculus.

C = 2piR

dC = 2pi*(dR)

delta C = 2pi*(deltaR)

So the change in circumference wrt the change in radius is independent of the radius!!!

Thus the comment, about using the Earth, Moon, Jupiter, Solar System and Universe on one extreme, and a beachball, basketball, baseball, pingpong ball and atom on the other.

Best,
Curiosity killed the cat, also depleted my wallet!
~Wyatt Labs by Mike~
 
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Offline basinstreetdesign

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Re: The sometimes 'Beauty' of mathematics???
« Reply #38 on: August 09, 2021, 03:40:12 am »
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:

Of course there is a correct formula.  The relevant observation is found on this page: http://www.numericana.com/answer/ellipse.htm#elliptic

"There is no simple exact formula:  There are simple formulas but they are not exact, and there are exact formulas but they are not simple."

Also here: https://wj32.org/wp/2012/12/15/formula-for-the-circumference-of-an-ellipse/

I'm surprised that no one has mentioned that a simple extension of your sequence is that the "diagonal" (space-diagonal) of a 1 x 1 x 1 x 1 hypercube or tesseract is root(4) = 2
STAND BACK!  I'm going to try SCIENCE!
 
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Offline RJSV

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Re: The sometimes 'Beauty' of mathematics???
« Reply #39 on: August 09, 2021, 04:05:42 am »
Magic and Mathsquid:
   Just so I can sleep tonight:.       1/ 2 * pi
ntimeters
So that mouse has easy time, right ?  That's  around 6 inches of clearance, right ? Uhh 20 centimeters ?
 

Offline magic

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Re: The sometimes 'Beauty' of mathematics???
« Reply #40 on: August 09, 2021, 06:42:27 am »
Yep. Even a cat would pass :D
 

Online TimFox

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Re: The sometimes 'Beauty' of mathematics???
« Reply #41 on: August 09, 2021, 01:41:02 pm »
You would be surprised how small a hole a standard household mouse can fit through.
 
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Offline mathsquid

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Re: The sometimes 'Beauty' of mathematics???
« Reply #42 on: August 09, 2021, 09:04:04 pm »
Yes.  The rope is \$\frac{1}{2\pi}\$ meters above the ground because the additional length was 1 meter.
 

Offline RJSV

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Re: The sometimes 'Beauty' of mathematics???
« Reply #43 on: August 10, 2021, 01:12:02 am »
Thanks, phew that ALGEBRA  is hard, for me.  Seriously, I've come to realize: College level Calculus got built on foundation of 'crappy' high school Algebra.

   Heck, I've heard Einstein lamented, "...couldn't  factor a polynomial, to save my life!..."

Oh, and those smaller four legged 'Rodents' can run straight up a slightly roughened exterior wall.
 

Offline CatalinaWOW

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Re: The sometimes 'Beauty' of mathematics???
« Reply #44 on: August 10, 2021, 02:38:26 am »
Sometimes the beauty comes when an (almost) totally non-math explanation is available.  The following is apparently related to a problem in topology, but has practical applications, and has delighted me ever since I came across it. 

It is the proof that you can place a square table on any uneven floor that meets a couple of simple criteria.  The simple criteria: 1.  No discontinuities in the floor.  2.  Irregularities have to be a fair amount smaller than the length of the legs.  3.  The legs are equal length.

The proof.  First, by inspection, you can get three legs to touch by putting two in contact with the floor and rotating the table about the line between the contact points until another leg touches.

Now the pretty part.  Since the table is symmetric for 90 degree rotations about its center there are three other rotational positions where three legs touch the same three points.  But they aren't the same three legs.  And if you keep the legs in contact with the surface as you rotate between the four positions there will be a point at which the current three legs transition to the next three.  At that point four legs are touching the ground.
 

Offline AntiProtonBoy

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Re: The sometimes 'Beauty' of mathematics???
« Reply #45 on: August 10, 2021, 02:38:59 am »
One particular construct I find super elegant in mathematics is transformation matrices used in visualising geometry and inverse kinematics. These matrices are almost like magic. You can combine a sequence of transform operations into a single matrix and get the correct result in a single operation.
 

Offline Circlotron

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Re: The sometimes 'Beauty' of mathematics???
« Reply #46 on: August 10, 2021, 03:22:16 am »
Can someone please explain how these cool mathematical principles came about as a consequence of a mindless universe beginning with a whopping big explosion? Or did they exist beforehand?
 

Offline AntiProtonBoy

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Re: The sometimes 'Beauty' of mathematics???
« Reply #47 on: August 10, 2021, 03:42:09 am »
Most mathematical principles is a form of "abstract logic" (for a lack of a better term), which may not need be based on the properties of the universe. Of course the universe needs to exist for us to have the mind to think 1+1=2, but the math alone does not have to define anything about the universe.
 
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Offline GlennSpriggTopic starter

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Re: The sometimes 'Beauty' of mathematics???
« Reply #48 on: August 10, 2021, 01:18:26 pm »
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:

Of course there is a correct formula.  The relevant observation is found on this page: http://www.numericana.com/answer/ellipse.htm#elliptic

"There is no simple exact formula:  There are simple formulas but they are not exact, and there are exact formulas but they are not simple."

Also here: https://wj32.org/wp/2012/12/15/formula-for-the-circumference-of-an-ellipse/

I'm surprised that no one has mentioned that a simple extension of your sequence is that the "diagonal" (space-diagonal) of a 1 x 1 x 1 x 1 hypercube or tesseract is root(4) = 2

Ah!! Yes... I've thought of that... and although 'logical', it is some-what impossible to 'imagine' such a 'diagonal'.  Here is a Mental 'representation' of
a HyperCube/Tesseract, as it can be perceived in 3-Dimesions in our mind, before going on here...

A 'square' has 4 right-angles. a 'Cube' has all right-angles too!, but we 'draw' one with distorted angles to represent it in 3-D.  A 'HyperCube' consists
of technically a Cube within a Cube, with lines joining the cubes too, and even THESE are at right-angles to everything else within!  Of course, our
brains can't fathom/picture that, so like with the 'cube' drawing, we can only imagine it as a 3-D 'shadow, and seeing distorted angles, as above!
We can't even IMAGINE where to place such a 'diagonal', within such an elusive concept!!!    :palm:

P.S.   I haven't finished with orig/prior discussions yet, though !!!   8)
« Last Edit: August 10, 2021, 01:22:26 pm by GlennSprigg »
Diagonal of 1x1 square = Root-2. Ok.
Diagonal of 1x1x1 cube = Root-3 !!!  Beautiful !!
 

Online tggzzz

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Re: The sometimes 'Beauty' of mathematics???
« Reply #49 on: August 10, 2021, 02:15:45 pm »
At this point, Flatland: A Romance of Many Dimensions by George Abbott is relevant.
There are lies, damned lies, statistics - and ADC/DAC specs.
Glider pilot's aphorism: "there is no substitute for span". Retort: "There is a substitute: skill+imagination. But you can buy span".
Having fun doing more, with less
 
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