Higher Dimensional Beings

[drone77]

Assuming that higher or lower dimensional beings exist, how would a 2-dimensional being living on a 3 dimensional sphere with a finite radius perceive the sphere.

Will the being perceive the sphere as a finite radius circle that is a projection of the sphere or will the being perceive it as an infinitely large plane?

 

Edited March 19, 2019 by drone77

[studiot]

The being would be able to determine the fact that he was on a sphere, and the size of it, by measuring spherical excess.

http://mathworld.wolfram.com/SphericalExcess.html

[drone77]

8 minutes ago, studiot said:

The being would be able to determine the fact that he was on a sphere, and the size of it, by measuring spherical excess.

http://mathworld.wolfram.com/SphericalExcess.html

Spherical Excess will help to calculate the size of the sphere, given that the being is aware of the fact that its universe is 3 dimensional. 

As far as perception is concerned a 2 dimensional being should not be able to directly perceive the third dimension unless there is some procedure for determining the same.

Can you elaborate on how the being will determine the number of dimensions of its universe or how will the being determine the  fact that it is on a sphere ?

 

[QuantumT]

This is how I imagine a (lower and) higher dimension:

2D (+ time) is out of the question. They do not have atoms and molecules to make them exits, or light to make their world visible.

4D (+ time) would probably be like our 3D, but without geometric laws of volume as we know them.
One could imagine carrying your house in your backpack. Like a snail. But the house would be live size once you pull it out.

[drone77]

19 minutes ago, QuantumT said:

This is how I imagine a (lower and) higher dimension:

2D (+ time) is out of the question. They do not have atoms and molecules to make them exits, or light to make their world visible.

4D (+ time) would probably be like our 3D, but without geometric laws of volume as we know them.
One could imagine carrying your house in your backpack. Like a snail. But the house would be live size once you pull it out.

My question actually is: Consider ourselves. We live in a 4 dimensional space considering time also as a dimension. Let us assume that in reality our universe is finite and 5 dimensional. So, ofcourse, we wouldn't be able to perceive the fifth dimension directly. (Not speaking about any abnormalities in the physical laws which may point towards a higher dimensional universe and thereby help us to conclude that our universe is in fact 5 dimensional. ) 

So the universe that we will be perceiving will be just a finite projection of the 5 dimensional universe (for better visualization I used a sphere(3D))  or will the universe be perceived as something 4 dimensional and infinite?

Edited March 19, 2019 by drone77

[Sensei]

28 minutes ago, QuantumT said:

2D (+ time) is out of the question. They do not have atoms and molecules to make them exits, or light to make their world visible.

Game of life is working fine in 2D (+time)..

https://en.wikipedia.org/wiki/Conway's_Game_of_Life

https://en.wikipedia.org/wiki/Life_simulation_game

 

 

Edited March 19, 2019 by Sensei

[QuantumT]

3 minutes ago, Sensei said:

Life simulation game is working fine with 2D (+time)..

https://en.wikipedia.org/wiki/Life_simulation_game

Excellent point! But isn't it a projection from a 3D (+t) reality?

If you pull the computer's plug it disappears.

[studiot]

1 hour ago, drone77 said:

My question actually is: Consider ourselves. We live in a 4 dimensional space considering time also as a dimension. Let us assume that in reality our universe is finite and 5 dimensional. So, ofcourse, we wouldn't be able to perceive the fifth dimension directly. (Not speaking about any abnormalities in the physical laws which may point towards a higher dimensional universe and thereby help us to conclude that our universe is in fact 5 dimensional. ) 

So the universe that we will be perceiving will be just a finite projection of the 5 dimensional universe (for better visualization I used a sphere(3D))  or will the universe be perceived as something 4 dimensional and infinite?

I recomment you stick with your 3D sphere analogy.

It is much easier to get your head around.

Further please note it is impossible to map a sphere onto a plane so beware of 'projection'.

2 hours ago, drone77 said:

Spherical Excess will help to calculate the size of the sphere, given that the being is aware of the fact that its universe is 3 dimensional. 

As far as perception is concerned a 2 dimensional being should not be able to directly perceive the third dimension unless there is some procedure for determining the same.

Can you elaborate on how the being will determine the number of dimensions of its universe or how will the being determine the  fact that it is on a sphere ?

 

There are two types of 'higher dimensional' curvature.

Intrinsic curvature which is the type I was describing. This has direct effects, such as spherical excess, within the lower dimension.
Any being of the lower dimension would be able to detect these effects, just as surveyors and navigators detect spherical excess on the surface of the Earth.

The other type is called extrinsic curvature and can't be detected by any means available to the lower dimensional being.

 

Have you understood spherical excess or do you need more help with it?

 

[Sensei]

5 hours ago, QuantumT said:

If you pull the computer's plug it disappears. 

..you should rather beg for not "pulling out of the plug".. I think...

 

Edited March 20, 2019 by Sensei

[MigL]

Are you sure that isn't the other way around Studiot ?

In the case of extrinsic curvature, where we have 'embedding', the Pythagorean theorem doesn't hold, and can be used to verify deviation from Euclidian. But the relation can be restored by adding an additional dimension extending into the embedding space.

IE     if      dx^2 + dy^2 =/= dh^2       then
                 dx^2 + dy^2 + de^2 == dh^2

where h is the hypotenuse, and e is the embedding dimension

[studiot]

9 hours ago, MigL said:

Are you sure that isn't the other way around Studiot ?

In the case of extrinsic curvature, where we have 'embedding', the Pythagorean theorem doesn't hold, and can be used to verify deviation from Euclidian. But the relation can be restored by adding an additional dimension extending into the embedding space.

IE     if      dx^2 + dy^2 =/= dh^2       then
                 dx^2 + dy^2 + de^2 == dh^2

where h is the hypotenuse, and e is the embedding dimension

Yes I'm quite sure.

We are talking about Gaussian curvature here; Gaussian curvature is intrinsic.

Here is a much better article than Wiki, with lots of explanatory references and examples.

https://www.maths.ox.ac.uk/about-us/departmental-art/theory/differential-geometry

 

Note that curvature is not the only property with extrinsic /intrinsic classification.

Intrinsic Property

Def: Relating only to the bearer of the property, not to the space in which it is embedded.

Extrinsic Property

Def: Relating only to the space in which it is embedded, not to the bearer of the property.

 

Chirality or handedness is one such, which also shows how something (a plane in the example) can have one property extrinsic, but not another.

The 'intrisic equations' of a plane curve can distinguish one curve from another.

But they cannot distinguish chirality for instance for a left and right handed spirals.

Chirality is a property of the space (Effectively because the direction of one of the axes is reversed).

[MigL]

Let's stick with curvature for the moment, without getting further confused by chirality.

Consider a donut shape ( torus ), embedded in a higher dimension, such that its curvature is extrinsic.
On the inner part, next to the hole, the curvature is negative, and a triangle will have angles that sum to LESS than 180 deg.
While on the outer periphery, where curvature is positive, the triangle's angles sum to MORE than 180 deg.

Now consider a flat torus, where curvature is intrinsic, and, like the old Asteroids game, the points on one side identify with the points on the other side. When you travel to one end of the space, you reappear ( without rotation or reflection ) on the opposite side, and continue on.
Any triangle drawn on this intrinsically curved space will have 180 deg even though it is topologically curved ( that's where 'flat' comes from in flat torus ).

Which case has discernible curvature ?
( topology is not my strongest field, but the above is my understanding, and I'm open to correction )

[studiot]

2 hours ago, MigL said:

Let's stick with curvature for the moment, without getting further confused by chirality.

Consider a donut shape ( torus ), embedded in a higher dimension, such that its curvature is extrinsic.

Did you actually read my link ?

So I thought with your first line you wanted to simplify.
But then you introduced a torus.

 

What makes you think a torus curvature is extrinsic?

 

Let me simplify.

 

Consider an annulus, which is a plane figure.

Can you calculate the curvature of that anulus using only data from the plane?

Now embed that plane in a three dimensions.

What has changed?

Does the addition of the third dimension allow calculation of a different curvature?

 

Edited March 20, 2019 by studiot

[MigL]

Ahh, I see.
If you are 'part' of the curvature, you can only measure it if it is intrinsic ( using the accepted methods, metrics/geodesics ); you cannot 'step outside' into the embedding dimension to measure extrinsic.