The Spat (unit) is the 3D counterpart of the turn (i.e. the solid angle of a sphere), equivalent to 4π (aka 2τ) steradians.
Relevant notes:
-
Lorentz–Heaviside units#Rationalization:
- Rationalized equations in physics generally have a factor related to the effective spatial symmetry:
- 2 for planar symmetry, 2π for cylindrical symmetry and 4π for spherical symmetry.
-
Planck units#Alternative choices of normalization:
- The factor 4π is ubiquitous in theoretical physics because the surface area of a sphere of radius r is 4πr2.
- If space had more than three spatial dimensions, the factor 4π would have to be changed according to the geometry of the sphere in higher dimensions.
-
Steradian#Analogy to radians:
- In two dimensions, an angle is related to the length of the circular arc that it spans.
- Similarly in three dimensions, the solid angle is related to the area of the spherical surface that it spans.
- In higher dimensional mathematical spaces, units for analogous solid angles have not been explicitly named.
When they are used, they are dealt with by analogy with the circular or spherical cases,
that is, as a proportion of the relevant unit hypersphere taken up by the generalized angle.
Note: this might be related to spinors in quantum mechanics
Conclusions: while 2τ (4π) does have more significance in 3D whereas τ (2π) is more significant in 2D, there is no purely mathematical reason to prefer the 3D instantiation of the constant, out of all the possible number of dimensions, except for the fact that our universe happens to be (apparently) 3D. This is in itself not insignificant, but it is circumstantial, and it's worth considering that these are symbols for human use, and humans have had many more applications for the 2D version than for the 3D one (IIUC). Besides, using τ and 2τ at least leaves the 2D unit "clean", without a multiplier.
The Spat (unit) is the 3D counterpart of the turn (i.e. the solid angle of a sphere), equivalent to 4π (aka 2τ) steradians.
Relevant notes:
-
Lorentz–Heaviside units#Rationalization:
- Rationalized equations in physics generally have a factor related to the effective spatial symmetry:
- 2 for planar symmetry, 2π for cylindrical symmetry and 4π for spherical symmetry.
-
Planck units#Alternative choices of normalization:
- The factor 4π is ubiquitous in theoretical physics because the surface area of a sphere of radius r is 4πr2.
- If space had more than three spatial dimensions, the factor 4π would have to be changed according to the geometry of the sphere in higher dimensions.
-
Steradian#Analogy to radians:
- In two dimensions, an angle is related to the length of the circular arc that it spans.
- Similarly in three dimensions, the solid angle is related to the area of the spherical surface that it spans.
- In higher dimensional mathematical spaces, units for analogous solid angles have not been explicitly named.
When they are used, they are dealt with by analogy with the circular or spherical cases,
that is, as a proportion of the relevant unit hypersphere taken up by the generalized angle.
Note: this might be related to spinors in quantum mechanics
Conclusions: while 2τ (4π) does have more significance in 3D whereas τ (2π) is more significant in 2D, there is no purely mathematical reason to prefer the 3D instantiation of the constant, out of all the possible number of dimensions, except for the fact that our universe happens to be (apparently) 3D. This is in itself not insignificant, but it is circumstantial, and it's worth considering that these are symbols for human use, and humans have had many more applications for the 2D version than for the 3D one (IIUC). Besides, using τ and 2τ at least leaves the 2D unit "clean", without a multiplier.