Articles such as this one follow in the footsteps of great educators such as David Hilbert. Your work here might inspire someone to invent a 3D interferometer... and with that comes the discovery of a space-time structure to fit a unified field.
A Mobius band will be oriented either left or right handed depending on which way the 1/2 twist is made. As with a spring, the twist and the closing of the loop locks in energy; in this respect the Mobius band resembles a unit of 1/2 spin (as it pertains to subatomic physics). If we cut this "spin unit" down the middle it becomes a spin of 1 (a full twist). Cutting the band off center makes a spin of 1 and 1/2. Gluing two bands together should make a spin of net zero or plus/minus 1... my math skills fall short at that point.
The Penrose Twistor is envisioned as an nth dimensional extension of vectors & tensors, and seems to become a Mobius/Klein type of topology; this could be the unified field model long sought by physicists, yet the math proved too difficult even for Roger Penrose.
I might have a way of simplifying the math in either case. This requires a redefinition of pi as the square root of 6 times Reimann's Zeta(1) value. However, the term 6 represents a structure of six Klein bottles interconnected/looped at their handles as to form a type of xyz axial system that allows all three axes to represent complex numbered coordinates. This pushes the difficult/impossible math toward the center (at the handles), and makes the exterior (pseudo-spheres of the Klein bottles) as simple as adding complex numbers.
With three values, 81, the speed of light, and a secret "uncertainty" number, I can make this 6-lobed structure mathematically represent any Lepton or Quark (by spin, mass, and charge). Pretty neat, huh? The only great difficulty is finding empirical evidence of such a 6-lobed structure in nature (doh!). That won't happen until someone invents the detector, a similarly 6-lobed interferometer!
PS a couple of other difficulties are we need 3.14 space dimensions and the value of pi gets smaller (becomes 2) at the center of the structure.