Library · 2. Core Claims · mark O

The tile is a domino

Claim

Every legal placement covers exactly one circle cell and one adjacent diamond cell.

Construction

A tile is one circle fused to one diamond. On the lattice that is one cell of each type, orthogonally adjacent, and there is no other way to place it.

Mark the six points where consecutive arcs meet, counting the head's own quarter divisions. They are not scattered: they outline a rectangle s wide and 2s tall, with the midpoints of the two long sides marked. That rectangle is the tile's own cell — one square for the circle, one for the diamond, sharing an edge.

Fig. 1 — the six junctions sit on the perimeter of the tile's own cell.

Fig. 2 — one tile on the lattice, its cell outlined: it covers exactly two adjacent cells, which is a domino on a checkerboard. Which shade holds the circles is Book G's open question and does not change the figure

Proof

By Claim F the only cells at distance s from a circle cell are diamond cells, and conversely. By Claim G the lattice is bipartite. A tile therefore spans exactly one edge of the grid graph, which is the definition of a domino on a checkerboard, and the four orientations of Claim N are the four incident edges.

The cell gives the area a third time. Three of the six arcs bulge outward and three bite inward, each a circular segment cut by a 90° chord of radius r, so they cancel exactly:

A = 2s² = 4r² = d²

Why it is worth its own page

This is the bridge claim. Alone it says almost nothing; through it the whole literature of domino tilings becomes available to this shape. Claim H is its first consequence and the badge count of Claim I its second — the sixteen arrangements are sixteen edges, and the exclusion rule is the matching condition.

Note

Aaron's slot.

Open questions

One of the four open marks.

Checks

What	How	Source
Six junctions on a cell	Their bounding box measures 141.422 × 282.843 against s × 2s = 141.421 × 282.843	_src/core.py
Equal chords	All six consecutive distances are r√2	_src/core.py
Segment area	πr²/4 − r²/2 = 0.2854r², three added and three subtracted	reference/geometry-constants.md
Consequence holds	Claim H — the published sequence would not be reproduced if the correspondence were inexact	OEIS A004003