Perfect Sphere
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Perfect Sphere
This has mostly been something I've been thinking about for the past week or so and I wanted to get opinions on how one would tackle such a problem. So lets say you are standing in a room and given a large (we'll say greater than 10m diameter) sphere. The claim is that it is a perfect sphere, how would you go about proving that it is so? I should say this is ignoring the whole "there is no such thing as a perfect sphere in nature" and your access to tools/equipment is not constrained.
Re: Perfect Sphere
atomic microscopy
or hit with something hard enough to put a dent in it.
or hit with something hard enough to put a dent in it.
Re: Perfect Sphere
Measure its purported diameter by balancing a flat sheet on top of it and measuring the height of the sheet. Then go to your lab and construct two perfect, rigid, impermeable hemispherical shells of the same purported radius as the "sphere". Fill the room with <insert favorite gas here>. Enter the room and seal the shells around the "sphere" (if it's not possible to do this, then it's not spherical, so stop right now). Leave the room. Empty the room of <favorite gas>. Enter the room. Remove the shells. If the object was not spherical, then you will be able to detect some gas in the room. Otherwise, the room will be completely free of <favorite gas>.
Re: Perfect Sphere
I’d pick a point on the object, set that point on the floor, find the highest point on the object, and measure its height. Then I’d construct a perfectly cylindrical tube of exactly that interior diameter (perhaps plus some arbitrarily small epsilon) and roll the object through the tube three times on mutually perpendicular axes. Only a sphere should be able to pass this test without hitting or moving away from the tube walls. There are other shapes of constant width, but I’m pretty sure the sphere is the only one with a constant circular crosssection of rotation. And the second and third rolls are just to double and triple check, really.
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Re: Perfect Sphere
If it's a perfect sphere, wouldn't it damage everything it touches or itself, as whatever it touches would be tangential to it, and give that pressure is Force/Area, wouldn't that do something destructive to the sphere or object it touches? If so, we could use this to prove that it's perfect?
Re: Perfect Sphere
kvn.l wrote:If it's a perfect sphere, wouldn't it damage everything it touches or itself, as whatever it touches would be tangential to it, and give that pressure is Force/Area, wouldn't that do something destructive to the sphere or object it touches? If so, we could use this to prove that it's perfect?
The other stuff would just compress around it, like it does normally.
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Re: Perfect Sphere
Determine any diameter of the alleged sphere by any suitable method. Use this to calculate the radius, then use this to calculate the volume of an ideal sphere of the same radius.
Using a suitably large volume of an appropriate liquid, submerge the sphere and measure the volume displaced.
Compare the displacement with the calculated ideal. Any discrepancy within the limits of measurement/calculation indicates an imperfect sphere.
Using a suitably large volume of an appropriate liquid, submerge the sphere and measure the volume displaced.
Compare the displacement with the calculated ideal. Any discrepancy within the limits of measurement/calculation indicates an imperfect sphere.
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 telcontar42
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Re: Perfect Sphere
This wouldn't prove that it was a perfect sphere, just that it has the same volume as a perfect sphere. It could be slightly deformed and have the same volume and the same diameter at the point you measured.
Re: Perfect Sphere
How about relativity? Since it's traveling at some speed through the universe, it must be elongated (however slight) along its vector. So it can only be a prefect sphere within the same reference frame.
In general, I don't think you can prove it's a perfect sphere. Any method you use to measure it would itself subject to measurement error.
In general, I don't think you can prove it's a perfect sphere. Any method you use to measure it would itself subject to measurement error.
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Re: Perfect Sphere
If it were so perfect it would give you a blowjob and a bunch of money.

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Re: Perfect Sphere
Hmm I didn't think of considering relativistic effects. Although as far as measurements go, I would think that because we are ignoring the idea that no perfect sphere can exist in nature, it would be safe to assume that you are capable, through some unknown method, to have completely accurate measurements.
Re: Perfect Sphere
Expanding on the above, to prove that you're in the same reference frame, you have to measure speed. To measure if it's a sphere, you have to measure the position of particles along the surface. Since you must measure closely enough for quantum effects to matter, and you can't measure both speed and position, you now have both Einstein and Heisenberg waving fingers at you.
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 BlackSails
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Re: Perfect Sphere
Physical spheres are made of discrete points (ie, molecules) and therefore cannot actually form a perfect sphere.
QED.
QED.
Re: Perfect Sphere
BlackSails wrote:Physical spheres are made of discrete points (ie, molecules) and therefore cannot actually form a perfect sphere..
crookedeye wrote: I should say this is ignoring the whole "there is no such thing as a perfect sphere in nature" and your access to tools/equipment is not constrained.
Yeah, we know. Anyways I was assuming this thread was in general about measuring the sphereicity of an object, and then describing how far a given circular thing deviates from being an actual sphere. In which case you might want to ask some geographers* because they've been measuring this big almostbutnotquite spherical thing for some time, and have had a great deal of success.
* There's a more specific word for this, but it escapes me at the moment...

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Re: Perfect Sphere
Using your quantum tunneling device, teleport yourself inside and attach weight (a) at any point on the sphere.
Fill the sphere with water now attach a piece of string to the weight (a) and attach a float (b) to the other end that transmits an electircal signal whenever it tocuches the spere.
Allow gravity to roll the sphere so that the Weight(a) is at the lowest point possible due to gravity.
Now release the float so that due to viscosity and gravity the float now rises to the highest point in the sphere possible(checking with the transmitter) you now have theoretical limit for the spheres radius
now you can anchour both (a) & (b) so they cannot move and attach a spinning weight (c) to the radius
Fix a rigid rod of the equivalent diameter and now detach the float and drain the water. Now swing the float around in a rotaional pattern for the centre of the sphere via measured radius.
via induction you can now demonstrate that so long as the float is moving fast enough to never become less that the radius you can use the electircal transmitter to falsify the claim that it is not a perfect sphere.i.e. if the float ever loses contact inside.
The mathematicians will surely be up in arms but screw em. So long as you vary angle of rotation you can technically cover ANY equatorial of the sphere and thus test the hypothesis.
Do I win?
Fill the sphere with water now attach a piece of string to the weight (a) and attach a float (b) to the other end that transmits an electircal signal whenever it tocuches the spere.
Allow gravity to roll the sphere so that the Weight(a) is at the lowest point possible due to gravity.
Now release the float so that due to viscosity and gravity the float now rises to the highest point in the sphere possible(checking with the transmitter) you now have theoretical limit for the spheres radius
now you can anchour both (a) & (b) so they cannot move and attach a spinning weight (c) to the radius
Fix a rigid rod of the equivalent diameter and now detach the float and drain the water. Now swing the float around in a rotaional pattern for the centre of the sphere via measured radius.
via induction you can now demonstrate that so long as the float is moving fast enough to never become less that the radius you can use the electircal transmitter to falsify the claim that it is not a perfect sphere.i.e. if the float ever loses contact inside.
The mathematicians will surely be up in arms but screw em. So long as you vary angle of rotation you can technically cover ANY equatorial of the sphere and thus test the hypothesis.
Do I win?
 Antimony120
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Re: Perfect Sphere
letterX wrote:BlackSails wrote:Physical spheres are made of discrete points (ie, molecules) and therefore cannot actually form a perfect sphere..crookedeye wrote: I should say this is ignoring the whole "there is no such thing as a perfect sphere in nature" and your access to tools/equipment is not constrained.
Yeah, we know. Anyways I was assuming this thread was in general about measuring the sphereicity of an object, and then describing how far a given circular thing deviates from being an actual sphere. In which case you might want to ask some geographers* because they've been measuring this big almostbutnotquite spherical thing for some time, and have had a great deal of success.
* There's a more specific word for this, but it escapes me at the moment...
if you mean for the almostbutnotquitespherical thing, it's called a geoid. Yeah, turns out the earth is an earthshaped objects. In other news tautologies are tautological.
Anyhow, I would shine a laser onto it at some point, use a beam spltter and recombination intereformetre setup, and then spin the "sphere" on some axis. If it's perfectly spherical there will be a static interference pattern. Repeat for a couple random other axes to be certain. That'll give you a pretty good error of measurement (a couple hundred nm). If you really want accuracy, and proove that it breaks physics by being more spherical than anything made out of atoms could be, use a suitably energetic electron interferometre in the same fashion. This of course assumes that you're capable of finding an axis to a great degree of accuracy (though there are some cute ways of doing this).
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 uncivlengr
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Re: Perfect Sphere
Since the sphere is phyically impossible, it's going to necessarily require to be measured by some physically impossible means, or not at all.
Once you've crossed that line, the sky's the limit  you can take your set of "perfect" calipers and hook it up to your infinitely fast robotic arm, that can measure the diameter from an infinite number of points on the surface of the sphere in a finite amount of time, stopping whenever a discrepancy in the readings is found, or when every infintessimal point on the surface has been measured.
It's actually a more interesting question when you impose real world contraints on the problem.
Once you've crossed that line, the sky's the limit  you can take your set of "perfect" calipers and hook it up to your infinitely fast robotic arm, that can measure the diameter from an infinite number of points on the surface of the sphere in a finite amount of time, stopping whenever a discrepancy in the readings is found, or when every infintessimal point on the surface has been measured.
It's actually a more interesting question when you impose real world contraints on the problem.
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 BlackSails
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Re: Perfect Sphere
Yeah, if the sphere can be perfect, you cannot tell, since your instruments are not perfect. And if your instruments are perfect, then its easy, you just measure the sphere.
Re: Perfect Sphere
BlackSails wrote:And if your instruments are perfect, then its easy, you just measure the sphere.
But how do you do it with a finite number of measurements?
Re: Perfect Sphere
Carnildo wrote:BlackSails wrote:And if your instruments are perfect, then its easy, you just measure the sphere.
But how do you do it with a finite number of measurements?
Don't, but make each measurementequivalent take an infinitely small amount of time/make your measurements continuous rather than discrete.
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Re: Perfect Sphere
I think there has to be some kind of macroscopic limit here for where we can draw the line at what constitutes perfect given that at some point we'll be dealing with oscillating wave functions and not anything that remotely resembles an actual sphere.
Also in relation to my own ego,I think I came up with best answer using physical principles (other than people invoking the 'how to measure the height of a building using a barometer' urban)
i.e finding a way to use gravity to find the spheres radius and then using angular momentum to gauge if at any point the spheres radius ceases to be a constant.
I toyed with using a singularity at the centre with the event horizon being on the very limit of the radius thus making any imperfections pressing inwards cause the sphere to collapse inwards but the simpler the better
Also in relation to my own ego,I think I came up with best answer using physical principles (other than people invoking the 'how to measure the height of a building using a barometer' urban)
i.e finding a way to use gravity to find the spheres radius and then using angular momentum to gauge if at any point the spheres radius ceases to be a constant.
I toyed with using a singularity at the centre with the event horizon being on the very limit of the radius thus making any imperfections pressing inwards cause the sphere to collapse inwards but the simpler the better
 BlackSails
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Re: Perfect Sphere
Carnildo wrote:BlackSails wrote:And if your instruments are perfect, then its easy, you just measure the sphere.
But how do you do it with a finite number of measurements?
Charge the sphere, stick a charged ball in the exact middle (as measured by your perfect ruler) and see if it moves.
Spin the sphere, measure the moment of inertia tensor.

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Re: Perfect Sphere
so am I the only one who is leaning towards.. eyeball it.. proclaim it perfect, then challenge someone to prove you wrong?
I'm bad at this arent I?
I'm bad at this arent I?
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Re: Perfect Sphere
Then we'd have to extract your eyeball and find a way to prove your eyeball is a perfect optical insturment.
I thought about the moment of inertia tensor but remember it doesn't have to be a sphere in order to give a result equal to a spherical radius
Unless of course you do it for all 3 axis. I don't think you can get a shape that creates a symmetrical result there unless there's some mathematician who knows of one.
I thought about the moment of inertia tensor but remember it doesn't have to be a sphere in order to give a result equal to a spherical radius
Unless of course you do it for all 3 axis. I don't think you can get a shape that creates a symmetrical result there unless there's some mathematician who knows of one.
Re: Perfect Sphere
Antimony120 wrote:if you mean for the almostbutnotquitespherical thing, it's called a geoid. Yeah, turns out the earth is an earthshaped objects. In other news tautologies are tautological.
Or perhaps he was thinking of an oblate spheroid. There can be more than one way to describe things. You know that, right? Roundish, geoidal, yourmammaesque. All the same thing, though "roundish" doesn't connote the enormous size as well.
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Re: Perfect Sphere
Velifer wrote:Antimony120 wrote:if you mean for the almostbutnotquitespherical thing, it's called a geoid. Yeah, turns out the earth is an earthshaped objects. In other news tautologies are tautological.
Or perhaps he was thinking of an oblate spheroid. There can be more than one way to describe things. You know that, right? Roundish, geoidal, yourmammaesque. All the same thing, though "roundish" doesn't connote the enormous size as well.
well of course not.. thats the a vague approximation of shape.. size is conferred from yourmammaesque.
EDIT: oh wait my bad.. He probably meant at least 10 meters not 10 miles.. so no it isn't a very appropriate term.
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Re: Perfect Sphere
Seems like some are trying to avoid the question by throwing smart sounding ideas at it that make it seem impossible to answer. How about we pose the question as about a measure of imperfection rather than judging whether or not it's perfect. A perfect sphere would have the property x^{2}+y^{2}+z^{2}=R everywhere on the surface, for some coordinate system centred in the middle. So if we measure any point on the surface, we can calculate its distance from the centre, and subtract R to give r. If we do this for multiple points, stacking into a vector r we have the meansquare error, r^{T}r. This is the measure of imperfection.
If you're concerned that you don't know the centre, first assume the sphere is perfect, pose the problem as a leastsquares approximation of the unknowns x_{0}, y_{0} and z_{0}. Then take loads of measurements and find the most likely centre.
As for practicalities of measuring position of a point on the surface, how about a laser range finder?
Edit: Oops, some squares missing in there.
If you're concerned that you don't know the centre, first assume the sphere is perfect, pose the problem as a leastsquares approximation of the unknowns x_{0}, y_{0} and z_{0}. Then take loads of measurements and find the most likely centre.
As for practicalities of measuring position of a point on the surface, how about a laser range finder?
Edit: Oops, some squares missing in there.
Last edited by martin878 on Thu Oct 28, 2010 10:20 am UTC, edited 1 time in total.
Re: Perfect Sphere
Construct a shape out of all possible tangent lines, examine the created shape to see if it is entirely space filling (well, except for a perfect sphere.)
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Re: Perfect Sphere
Velifer wrote:Construct a shape out of all possible tangent lines, examine the created shape to see if it is entirely space filling (well, except for a perfect sphere.)
I think that would only tell you that you've got a closed object with no sharp edges.
Re: Perfect Sphere
1. Resurrect Giotto.
2. Buy Giotto lunch and ask him to sculpt you a perfect sphere.
3. Compare Giottosphere to contendersphere.
4. ???
5. Profit.
Since constructing an apparatus with perfect accuracy and precision is impossible, it would be impossible to determine whether a perfect sphere was in fact perfect. You could only measure its circularity/roundness to a ludicrously high confidence. With that said, I'm sure if we measured a macroscopic object to within +/1 Planck length, then I would go ahead and call that perfect. How to do it? Meh, not my job.
2. Buy Giotto lunch and ask him to sculpt you a perfect sphere.
3. Compare Giottosphere to contendersphere.
4. ???
5. Profit.
Since constructing an apparatus with perfect accuracy and precision is impossible, it would be impossible to determine whether a perfect sphere was in fact perfect. You could only measure its circularity/roundness to a ludicrously high confidence. With that said, I'm sure if we measured a macroscopic object to within +/1 Planck length, then I would go ahead and call that perfect. How to do it? Meh, not my job.
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Re: Perfect Sphere
Build a continuous flat ring around the earth's surface in an evacuated chamber, using the same material the sphere is made of. Roll the sphere. It should roll forever.
This. Should be the same number repeated across the diagonal and zeros elsewhere iirc. You do it by spinning it up along each of three orthogonal axes, with constant force, and measuring the resulting rotational acceleration. Maybe strap some perfect thrusters onto it.
My favorite method would be to put it out in deep space, then orbit something around it. It should take a very very long time to drift out of orbit. Actually this might be better suited as a method of measuring the cumulative gravitational force exerted by the universe on a single region of space, assuming your sphere really is perfect.
BlackSails wrote:Spin the sphere, measure the moment of inertia tensor.
This. Should be the same number repeated across the diagonal and zeros elsewhere iirc. You do it by spinning it up along each of three orthogonal axes, with constant force, and measuring the resulting rotational acceleration. Maybe strap some perfect thrusters onto it.
My favorite method would be to put it out in deep space, then orbit something around it. It should take a very very long time to drift out of orbit. Actually this might be better suited as a method of measuring the cumulative gravitational force exerted by the universe on a single region of space, assuming your sphere really is perfect.
"Welding was faster, cheaper and, in theory,
produced a more reliable product. But sailors do
not float on theory, and the welded tankers had a
most annoying habit of splitting in two."
J.W. Morris
produced a more reliable product. But sailors do
not float on theory, and the welded tankers had a
most annoying habit of splitting in two."
J.W. Morris
Re: Perfect Sphere
Solt wrote:Build a continuous flat ring around the earth's surface in an evacuated chamber, using the same material the sphere is made of. Roll the sphere. It should roll forever.
Actually, a perfect sphere set rolling on a level surface with no air resistance still slows down. I wish I could remember the place I read about this effect, but it has to do with the normal force from the ground being stronger slightly in front of the center of the point of contact than slightly behind, due to the rolling motion, so there is a net torque acting to slow rotation. And in any reallife scenario it will eventually come to a stop.
Solt wrote:BlackSails wrote:Spin the sphere, measure the moment of inertia tensor.
This. Should be the same number repeated across the diagonal and zeros elsewhere iirc. You do it by spinning it up along each of three orthogonal axes, with constant force, and measuring the resulting rotational acceleration. Maybe strap some perfect thrusters onto it.
I admit I don’t know much about tensors, but are you presupposing the object to be made of a material with uniform (or at least rotationally symmetric) density?
Solt wrote:My favorite method would be to put it out in deep space, then orbit something around it. It should take a very very long time to drift out of orbit. Actually this might be better suited as a method of measuring the cumulative gravitational force exerted by the universe on a single region of space, assuming your sphere really is perfect.
I think you already see why this doesn’t work.
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Re: Perfect Sphere
Solt wrote:My favorite method would be to put it out in deep space, then orbit something around it. It should take a very very long time to drift out of orbit. Actually this might be better suited as a method of measuring the cumulative gravitational force exerted by the universe on a single region of space, assuming your sphere really is perfect.
You mean something like this?

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Re: Perfect Sphere
uncivlengr wrote:Since the sphere is phyically impossible, it's going to necessarily require to be measured by some physically impossible means, or not at all.
Once you've crossed that line, the sky's the limit  you can take your set of "perfect" calipers and hook it up to your infinitely fast robotic arm, that can measure the diameter from an infinite number of points on the surface of the sphere in a finite amount of time, stopping whenever a discrepancy in the readings is found, or when every infintessimal point on the surface has been measured.
It's actually a more interesting question when you impose real world contraints on the problem.
I want to see that robot!
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Re: Perfect Sphere
Carnildo wrote:Velifer wrote:Construct a shape out of all possible tangent lines, examine the created shape to see if it is entirely space filling (well, except for a perfect sphere.)
I think that would only tell you that you've got a closed object with no sharp edges.
Topology. How does that work?
Fine. Cut it into pieces and see if you can reassemble it into two or more identical spheres.
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 Antimony120
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Re: Perfect Sphere
Unfortunatly that won't work either, you're most likely referring to the BanachTarski paradox which has nothing to do with a perfect sphere, though using it on a perfect sphere is by far the easiest to work out example of the problem, see for example Tarski's circlesquaring problem, althought that isn't quite the same principal. In any case the point is that you haven't proven it's a perfect sphere, just some that it possesses some characteristics.
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 uncivlengr
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Re: Perfect Sphere
That raises another problematic issue with using the notion "perfection"  do external forces deform a "perfect" sphere? Are gravitational forces going to deform the sphere, spoiling the testing?Qaanol wrote:Actually, a perfect sphere set rolling on a level surface with no air resistance still slows down. I wish I could remember the place I read about this effect, but it has to do with the normal force from the ground being stronger slightly in front of the center of the point of contact than slightly behind, due to the rolling motion, so there is a net torque acting to slow rotation. And in any reallife scenario it will eventually come to a stop.
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Re: Perfect Sphere
Qaanol wrote:Actually, a perfect sphere set rolling on a level surface with no air resistance still slows down. I wish I could remember the place I read about this effect, but it has to do with the normal force from the ground being stronger slightly in front of the center of the point of contact than slightly behind, due to the rolling motion, so there is a net torque acting to slow rotation. And in any reallife scenario it will eventually come to a stop.
Yes, because in any real life scenario it would NOT be a perfect sphere, and it would deform. You are actually describing one of the two mechanisms by which a perfectly round real wheel experiences friction, the second being through continuous physical deformation at the point of contact (deformation requires energy).
A perfect sphere must be infinitely rigid which creates many other problems, clearly, but one of those problems would NOT be deformation and thus the mechanism you describe would not happen, because the sphere would only touch the surface at a single point. Literally, a single geometric point. Or maybe the area of contact would be a planck's length squared or something. I don't know that part should be answered by the physicists.
"Welding was faster, cheaper and, in theory,
produced a more reliable product. But sailors do
not float on theory, and the welded tankers had a
most annoying habit of splitting in two."
J.W. Morris
produced a more reliable product. But sailors do
not float on theory, and the welded tankers had a
most annoying habit of splitting in two."
J.W. Morris
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Re: Perfect Sphere
Solt wrote:I don't know that part should be answered by the physicists.
The physicists are staying the hell away from this problem until people stop treating it like a math problem and pick a sane, reallife definition for "perfection".
Re: Perfect Sphere
Yea it can never really be mathematically round.
And I can't think of a good way to define it as perfectly round using physical terms. Calculus shows us that a finite collection of small segments can never be a continuous curve, no matter how small you take the segments to be, unless you take an infinite number of infinitely small segments. So what is the most round an object in nature can actually be? How round is a star? How round is a collection of phospholipids immersed in water? In both cases, not very. How round is a black hole?
If I had to choose I guess I'd say a perfect sphere is actually a spherical shell exactly 1 atom thick, with a radius chosen such that the gaps between every pair of neighboring atoms is equal. Of course such a sphere could only exist in a vacuum and with no external forces on it, and I don't know if interatomic bonding would even be strong enough.
For some reason this makes me want to build a sphere in Minecraft.
And I can't think of a good way to define it as perfectly round using physical terms. Calculus shows us that a finite collection of small segments can never be a continuous curve, no matter how small you take the segments to be, unless you take an infinite number of infinitely small segments. So what is the most round an object in nature can actually be? How round is a star? How round is a collection of phospholipids immersed in water? In both cases, not very. How round is a black hole?
If I had to choose I guess I'd say a perfect sphere is actually a spherical shell exactly 1 atom thick, with a radius chosen such that the gaps between every pair of neighboring atoms is equal. Of course such a sphere could only exist in a vacuum and with no external forces on it, and I don't know if interatomic bonding would even be strong enough.
For some reason this makes me want to build a sphere in Minecraft.
"Welding was faster, cheaper and, in theory,
produced a more reliable product. But sailors do
not float on theory, and the welded tankers had a
most annoying habit of splitting in two."
J.W. Morris
produced a more reliable product. But sailors do
not float on theory, and the welded tankers had a
most annoying habit of splitting in two."
J.W. Morris
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