Library · 2. Core Claims · mark N

Symmetry

Claim

One mirror axis, no rotational symmetry — therefore four distinct orientations on the lattice.

Construction

Stand the shape upright. The vertical line through the head's centre and the diamond's centre is a mirror: reflecting across it maps the shape onto itself. No other line does, and no rotation short of a full turn does.

Fig. 1 — the single mirror axis.

Fig. 2 — the four orientations, each a quarter turn of the tile about its own circle. The mirror produces nothing new, so these four are all there are

Proof

The shape's symmetry group is therefore of order 2, the identity and that one reflection.

The lattice's own symmetry group is D₄, of order 8. The number of distinct placements, counted up to the lattice's symmetries, is the orbit size: 8 / 2 = 4.

Those four are the quarter-turns. Reflections produce nothing new, because the shape already contains its own mirror — which is why a tile needs no handed version, and why a two-sided physical tile is the same object on both faces.

Consequence

Four orientations is what makes the arrangement counts come out as they do. A physical lock profile that fixes orientation reduces four to one and closes off most of the pattern space, which is the difficulty recorded on the applied shelf.

The cousins dataset, built separately and for another purpose, records nuna as family 4·52: self-mirror, four placements. Two pieces of work, the same fact.

Note

Aaron's slot.

Open questions

This mark is one of the four still open, so the page is written but not filed.

Checks

What	How	Source
Mirror, by construction	The tile path is symmetric about x = 0 at every sampled point	_src/core.py
Orbit arithmetic	\|D₄\| / \|symmetry group\| = 8 / 2 = 4	reference/geometry-constants.md
Independent record	4·52, self-mirror, four placements	nuna-cousins-reference.md