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#1 2008-04-09 10:43:04

mikau
Member
Registered: 2005-08-22
Posts: 1,504

does anyone else find it spooky..

that

Area of Circle = f(r), and Circumference of Circle = f'(r)

AND

volume of sphere = g(r) and surface area of sphere = g'(r)

is this a coincidence?


A logarithm is just a misspelled algorithm.

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#2 2008-04-09 10:46:50

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,561

Re: does anyone else find it spooky..

I noticed that myself about 5 years ago,
but I know nothing about it really.


igloo myrtilles fourmis

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#3 2008-04-09 10:58:41

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,534

Re: does anyone else find it spooky..

Nice! It also works for a 4D hypersphere (called a "3-sphere"):


Any more like that I wonder? Or is it special to spheres?


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#4 2008-04-09 11:53:30

Ricky
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Registered: 2005-12-04
Posts: 3,791

Re: does anyone else find it spooky..

Well, now that your bought hyperspheres into it...

Given "hypercube" with side of length > 1, it's area diverges as you go up in dimensions.
Given a hypersphere with radius of any length, it's area goes to 0 as you go up in dimensions.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#5 2008-04-10 13:48:50

mikau
Member
Registered: 2005-08-22
Posts: 1,504

Re: does anyone else find it spooky..

I actually realized the relation with the surface area and area/volume of a circle and sphere make perfect sense!

Consider the change in volume of the volume of a sphere dv coresponding to dr, this should be like adding a thin coating of paint onto the surface, increasing its volume ever so slightly. The total area of this coating would be the surface area, and the thickness would be dr.

Likewise, consider a circle thats getting wider. The slight change in the surface area should be like wrapping a string once around the outside to increase the area. The width of the string would be dr, and the length would be the circumference of the circle.

Now, who wants to explain why this makes sense in 4 dimensions? tongue

Last edited by mikau (2008-04-10 13:57:59)


A logarithm is just a misspelled algorithm.

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#6 2008-04-10 14:36:55

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,561

Re: does anyone else find it spooky..

?Ring Method for circle?


igloo myrtilles fourmis

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#7 2008-04-10 15:29:13

bossk171
Member
Registered: 2007-07-16
Posts: 301

Re: does anyone else find it spooky..

We used something like this in Physics... I wish I could remember the details, something to do with linear density. But it does make a certain kind of intuitive sense, Mikau's description of paint is genius. I always "felt" that explanation, but could never verbalize it, I really appreciate that description  thanks.


There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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#8 2014-02-23 15:40:52

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 15,067
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Re: does anyone else find it spooky..

mikau wrote:

that

Area of Circle = f(r), and Circumference of Circle = f'(r)

AND

volume of sphere = g(r) and surface area of sphere = g'(r)

is this a coincidence?

Ofcourse not!

You get a sphere by integrating many surfaces and a circle by integrating many circumferences


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

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#9 2014-03-18 13:23:53

eigenguy
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Registered: 2014-03-18
Posts: 78

Re: does anyone else find it spooky..

Agnishom has it right, of course. But this only works for very symmetric shapes. No matter how you try to work it on ellipses and ellipsoids, for example, it fails.


"Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions." - Bröcker & Jänich

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