Patterns Free to Point in Any Direction: Slices, Traces, Transits and Two Cuts of One Object

A direction is something a pattern vibrates in. A particle pattern of depth d vibrates in d bound directions, and those directions can point in any direction of space, including out of the three we observe. A photon is the two-direction case, which is why light travels in every direction of our three. This paper works out what an observer confined to three directions sees of such a pattern. The core of a depth-d pattern meets our three in a set of dimension 3 − d, and in space and time 4 − d: depth two leaves a line, depth three a lasting point, depth four a single instant. The average squared share of an observed direction lying in the pattern is d/D. A tilted free direction moves the trace to distance h tan φ, through infinity and back. A bent free line crosses our three in pairs that are born together and separate as √t. A field packet moving along a hidden direction appears in our three and vanishes again while its total energy stays fixed, with the slice energy balanced exactly by the flux through the hidden direction. One vibration plane cut at two places gives the correlation −cos θ and the CHSH value 2√2. Each count of directions m carries its own sphere Sm−1, and the right-angle field counts are 1, π/2, 2, 3π/4, 8/3 for m = 2 to 6.