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Geometry:what is the opposite of a sphere?& what shape maximises surface area/volume?
01-31-2019, 06:07 PM (This post was last modified: 01-31-2019 06:15 PM by StephenG1CMZ.)
Post: #1
Geometry:what is the opposite of a sphere?& what shape maximises surface area/volume?
Over on quora I just saw an interesting question: What is the opposite of a sphere?

My non-technical guess was all the space outside the sphere (like a torus but in 3D). That has the advantage of being easily visualisable in 3D.

But what is the defining characteristic of a sphere that we want the opposite of?
Some answers suggested a negative curvature, and that sounds mathematically interesting, but I'm not yet visualising how that looks.
Other answers picked on the number of corners, but I don't think it has a definite number. It seems to me you can make a case for both zero and infinity.

Another answer suggested that a sphere minimises suface area/volume, so its opposite should instead maximise that ratio...But what shape has that characteristic?

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01-31-2019, 06:47 PM
Post: #2
RE: Geometry:what is the opposite of a sphere?& what shape maximises surface area/...
.
Well, the mathematical 3D object called Gabriel's Horn has finite volume but infinite area.

That would surely maximize area/volume.

V.
.

  
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01-31-2019, 06:56 PM
Post: #3
RE: Geometry:what is the opposite of a sphere?& what shape maximises surface area/...
(01-31-2019 06:07 PM)StephenG1CMZ Wrote:  What is the opposite of a sphere?

The pseudosphere has constant negative Gaussian curvature (rather than the constant positive curvature of the sphere), leading to its name.

Quote:My non-technical guess was all the space outside the sphere (like a torus but in 3D). That has the advantage of being easily visualisable in 3D.

This still has positive curvature.

Quote:But what is the defining characteristic of a sphere that we want the opposite of?
Some answers suggested a negative curvature, and that sounds mathematically interesting, but I'm not yet visualising how that looks.

[Image: Pseudosphere_700.gif]

Locally it looks like a saddle:

[Image: universe_geometry.gif]

Quote:Other answers picked on the number of corners, but I don't think it has a definite number. It seems to me you can make a case for both zero and infinity.

Another answer suggested that a sphere minimises surface area/volume, so its opposite should instead maximise that ratio...But what shape has that characteristic?

Gabriel's Horn has finite volume but infinite surface area:

[Image: GabrielsHorn_800.gif]

Cheers
Thomas
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01-31-2019, 09:29 PM
Post: #4
RE: Geometry:what is the opposite of a sphere?& what shape maximises surface area/...
(01-31-2019 06:47 PM)Valentin Albillo Wrote:  Well, the mathematical 3D object called Gabriel's Horn has finite volume but infinite area.

That would surely maximize area/volume.

(01-31-2019 06:56 PM)Thomas Klemm Wrote:  Gabriel's Horn has finite volume but infinite surface area:

[Image: GabrielsHorn_800.gif]

Whoa, that's the coolest thing, that I can't fully process, that I've heard all week.

From the linked article:

Quote:This leads to the paradoxical consequence that while Gabriel's horn can be filled up with pi cubic units of paint, an infinite number of square units of paint are needed to cover its surface!

Thanks for the thought provoking question and these answers.

And now... how the heck do ya hold this thing up to paint it....

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02-01-2019, 07:34 AM
Post: #5
RE: Geometry:what is the opposite of a sphere?& what shape maximises surface area/...
(01-31-2019 09:29 PM)rprosperi Wrote:  
(01-31-2019 06:47 PM)Valentin Albillo Wrote:  Well, the mathematical 3D object called Gabriel's Horn has finite volume but infinite area.

That would surely maximize area/volume.

(01-31-2019 06:56 PM)Thomas Klemm Wrote:  Gabriel's Horn has finite volume but infinite surface area:

[Image: GabrielsHorn_800.gif]


Whoa, that's the coolest thing, that I can't fully process, that I've heard all week.

From the linked article:

Quote:This leads to the paradoxical consequence that while Gabriel's horn can be filled up with pi cubic units of paint, an infinite number of square units of paint are needed to cover its surface!

Thanks for the thought provoking question and these answers.

And now... how the heck do ya hold this thing up to paint it....

Well, the inner surface should be already painted up once you fill it, no?

Greetings,
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02-01-2019, 10:23 AM (This post was last modified: 02-01-2019 10:24 AM by Pekis.)
Post: #6
RE: Geometry:what is the opposite of a sphere?& what shape maximises surface area/...
I would'nt try to blow in this horn ... Smile
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02-02-2019, 05:25 AM (This post was last modified: 02-02-2019 05:26 AM by Mark Hardman.)
Post: #7
RE: Geometry:what is the opposite of a sphere?& what shape maximises surface area/...
(01-31-2019 06:56 PM)Thomas Klemm Wrote:  Gabriel's Horn has finite volume but infinite surface area:

[Image: GabrielsHorn_800.gif]

Cheers
Thomas

In the Prime's Graph 3D App:

[Image: attachment.php?aid=6882]

√(1/(X^2+Y^2))

X=-1..1
Y=-1..1
Z= 1..10


Attached File(s) Thumbnail(s)
   

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02-02-2019, 11:04 PM (This post was last modified: 02-02-2019 11:12 PM by StephenG1CMZ.)
Post: #8
RE: Geometry:what is the opposite of a sphere?& what shape maximises surface area/...
A different thread on Quora attempts to explain the apparent paradox
https://www.quora.com/Why-does-Gabriels-...m-possible

Whereas mathematicians assume the surface is infinitely thin, physicists know that paint has a thickness, and if the paint gets thinner as x gets smaller, that helps... Apparently.

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02-05-2019, 11:04 PM (This post was last modified: 02-05-2019 11:20 PM by Vtile.)
Post: #9
RE: Geometry:what is the opposite of a sphere?& what shape maximises surface area/...
That horn thing is interesting, kind of what diracs delta is as function with area of 1, but value on infinity.

Similar paradox as this horn is ie. with complex infinities. There is ie. real part as constant, but its significance coverges to zero, while the complex part go to infity. Again don't reference me to your phd research.

The obvious answer for OP is cube. Tongue


PS. Give me gabriels horn and I will show how to paint it with finite amount of paint.. Easy just fill it with exactly the volume amount of paint and all the surface area is painted. Prove that I'm wrong.
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02-06-2019, 07:08 AM (This post was last modified: 02-06-2019 07:09 AM by Massimo Gnerucci.)
Post: #10
RE: Geometry:what is the opposite of a sphere?& what shape maximises surface area/...
(02-05-2019 11:04 PM)Vtile Wrote:  PS. Give me gabriels horn and I will show how to paint it with finite amount of paint.. Easy just fill it with exactly the volume amount of paint and all the surface area is painted. Prove that I'm wrong.

See #5... ;)

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02-06-2019, 08:31 AM
Post: #11
RE: Geometry:what is the opposite of a sphere?& what shape maximises surface area/...
(02-05-2019 11:04 PM)Vtile Wrote:  PS. Give me Gabriel's horn and I will show how to paint it with finite amount of paint. Easy just fill it with exactly the volume amount of paint and all the surface area is painted. Prove that I'm wrong.

Quote:This leads to the paradoxical consequence that while Gabriel's horn can be filled up with \(\pi\) cubic units of paint, an infinite number of square units of paint are needed to cover its surface!

You're not wrong. Still the area is infinite. Which makes it paradox as it contradicts our everyday experience of how paint works.
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02-06-2019, 08:54 PM
Post: #12
RE: Geometry:what is the opposite of a sphere?& what shape maximises surface area/...
(02-06-2019 08:31 AM)Thomas Klemm Wrote:  
(02-05-2019 11:04 PM)Vtile Wrote:  PS. Give me Gabriel's horn and I will show how to paint it with finite amount of paint. Easy just fill it with exactly the volume amount of paint and all the surface area is painted. Prove that I'm wrong.

Quote:This leads to the paradoxical consequence that while Gabriel's horn can be filled up with \(\pi\) cubic units of paint, an infinite number of square units of paint are needed to cover its surface!

You're not wrong. Still the area is infinite. Which makes it paradox as it contradicts our everyday experience of how paint works.

You just need an infinitesimal layer of paint.

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02-06-2019, 09:53 PM (This post was last modified: 02-06-2019 10:06 PM by Vtile.)
Post: #13
RE: Geometry:what is the opposite of a sphere?& what shape maximises surface area/...
Haha, One should not post anything at half sleep way over the bed time. Tongue So why not continue when I again broke the rule anyway.

But... How much paint you can pour out from the horn?


Hmm... The Horn if such could somehow could exist it wouldn't have infinite surface and finite volume if the blanks length is really the smallest unit in universe (Both would be infinite). Do my half sleep brains got it right. Smile ..zzzzZZZZZ
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02-09-2019, 10:43 PM
Post: #14
RE: Geometry:what is the opposite of a sphere?& what shape maximises surface area/...
Hello!

You can use the Koch curve (see https://en.wikipedia.org/wiki/Koch_snowflake) for finite area but with infinite perimeter in a 2d plane, if you start with a triangle.

I think you can expand this principe to 3d objects like the tetrahedron.
Each of the triangle faces can split in 4 similar but smaller triangle (https://upload.wikimedia.org/wikipedia/c...at.svg.png).

If you replace for the inner triangle a new smaller tetrahedron again, you get the a 3d version of the Koch curve.

And if we repeat this process infinite we get a solid with finite volume, but infinite surface.

I think you can do this with nearly spheric object like the football (https://en.wikipedia.org/wiki/Buckminsterfullerene) too. You must only divide the faces in similar pieces and replace it with a part of a smaller version of a 3d object. The only condition is: The smaller 3d object must have plane faces like the original object.

Don't hesitate to correct me, if I'm not right.

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02-17-2019, 05:10 PM
Post: #15
RE: Geometry:what is the opposite of a sphere?& what shape maximises surface area/...
I said earlier that one could argue that a sphere either had 0 edges/faces or an infinite number.
I was imagining starting with a line, and adding more sides - a tringle, square, polygon and circle, only in 3D.

That seemed iintuitive, but I have just now happened up some references to a regular polyhedron having a maximum of 120 sides, which surprised me...without having looked at the details I would have guessed one could always add more mathematically.

Right now I'm no longer certain that a sphere could have an infinite number of faces, or whether the maximum number of faces has any relevance to spheres. But I am sure someone here can explain.

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02-17-2019, 06:09 PM
Post: #16
RE: Geometry:what is the opposite of a sphere?& what shape maximises surface area/...
(02-17-2019 05:10 PM)StephenG1CMZ Wrote:  Right now I'm no longer certain that a sphere could have an infinite number of faces, or whether the maximum number of faces has any relevance to spheres. But I am sure someone here can explain.

You can still do a tessellation of the sphere:

[Image: Spherical-tessellation-grids-and-parent-...nitial.png]

However the triangles aren't regular.

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02-17-2019, 08:21 PM
Post: #17
RE: Geometry:what is the opposite of a sphere?& what shape maximises surface area/...
(02-17-2019 05:10 PM)StephenG1CMZ Wrote:  Right now I'm no longer certain that a sphere could have an infinite number of faces, or whether the maximum number of faces has any relevance to spheres. But I am sure someone here can explain.

One thing you can be sure of is that it will satisfy the Euler characteristic for a polyhedron: F + V = E + 2. For tessellating a sphere, the simplest case is one face and one vertex, but no edges.

— Ian Abbott
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