Naked Science Forum

Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Richard777 on 27/10/2019 15:06:08

Title: Can a spherical region in 3D be represented using four dimensions?
Post by: Richard777 on 27/10/2019 15:06:08
A spherical surface in 3D space (x,y,z) with radius (r) is represented as a simple equation.
   x^2 + y^2 + z^2 = r^2
The regions within and beyond the surface are undefined.
In order to define the inner region bounded by the surface, assume one more dimension (λ) must be included. A spherical region having a constant surface radius (a) and components (λ,x,y,z) may have an enclosed region defined as;
   λ^2 + x^2 + y^2 + z^2 = a^2
   λ^2 + r^2  = a^2
Where;   a is assumed to be constant 
r = 0 and; λ = a represents the center of the spherical region    
r = a and; λ = 0 represents the surface of the spherical region
r<a and; λ<a for any point within the surface
   r and λ are complex beyond the surface
The wave dimension (λ) may be written as;   λ = cT   (where; T is time and; c is the light constant)
A spherical region having a constant surface radius (a) and space-time components (cT,x,y,z) may be defined as;
   cT^2 + x^2 + y^2 + z^2 = a^2
Can a spherical region in 3D be represented using four dimensions?

Title: Re: Can a spherical region in 3D be represented using four dimensions?
Post by: alancalverd on 27/10/2019 17:05:27
No need for an additional dimension. Any point inside or outside the sphere will just have different coordinate values x' y' z'.

The "inner region bounded by the surface" is simply a volume containing all those points for which r' < r
Title: Re: Can a spherical region in 3D be represented using four dimensions?
Post by: yor_on on 27/10/2019 19:53:46
Looked at otherwise Richard. How would you define it without time?
Title: Re: Can a spherical region in 3D be represented using four dimensions?
Post by: evan_au on 28/10/2019 09:27:50
Quote from: OP
Can a spherical region in 3D be represented using four dimensions?
The Holographic Principle suggests that it is possible to represent a region of 3D space by using two dimensions, in certain specific instances such as:
- matter falling into a black hole, where all of the information is captured on the 2D surface of the black hole
- It may also apply at the level of the entire universe
- It was derived from considerations of thermodynamics and entropy
- But at this time, you would have to describe it as speculation, as it has not really been proven

See: https://en.wikipedia.org/wiki/Holographic_principle
Title: Re: Can a spherical region in 3D be represented using four dimensions?
Post by: Bored chemist on 08/12/2019 21:50:46
A spherical surface in 3D space (x,y,z) with radius (r) is represented as a simple equation.
   x^2 + y^2 + z^2 = r^2
The regions within and beyond the surface are undefined.

The regions within and beyond the surface are undefined as
x^2 + y^2 + z^2 < r^2 and
x^2 + y^2 + z^2 > r^2
respectively