Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started 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 spacetime 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?

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

Looked at otherwise Richard. How would you define it without time?

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

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