Shapes · 16-point badge

Combination shapes

The fixed centre tile fused with one to six of the sixteen tiles that share an arc with it, wherever the added tiles join into one unbroken shape. Of the 1,157 legal combinations, 717 are connected. They fall into 362 families, 20 of them a family of one.

Badge: The 16-point badge
Cards: All 362 families on /shapes
Books: Book J Book AD Book AJ
Patterns: Patterns/patterns?shape=combination-3-19
Lock: The lock
Elsewhere: 24-point combination shapes The cousins

How a shape is made

Lattice. Circles sit on lattice nodes whose coordinates sum to an even number, diamonds on the odd nodes, one step S = r√2 apart.
Tile. A circle plus one neighbouring diamond. The centre tile is fixed, upright: circle at the origin, diamond directly below.
The sixteen. Tiles A–P are the sixteen that share an arc with the centre tile. They are drawn on the badge.
Legal. Two tiles can appear together only if they share neither a circle nor a diamond. That leaves 1,157 legal sets of one to six tiles; no legal set holds seven.
Connected. A combination shape is a legal set whose added tiles touch one another in one piece. The centre tile is not counted when checking this. 717 of the 1,157 qualify.
Family. Shapes that match under the eight turns and mirrors about the centre circle form one family; the other members are its cousins. Families are numbered tier·nn, tier first, then alphabetically by code.

Contact: edge joins and point joins

Two tiles touch in one of two ways. An edge join is when one tile's diamond sits one step from the other's circle, so their outlines share a whole arc. A point join is when two circles sit on diagonal nodes, exactly tangent, or two diamonds sit on diagonal nodes and meet at a corner. Either kind counts towards connection. Each card records how many of each its shape has.

2·01 AC
1 edge join

2·02 AD
1 point join

3·13 AEM
2 point joins

6·02 ACFHKO
6 edge joins

Marks drawn from the badge. Every figure links to its card. Of the 717 shapes, 16 are single tiles with nothing to join, 660 contain at least one edge join, and 41 hold together on points alone.

The count

Tier	Legal	Shapes	Families	Unique
Tier 1	16	16	5	1
Tier 2	98	40	17	0
Tier 3	288	100	53	6
Tier 4	417	223	112	1
Tier 5	274	274	141	8
Tier 6	64	64	34	4
All	1,157	717	362	20

Recomputed from the rules for this page and matched to the technical reference row by row: every family's id, code, members and join counts agree. Tier links open the first card of each tier.

The twenty families of one

A unique family has no cousins: every turn and mirror of it is itself.

1·05 I

3·05 ACO

3·09 ADN

3·13 AEM

3·47 EIM

3·52 FIL

3·53 GIK

4·34 AEIM

5·18 ACFLO

5·29 ACGKO

5·48 ADFLN

5·56 ADGKN

5·131 CFILO

5·134 CGIKO

5·140 DFILN

5·141 DGIKN

6·08 ACFILO

6·11 ACGIKO

6·17 ADFILN

6·18 ADGIKN

362 or 317 disputed

This catalogue counts 362 families. Book AD carries 317. Both numbers come out of the same 717 shapes; they differ in what counts as "the same".

362 counts shapes as arrangements around the fixed centre tile: two shapes are one family only if a turn or mirror about the centre circle carries one onto the other.
Letting a shape also slide away from the centre, while keeping its tile boundaries, merges some families: 348.
Forgetting the tile boundaries as well, and comparing only the drawn outline of circles and diamonds, gives exactly 317, by tier 5, 12, 38, 93, 135, 34.

So 317 is reproducible as the count of distinct outlines. Whether that is what Book AD means, and which count the Library should lead with, is open.

In nuna life

Every family was run as a starting pattern in nuna life, alone on an empty board, under seven rules, until it repeated, died or sent something off the board, up to 400 generations. Cousins behave identically, so one row covers a whole family. The full result for each rule is on each card.

Rule	Neighbours, births and survival
Default	24 neighbours, born 3 or 4, stays 2-4, more neighbours wins
Both fail	same counts, both fail
16-neighbour default	16 neighbours, born 2, stays 1-2
Woven lines	16 neighbours, born 2, stays 2-3
Lines	16 neighbours, born 1-2, stays 2-3
Glider traffic	16 neighbours, born 2 or 4, stays 2-3
Wildfire	16 neighbours, born 2-3, stays 1-2

Under the default rule

Result	Families
Stamp: stays exactly itself	118
Settles into another still shape	217
Becomes another family's shape	22
Dies: the single-tile shapes	5

Breathers, period 41

Under the 16-neighbour default, these become a four-tile pattern that blooms to thirty tiles and returns every 41 generations.

3·12 AEJ
breather

3·19 BEH
breather

3·26 BIM
breather

5·03 ACFHM
breather

5·77 BDGIM
breather

Period 42 pulses

4·09 ACGM, 4·56 BEHM, 4·78 BHKO, 4·97 CGIM, 5·21 ACGIM, 6·20 BDFHKO.

The fast glider

5·55 ADGJN and 5·115 BFILO make the period-4 glider under woven lines and glider traffic. Under wildfire, 182 families send something off the board.

Becomes another family

Source: the nuna life index for dataset 02. Its 362 rows match this catalogue's ids and codes one for one.

Every family

Each code links to its card on /shapes. To start a pattern from any family, Patterns takes /patterns?shape=combination-tier-nn.

Tier 15 families · 16 shapes

Tier 217 families · 40 shapes

Tier 353 families · 100 shapes

Tier 4112 families · 223 shapes

Tier 5141 families · 274 shapes

Tier 634 families · 64 shapes

Open questions

362 or 317. Which count does Book AD mean, and which should the Library lead with? 317 is reproduced above as the number of distinct outlines.
Connection rule. The centre tile is left out when checking that a shape is one piece. That matches the reference; whether it is the intended definition, and not only a convention, is not stated anywhere in the sources.
Tiling. Nothing is yet established about how combination shapes fill the plane; Patterns is where that work sits.
Gen 0 changes. Several "becomes" results in the life index are reached at generation 0, for example 2·01 becoming 2·16. That reads as already being the other shape, which needs a note from the life chat.