Beyond the cube

In a column point set every point sits on the diagonal of its cell, at a height that depends on its column. Two neighbouring columns are close when their heights differ by about 2/3 of a period: then their points are only √(2/3) cells apart. If g is how far the height steps stay from 2/3 of a period, the minimum distance is δN^1/3 = √(2/3 + 3g²), measured in periods.

To beat the cube value 1 you need g > 1/3, so every step would have to lie strictly between 0 and 1/3 of a period. But three steps, along e_x, then e_y, then e_z, lead back to the same column, so they add up to a whole number of periods, and three numbers between 0 and 1/3 cannot do that. The cube ℤ³ (steps 0) and its half-turn about (1,1,1) (steps 1/3) sit exactly on the ceiling. Move the first two steps and watch the third follow.

In the two-coset net 𝒯_p, y and z read the same geometric digit, so they always wrap together, while x reads h(n_y + n_z) with h ≡ −1/2: neighbouring cells alternate between two heights half a cell apart. The slabs n_x = const are puckered square grids, sheared by 1/8 from one slab to the next, and the two kinds of contact balance at the squared length 33/32. A coordinate that can wrap alone cannot move by 1 + α with a small α ≠ 0 along its own axis, because where it wraps alone the same step moves it by α only. So the free coordinate x moves by exactly one cell along e_x, up to the jitter.

In the rank-one net 𝒲_p, z reads a digit ζ and x and y both read 2ζ. So z wraps only where x and y wrap too: all three wraps are locked. That buys room for a drift: the step along e_z is (4ε, 4ε, 1 + 2ε) with ε = ρ/(2p) close to 1/48, longer than one cell, and where z wraps it is still long because x and y wrap with it. The slab view puts the coordinate in question vertical, joins each point to the one in the next cell up, and colours the steps by how the second digits wrap. The third option shifts the second digit of z by 1, which unlocks the wraps (page 2, section 4).

Without the third digits, the value of 𝒲_p is set by two contacts: the diagonal contact, whose squared length q_a(ε) = 5/4 − 9ε + 81ε² falls as the drift ε grows, and the step along e_z, with q_b(ε) = 1 + 4ε + 36ε², which rises. They balance at ε* = (13 − 2√31)/90, where both equal μ*². Among all rank-one layers with image (2, 2, 1), μ* is the least upper bound of the value without the third digits, and since ε* is irrational no prime attains it; 𝒲_p aims at ε = 1/48, just past the balance, and its limit √281/16 lies just below μ*.

In dimension five the net 𝒬*_p uses a layer whose entries are all half-integers, read modulo p through 1/2 ≡ (p + 1)/2. Up to the drift and the jitter, the point of cell n then sits at n or at n + (1/2, …, 1/2), according to the parity of n₁ + … + n₅: locally the packing D₅⁺, with one point per unit cube and minimum √5/2 = 1.118…. The slab shows one plane of cells; the colour gives the parity.

Odd sums of half-integers can never be whole, so every step to a neighbouring cell with an odd number of moving coordinates keeps all five coordinates about half a cell long. The drift shortens them a little: δN^1/5 ≈ (√5/2)(1 − 17/(5p)). The paper certifies δN^1/5 > 1 for every prime p ≥ 37 with a relaxation that allows every wrap of the first two offset digits. The browser computes exact distances only while N = p⁵ stays small; beyond that the chart shows the certified bound.