
Volume of a Sphere - ColinWright
https://www.solipsys.co.uk/new/VolumeOfASphere.html
======
magicalhippo
Generalizations to higher dimensions[1] also have some fun properties. For
one, the volume of a n-dimensional sphere (ball) goes to zero as n goes to
infinity.

But not in a trivial way: given a radius r, you can find the number of
dimensions n that maximizes the volume of the n-sphere. For r=1 it's 5
dimensions, but for r=3 it's 55 dimensions.

Numberphile[2] has a video on higher dimensional spheres which talks about
this weirdness.

[1]: [https://en.wikipedia.org/wiki/Volume_of_an_n-
ball](https://en.wikipedia.org/wiki/Volume_of_an_n-ball)

[2]:
[https://www.youtube.com/watch?v=mceaM2_zQd8](https://www.youtube.com/watch?v=mceaM2_zQd8)

~~~
Retric
I am going to sugest something about that analysis is flawed as r=1 is unit
less. Suggesting measurements in feet and nD feet would favor 55d, but yards /
nD yard would favor 3D.

Though I just woke up so I may be missing something obvious.

~~~
vitus
A 5-dim yard and a 6-dim yard convert to a different number of n-dim feet (243
vs 729).

\- A 5-d sphere with R=1yd would have volume of ~5.264 yd^5 or 1279 ft^5.

\- A 6-d sphere with R=1yd would have volume of ~5.168 yd^6 or 3767 ft^6.

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gus_massa
Nice post. The graph of the sphere, the cone and the cylinder probably needs a
3D version just before it, so it's clear that they are 3D objects. (With a
small doll or plastic eye to show the point of view?)

[Request for someone with a lot of Blender capabilities: Make a 3D version of
the graph, where the back half of the objects are opaque and the front half
are transparent. Since you already have the 3D model, bonus points for another
version where the top third of the objects is transparent and the bottom two
thirds are opaque.]

~~~
ColinWright
> _Nice post._

Thank you.

> _The graph of the sphere, the cone and the cylinder probably needs a 3D
> version just before it, so it 's clear that they are 3D objects._

Sadly, my skills aren't up to that, and I don't currently have time to acquire
them 8-(

> _Request for someone with a lot of Blender capabilities: Make a 3D version
> of the graph, where the back half of the objects are opaque and the front
> half are transparent._

That would be cool.

> _... bonus points for another version where the top third of the objects is
> transparent and the bottom two thirds are opaque._

Even cooler.

Sadly, the post is sinking without trace, so I doubt anyone will see the
request.

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FabHK
As an aside, if you take the area of a circle, A = pi r^2, and take the
derivative wrt r, you get the circumference 2 pi r. Similarly, the derivative
of the volume of a sphere, V = 4/3 pi r^3, is the surface area A = 4 pi r^2.

It's sort of obvious, when you think about it, but it wasn't obvious to me
before thinking about it :-)

~~~
fctorial
Isn't it the case that integral of surface area is volume, that seems more
obvious to me. The derivative happens to be reverse of integral and hence area
happens to be derivative of volume.

------
howling
> It was only 282 years ago that Euler presented in his textbooks the exact
> formula for the volume of a sphere

It's strange that Fermat's Library stated this since the formula for volume of
sphere was discovered by Archimedes more than 2000 years ago.

[1]:
[https://en.wikipedia.org/wiki/On_the_Sphere_and_Cylinder](https://en.wikipedia.org/wiki/On_the_Sphere_and_Cylinder)

~~~
ColinWright
Yes and no.

My understanding is that Archimedes didn't, and never could have, produced the
actual formula, because that way of writing formulas didn't exist. That's
discussed in the sub-thread here:

[https://twitter.com/daveinstpaul/status/1213469023461138432](https://twitter.com/daveinstpaul/status/1213469023461138432)

Additionally, the result was probably known before Archimedes, but again,
writing that formula in that form probably didn't happen until Euler.

~~~
quietbritishjim
That tweet is merely about the way the formula was expressed, not about its
mathematical content:

> Archimedes discovered the formula many centuries before Euler, although he
> could not express it in modern algebraic notation.

If Archimedes was able to express the formula in natural language (e.g. "take
the radius of the hypersphere, multiply it by itself as many times as there
are dimensions, ..." etc.) in a way that was unambiguously understood by other
mathematicians (or natural philosophers) of the time, then that formula has
definitely been discovered. Writing it in modern notation with one-character
symbols for addition, multiplication (typically zero symbols!) etc. is nowhere
near required for it to count as being discovered.

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ilovepyramids
Very interesting! It's not clear to me why it's so obvious that doubling the
height of a pyramid while keeping the side constant would double the volume of
the pyramid. The reason it is not obvious to me is that the proportions of the
pyramid would change. The author of the post seems to think this is so obvious
that it doesn't bear explanation. Can anyone explain to me why this works?

~~~
chmod775
Its not specific to pyramids - its true for stretching any shape in 2D and 3D
or lines 1D. Stretch it to twice its former size across a single axis and you
get double the length/area/volume.

It becomes obvious in 2D if you look at how the area of a rectangle is
calculated (x*y) and imagine your shape is just projected/drawn on that
rectangle and stretches with it.

~~~
ilovepyramids
I did consider this, and it seemed to make it feel a little clearer—but when I
first posted, I didn't feel it was obvious that the _fraction_ of the
rectangle occupied by the triangle would remain constant if you stretched it.
But I think the other replies prove that that is true. Thank you.

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cmurf
Makes me think of the steradian. This is used in photometery for basic things
like the lumen (1 lumen = 1 candela per steradian), where there's the concept
of a light source emitting in all directions uniformly, and also what amounts
of that arrive at what distances onto a surface, and what amounts are absorbed
or reflected, and so on.

[https://en.wikipedia.org/wiki/Steradian](https://en.wikipedia.org/wiki/Steradian)

[https://en.wikipedia.org/wiki/Lumen_(unit)](https://en.wikipedia.org/wiki/Lumen_\(unit\))

------
zokier
Hammer and nails, but to me the intuitive derivation of the volume formula
would be through integration. Approximate slice of sphere as a cylinder, and
sum up the volumes of those cylinders as their thickness approaches zero

~~~
ColinWright
This was all done before calculus had been developed, and the ideas here are
clear precursors of the ideas later made precise by Newton and Leibniz.

The intuitive derivation is by thin slices that are thin enough to see why
it's true. The trained and sophisticated version is "Well, it's just calculus,
isn't it."

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arh68
Wow, I might finally grok this. At _h_ =0, the area of the dome's slice = the
cylinder's slice ( _πRR_ ). As h increases, the dome slice gets smaller, but
by _exactly πhh_. I picture that smaller circle in the plane of my screen,
perp to the dome slice. Somehow, for any given _h_ , that _πhh_ makes up for
the lost area at the edge.

The bit about the Hat Box theorem still just makes me blink.

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lurquer
I like his proof of a the volume of a cone. I usually considered it via Euclid
XII, prop 7, which was always hard to visualize. Starting with 6 pyramids in a
cube is ingenious.

[https://mathcs.clarku.edu/~djoyce/java/elements/bookXII/prop...](https://mathcs.clarku.edu/~djoyce/java/elements/bookXII/propXII7.html)

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bumbledraven
The volume and surface area of a ball are about 1/2 that of the box it came
in.

This comes in handy for back-of-the-envelope calculations.

Source:
[http://www.vendian.org/envelope/dir0/ballbox.html](http://www.vendian.org/envelope/dir0/ballbox.html)

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suifbwish
The simplest way to get the volume of a sphere or any other solid 3d object
under a certain size is to fill a bucket of water to the brim, put the object
in until it is completely submerged, take it out, then measure how much water
is now missing. Obviously only works with smaller objects.

~~~
commandlinefan
But you’d never get an exact measurement that way, since pi is irrational.

~~~
bmm6o
The precision of the instruments is a much bigger obstacle than the
representation of the value. You would be just as uncertain that the unit cube
has volume 1.000000...

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censorshiptest
4/3 _pi_ r^3

