# How can a sphere be transformed using differential geometry?

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• James1355
James1355
TL;DR Summary
Transformation of a sphere
Any given sphere surface consists of a finite number of fixed points. If all these points on the surface were to rotate/flip in their locations by 180° in respect to the centre of the sphere simultaneously and hence making the entire sphere turn outside in, how do you go about formulating this through differential geometry?

James1355 said:
TL;DR Summary: Transformation of a sphere

Any given sphere surface consists of a finite number of fixed points.
No. an infinite number of points
If all these points on the surface were to rotate/flip in their locations by 180° in respect to the centre of the sphere simultaneously and hence making the entire sphere turn outside in
You can't 'flip a point'. it has no front or back.
how do you go about formulating this through differential geometry?

##\ ##

My understanding is that any sphere with a fixed radius can only have finite number of points on its surface. Since infinite number of points (as you pointed out), would imply infinitely growing radius?

Although a point doesn't have a front or back, it can be flipped by swapping the X and Y and keeping Z

James1355 said:
My understanding is that any sphere with a fixed radius can only have finite number of points on its surface. Since infinite number of points (as you pointed out), would imply infinitely growing radius?
I most certainly did not ! Any finite area contains an infinite number of points !

Although a point doesn't have a front or back, it can be flipped by swapping the X and Y and keeping Z
Ah, that's what you call rotate/flip ? I would call it two reflections
And there's no 'turn outside in' !

##\ ##

Ibix
An inversion with respect to the surface of the sphere will send $(r,\theta,\phi) \mapsto (a^2r^{-1},\theta,\phi)$; this acts as the identity on the sphere $r = a$.

Ishika_96_sparkles and James1355
BvU said:
I most certainly did not ! Any finite area contains an infinite number of points !Ah, that's what you call rotate/flip ? I would call it two reflections
And there's no 'turn outside in' !

##\ ##
Oh yes there's, absolutely.
I actually found the answer! In fact Stephen Smale PhD has done a fantastic work called "eversion of a sphere" that shows this is possible. I suggest you have a read too, it would greatly enhance your outlook towards solving problems and soften the defensiveness.
And lastly I hope you enjoy your delicious humble pie

We are shifting from differential geometry to differential topology. Fine with me, but it feels like changing the subject to me.

Did Smale PhD by happenstance also corroborate your
James1355 said:
Any given sphere surface consists of a finite number of fixed points.

Seriously: I liked the topological eversion lemma. But it's a long stretch () away from your post #1 BvU PhD
##\ ##

James1355 said:
TL;DR Summary: Transformation of a sphere

Any given sphere surface consists of a finite number of fixed points.
How many?

n

On a finite real number line interval [0,1] , how many integral ##n## jumps would it take to reach from 0 to 1?
does this question make sense? Since, the interval is defined over the reals ##\mathbb{R}## , there are no integers in between that could be put in correspondence with ##\mathbb{Z}##.
In a unit sphere in 3D real Euclidean space, the total curved surface area is given by ##4\pi##... can we put every point on this sphere in 1-to-1 correspondence with the ##\mathbb{Z}##? as we have already define the space to be real.

Suppose, we have a 2D plane embedded in a 3D volume. is there a difference in coordinates if i look from above or look from below the sheet?

Does the flip by ##\pi## radians along the horizontal axes passing through the plane differentiate between the points lying "above" the sheet and "below"?

Please correct me if I am wrong. I just wanted to take part in the discussion.

Thanks for jumping in !

I'm an experimental physicist, and understand you mathematician folks here are looking at this problem from your angle - quite rightly so.

In a practical world any given hollow sphere can only have a Finite number of particles forming its surface. Let's imagine this proverbial sphere does consist of a lattice of one layer thick atoms. Given the void between the individual atoms, it must be possible to formulate a topological expression (as @BvU corrected me in the thread above), whereby all the atoms of the surface collapse diametrically towards the core, slipping past each other, and emerge inside out (outside in).

I know there's a solution to this in differential topography proposed by Stephen Smale in 1959 called "Eversion of a Sphere".

But the reason why I posted my question on this forum was to pick the brain of the mathematicians like yourself to explore a version of this Eversion that is closer to the finite number of constituent particles.

James1355 said:
I'm an experimental physicist
Nice!

Alright, then it is agreed that in our real world, all material objects are formed from a finite number of particles (atoms or molecules). Thus, a pure metallic sphere would have finite atoms spanning its curved surface area and we could in fact put these atoms in correspondence with the set of integers ##\mathbb{Z}##.

Suppose, our sphere is composed of a thin layer of material, say Graphene, and hence a finite number of carbon atoms i.e., ##n## form a hexagonal (ortho) lattice structure. What would an eversion of this material sphere entail as far as physical observations are concerned? Is this what interests you?

Sorry for the off topic, but how can someone say that a sphere has finitely many fixed points, and then that he has read Smales PhD thesis!? Here is a pdf.

martinbn said:
Sorry for the off topic, but how can someone say that a sphere has finitely many fixed points, and then that he has read Smales PhD thesis!? Here is a pdf.
He has mentioned he is an experimental physicist and made a point that in real world we cannot have infinite atoms on a spherical ball unless the radius is infinite. His interest seems to be in conflict with the Smale's work (of which i have no understanding) and the real world objects.

James1355 said:
all the atoms of the surface collapse diametrically towards the core, slipping past each other, and emerge inside out
what physical marker would be present to tell the "inside" from the "outside" except the change in the shape of the sphere during the process? Suppose, we have a 2D "rubber-band" made up of single atom thin graphene layer. How would we be able to tell the difference, physically, between the non-inverted/inverted band state i.e., before and after? I assume that this change is happening in the physical space of the object. Do you look for the physical effects in configuration space and phase space as well?

martinbn said:
Sorry for the off topic, but how can someone say that a sphere has finitely many fixed points, and then that he has read Smales PhD thesis!? Here is a pdf.
No please, I genuinely welcome all points of view, in fact that's why I posted here - otherwise discussing this notion amongst my fellow physicists would only be the recital of what we already know/ how we think.

Ishika_96_sparkles said:
Nice!

Alright, then it is agreed that in our real world, all material objects are formed from a finite number of particles (atoms or molecules). Thus, a pure metallic sphere would have finite atoms spanning its curved surface area and we could in fact put these atoms in correspondence with the set of integers ##\mathbb{Z}##.

Suppose, our sphere is composed of a thin layer of material, say Graphene, and hence a finite number of carbon atoms i.e., ##n## form a hexagonal (ortho) lattice structure. What would an eversion of this material sphere entail as far as physical observations are concerned? Is this what interests you?
The material per-se is irrelevant to the point. It's the mathematical function that I'm interested in.

How about breaking this problem into two components; imagine we have a semisphere made of rubber, painted red on one side and blue on the opposite. We can turn it inside out/outside in regardless of the finite/infinite number of paint particles. This assumption is intuitively true. This surely can be expressed in the language of mathematics?

And if so, from here the approach could be replicated for the other half (semisphere) to produce the desired result.

Ishika_96_sparkles said:
He has mentioned he is an experimental physicist and made a point that in real world we cannot have infinite atoms on a spherical ball unless the radius is infinite. His interest seems to be in conflict with the Smale's work (of which i have no understanding) and the real world objects.
Aha! There shouldn't be. After all the logical sequence present itself as pure maths, theoretical physics, experimental, material science. To make it happen it has to make sense/work mathematically first

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