Library · 2. Core Claims · mark F

Lattice spacing

Claim

Adjacent lattice points sit exactly r√2 apart.

Construction

Take the grid of tangent circles and turn it 45°. Circle centres and diamond centres now alternate along both the horizontal and the vertical. Call the common spacing s.

Each lattice point is either a circle centre or a diamond centre, and a tile occupies one of each, adjacent.

Fig. 1 — circle nodes and diamond nodes alternating at step s = r√2.

Fig. 2 — a patch of the lattice. Nodes alternate between the two kinds, one tile covers one adjacent pair (both picked out in ink), and the step between neighbours is r√2. The glyphs name the two kinds of node, not their parity, which Book G leaves open

Proof

In the unturned picture, four circle centres form a square of side 2r with the diamond at its centre. The distance from a corner to the centre of a square of side 2r is half the diagonal: (2r√2)/2 = r√2.

s = r√2 ≈ 1.41421r

Note that s is larger than r but smaller than 2r. The tile is 2r wide on a lattice of pitch 1.414r, which is exactly why neighbouring tiles interlock rather than sitting side by side: each one overhangs its own cell.

Note

Aaron's slot.

Open questions

The project writes this step as both s and S. One letter would do.

Checks

What	How	Source
Junction spacing on the tile	All six junction chords measure 141.422 at r = 100, against r√2 = 141.4214	_src/core.py
Same-type pitch	Two circle centres of the same parity are 2s = 2.828r apart, measurable on any board	reference/geometry-constants.md
Lattice formula on file	P(a,b) = a·U + b·V with U = (r√2, r√2), V = (−r√2, r√2)	nuna_master_reference.md