Six-tile groupings
Claim
Six tiles is the ceiling, and all 64 six-tile arrangements are connected; they fall into 34 families, four of them one-member orbits.
Construction
A combination holds between one and six tiles docked around the centre. Six is not a convention but a consequence: no legal combination holds seven, because the exclusion rule runs out of circles and diamonds before a seventh tile can be placed.
That makes the six-tile tier the top of the system and the densest thing it produces. It is also the smallest tier — 64 arrangements against tier four's 417.
Proof
The tier counts come out of the same enumeration as the rest: 16, 98, 288, 417, 274 and 64, summing to 1,157, recomputed here from the exclusion key.
Every one of the 64 is connected. The technical reference gives tier six as 64 arrangements, 64 combination shapes, 34 families and 4 without cousins — the only tiers where legality and wholeness coincide completely are five and six.
Those 34 families are the most contact-rich shapes in the system. Five of them reach six edge joins with no point joins at all — ACFHKO, ACGIKO, BDFHKO, BDGIKO and BDGJLO — and those five are the six-tile shapes that hold together along arcs everywhere rather than pivoting on a point.
What is missing
A tool. The design space at six tiles is small enough to print and large enough to be unmanageable by eye, and nothing on file sifts it — no filter by join count, by symmetry, by silhouette, or by whether the resulting shape tiles.
The project has named this gap itself: a dedicated sifting and notation tool for groupings as the dataset grows.
Note
Four of the thirty-four are their own orbit entirely. Do any of them look like anything to you?
Open questions
- No sifting tool exists, so the tier is enumerated but not explored.
- Why tiers five and six are fully connected while tier two is barely half connected. Observed in the data; no argument on file.
- Whether any six-tile shape tiles the plane. Unknown, and it is the same gap as the cousin tiling classification.
Checks
| What | How | Source |
|---|---|---|
| Tier capacities | 16 · 98 · 288 · 417 · 274 · 64, summing to 1,157, recomputed | _src/core.py |
| Six is the ceiling | No independent set of size seven exists in the exclusion graph | _src/core.py |
| Tier six, in detail | 64 arrangements, 64 shapes, 34 families, 4 unique | nuna-combination-shapes-reference.md |
| Fully connected | 64 of 64, as at tier five's 274 of 274 | nuna-combination-shapes-reference.md |