Shapes · the cousins

The cousins

Cut a circle from a square and four offcuts are left over. Put one to four of them back against the circle, in any way they can join, and you get a cousin. There are 73 families behind 507 placements. Nuna is one of them, 4·52: all four offcuts gathered into the diamond and hung on the circle.

Cards: All 73 families on /shapes Nuna, 4·52
Books: Book AC
Patterns: Patterns, starting from nuna/patterns?shape=cousin-pieces-nn
Lock: The lock
Elsewhere: 16-point combination shapes 24-point combination shapes

1·01 A0
one piece

2·01 A0A1
two pieces

3·09 A0B1E0
three pieces

4·01 A0A1A2A3
the square

4·52 A0B1E0H0
nuna

How a cousin is made

The offcut. One r × r cell with a quarter disc of radius r taken from one corner: a concave arc and two straight legs. Its area is (4−π)r²/4, and it is its own mirror image, so it has four placements in a cell, one per turn.
Arc join. An offcut meets the circle along its arc. That only works in the four cells around the circle's centre, the home sites.
Leg join. An offcut can instead fix to a piece already placed, leg to leg, sharing a whole straight side. No other contact counts.
Cousin. The circle plus one to four offcuts, each reachable from the circle through arc and leg joins, with no overlaps. Its area is πr² plus a quarter of (4−π)r² for each piece; with all four, 4r² = d², the square it came from.
Family. Placements that match under the eight turns and mirrors form one family. Families are numbered pieces·nn, then alphabetically by code.

A code names the sites a cousin's pieces sit in: twenty sites, A to T, each with a digit counting quarter turns about the circle's centre. A0A1A2A3 is the square, every offcut back in its own corner. A0B1E0H0 is nuna: one home site and a chain of three.

The count

Pieces	Placements	Up to turn	Families	Own mirror	In two hands	Discarded
1	4	1	1	1	0	1
2	22	6	4	2	2	2
3	100	25	14	3	11	3
4	381	98	54	10	44	6
All	507	130	73	16	57	12

Families, placements and mirror counts are read from the shapes data and match the cousins reference. "Up to turn" and "discarded" are the reference's own figures. Discarded shapes are offcuts joined only to each other, with no circle; the curved diamond is one of them. Links open the first card of each group.

Mirror images

Sixteen families are their own mirror image; the other fifty-seven come in two hands, left and right, and each card links to its twin. Nuna is its own mirror. The square, 4·01, is the only cousin with a single placement: every turn of it is itself.

How they tile

Status	Families
Tiles the plane alone	35
Tiles only with a partner	2
No partner found	17
Always leaves gaps	19

A cousin with fewer than four pieces carries spare quarter discs that nothing can absorb, so it can never tile in any company. Each card shows its partners; Patterns takes any of them as a starting shape.

Every family

One piece1 families · 4 placements

1·01A0

Two pieces4 families · 22 placements

Three pieces14 families · 100 placements

Four pieces54 families · 381 placements

Open questions

Which badge. The cousins come from the circle and its offcuts, not from a badge, so this page links none. Should it link the 16-point badge anyway?
"The offcut". The cousins reference names 1·01 "the offcut", but 1·01 is the circle with one offcut; the offcut alone is d1·01, among the discarded shapes. Which name belongs where?
4·01's mirror. The reference lists 4·01 as UNIQUE in the mirror column, where every other symmetric cousin says "self". The data counts it as its own mirror. One word or the other should change.
Tiling source. The tiling statuses come from the cousin tiling data used to build /shapes. "No partner found" means none was found among the cousins, not that none exists.