wall to wall

[AntonioLao]

A complete or total occupation of space and time is a wall to wall covering of reality whose topology has zero genuses. It is defined as the maximum number a surface can be cut along simple closed curves without the surface separating into disconnected parts. In the theory of surfaces genus is a topological invariance that remains the same under continuous deformation and two surfaces of equal genus can be transformed into each other, for example, a cube into a sphere, vice versa but not a sphere into a torus or a torus into a sphere. Without giving a proof, the only caveat is that these surfaces are ordinary “two-sided” surfaces that satisfy the Euler characteristic invariance given by V-E+F=2-2G where V is the number of vertices, E is the number of arcs, F is the number of regions, and G is the genus. For a cube (it is a Platonic solid) with V=8, E=12, and F=6, G is zero. G is also zero for the other four Platonic solids: tetrahedron, octahedron, icosahedron, and dodecahedron.

On the other hand, the topological invariance of “one-sided” surfaces, the Euler characteristic is given by V-E+F=2-G. This is true for both the two dimensional Möbius strip and the three dimensional Klein bottle. A person living in the immediate vicinity of a Klein bottle universe can paint it entirely simply with one color, say red, from wall to wall using right-handed strokes but at the end of each cycle the strokes are miraculously transformed into left-handed strokes in plain sight of the painter’s bewilderment. This is analogous to the sudden appearance of the painter’s heart located on the right instead of on the left. Nevertheless, a heartless painter can paint forever from wall to wall without worrying the definite location of the heart as a consequence of directional invariance of infinitesimal squares of energy of quantum vacuum fluctuations of zero-point energies represented by symmetric singular Hadamard matrices of any order or dimension.