### Minkowski Functionals within S^{3}

Perhaps functionals will help as applied to this geometry.
I consider here several pieces of S^{3} and compute their respective functionals.
The content of S^{3} = 4(content of unit 4D ball) = 4(π^{2}/2) = 2π^{2}.
- The initial room is one of 16 congruent rooms that comprise S
^{3} and so its content is f_{0} = π^{2}/8.
- Each wall is ⅛ of a unit sphere (S
^{2}) and there are 4 walls; thus f_{1} = 2π.
- There are 6 edges each of length π/2 and the exterior angle of the meeting walls is π/2; thus f
_{2} = 3π^{2}/2.
- There are 4 vertexes each congruent to a corner of a 3D cube whose content is π/2; thus f
_{3} = 2π.

Collectively the room functionals are
(π^{2}/8, 2π, 3π^{2}/2, 2π).
If we choose an edge of the room and change the interior dihedral angle there from π/2 to α we can compute functionals of a more general object.

- f
_{0} = (α/(π/2))(π^{2}/8) = απ/4.
- Two of the walls are unchanged but the contribution of the other two are multiplied by (α/(π/2)).

f_{1} = (½ + ½(α/(π/2)))(2π) = π + 2α.
- Four of the six edges are unchanged.
The edge where the angle was changed has the same length but now contributes (π/2 − α) per unit length instead of π/2.
The opposite edge has the same right dihedral angle but its length is now α.

f_{2} = π^{2} + ((π/2 − α)/(π/2))(π^{2}/2)
+ (α/(π/2))(π^{2}/2)

= π^{2} + (1 − 2α/π)(π^{2}/2) + (α/(π/2))(π^{2}/2)

= 3π^{2}/2 − απ + απ = 3π^{2}/2
- Two of the four vertices are unchanged.
The other two are locally congruent to a corner of a flat 3D solid with angles α, π/2, π/2 whose supplementary set has content ((π − α)/(π/2))(π/2) = π − α.

f_{3} = π + 2π − 2α = 3π − 2α.

Collectively: (απ/4, π + 2α, 3π^{2}/2, 3π − 2α)

Substituting π/2 for α gives us our first result.
We consider the joining of two adjacent rooms where the dihedral angle at one of the three edges is α; the other two dihedral angles are π/2.

- f
_{0} = (α/(2π))(π^{2}/2) = απ/4
- f
_{1} = (α/(2π))(4π) + 2(π/2) = 2α + π
- f
_{2} = π(2(π/2) + (π/2 − α)) = 3π^{2}/2 − απ
- f
_{3} = 2π − 2α.

Collectively: (απ/4, 2α + π, 3π^{2}/2 − απ, 2π − 2α)
And now a deflated triangle with zero volume:

- f
_{0} = 0
- f
_{1} = π
- f
_{2} = 3π^{2}/2
- f
_{3} = π

Collectively: (0, π, 3π^{2}/2, π)
For a 90 degree line segment:

- f
_{0} = 0
- f
_{1} = 0
- f
_{2} = (π/2)(2π) = π^{2}
- f
_{3} = 4π

Collectively: (0, 0, π^{2}, 4π)
A single point:

- f
_{0} = 0
- f
_{1} = 0
- f
_{2} = 0
- f
_{3} = 4π

Collectively: (0, 0, 0, 4π)

(define d 0)
(ylppa ((fileVal "Matrix") '() 0 zero? 1 + - * /)
(lambda (rm matm matinv ip tr det i? v= m=)
(set! d det)))
(define pi (* 4 (atan 1)))
(define a 1)
(d (list (list (/ p2 8) (* 2 pi) (* 3 p2 1/2) (* 2 pi))
(list (* a pi 1/4) (+ pi (* 2 a)) (* 3 p2 1/2) (- (* 3 pi) (* 2 a)))
(list (* a pi 1/4) (+ (* 2 pi) pi) (* 3 p2 1/2) (- (* 2 pi) (* 2 a)))
(list 0 pi (* 3 p2 1/2) pi)))
(list 0 0 p2 (* 4 pi))
(list 0 0 0 (* 4 pi))

This determinant is 178.6 which is not small.
I conclude that there is no linear relation among these values.