Pieces

A gallery of pieces used by puzzles. Each piece is a polycube shape.

Representing Polycubes

Compact Notation

Polycubes are represented as delimited strings of numbers.

numbers represent stacks of cubes (in the z direction, coming out of viewing plane toward viewer)
row of numbers are adjacent stacks (in the x direction, left to right across viewing plane)
single slash / delimits rows (y direction, downward on viewing plane).
double slash // delimits planes (z direction, generally only necessary to represent overhangs)
0s represent empty spaces. Use a 0 in a empty row or plane when there is nothing to represent (for an example, see 'hollow cube' shape). This prevents two / delimiters from being mistaken for a single // delimiter.

Naming

Every polycube gets a shorthand name.

Name rods by length, prefixed with "I" ("I4")
Name rectangular solids by their dimensions, delimited with "x" ("1x2x3")
When possible, name 2D polycubes by a letter approximating their shape ("P")
When possible, name polycubes by overhang-free faces, with bigger faces first ("PQO")
Use + to indicate a shape where footprint has been extended, generally along the Z axis ("PLR+")
Use +N to indicate a shape where an N-long spike has been added, generally at the top left along the Z axis ("L+3", "knot+1")

Orientation

Compact notation implies a default orientation. Best practices for choosing an orientation:

Favor orientations that are easy to quickly assess and understand.
Minimize the number, size, and complexity of z-axis structures extruding from the xy plane. If there is a complex 2D shape, lay it out flat on the xy plane so the viewer can see it easily.
Don't worry about finding a canonical/minimal representation that can be algorithmically determined (eg, shortest or lexicographically earliest representation). Instead, orient in a way that is convenient/intuitive.
If the name of a polycube uses letters to describe its shape, try to match orientation and chirality of those letters. (TODO: match chirality? of footprint or piece when viewed head-on?)

Polycubes

empty

Key: empty
Compact Form: /
0 Points:
Bounds: undefined
Has overhangs: false
Footprint area: 0

Elevation

3D view

I1

Key: I1
Compact Form: 1
1 Points: 1,1,1
Bounds: 1x1x1 (1-1, 1-1, 1-1)
Has overhangs: false
Footprint area: 1

Elevation

3D view

I2

Key: I2
Compact Form: 11
2 Points: 1,1,1|2,1,1
Bounds: 2x1x1 (1-2, 1-1, 1-1)
Has overhangs: false
Footprint area: 2
Used by Tatami Tiling - Classic

Elevation

3D view

R

Key: R
Compact Form: 11/1
3 Points: 1,1,1|2,1,1|1,2,1
Bounds: 2x2x1 (1-2, 1-2, 1-1)
Has overhangs: false
Footprint area: 3

Elevation

3D view

I3

Key: I3
Compact Form: 111
3 Points: 1,1,1|2,1,1|3,1,1
Bounds: 3x1x1 (1-3, 1-1, 1-1)
Has overhangs: false
Footprint area: 3

Elevation

3D view

T

Key: T
Compact Form: 111/01
4 Points: 1,1,1|2,1,1|3,1,1|2,2,1
Bounds: 3x2x1 (1-3, 1-2, 1-1)
Has overhangs: false
Footprint area: 4
Used by Angel Cube, Clones Tile A Rectangle - Simple Version

Elevation

3D view

O

Key: O
Compact Form: 11/11
4 Points: 1,1,1|2,1,1|1,2,1|2,2,1
Bounds: 2x2x1 (1-2, 1-2, 1-1)
Has overhangs: false
Footprint area: 4

Elevation

3D view

I4

Key: I4
Compact Form: 1111
4 Points: 1,1,1|2,1,1|3,1,1|4,1,1
Bounds: 4x1x1 (1-4, 1-1, 1-1)
Has overhangs: false
Footprint area: 4

Elevation

3D view

L

Key: L
Compact Form: 1/1/11
4 Points: 1,1,1|1,2,1|1,3,1|2,3,1
Bounds: 2x3x1 (1-2, 1-3, 1-1)
Has overhangs: false
Footprint area: 4
Used by Clones Tile A Rectangle - Restricted Neighbors

Elevation

3D view

S

Key: S
Compact Form: 011/11
4 Points: 2,1,1|3,1,1|1,2,1|2,2,1
Bounds: 3x2x1 (1-3, 1-2, 1-1)
Has overhangs: false
Footprint area: 4

Elevation

3D view

S2

Key: S2
Compact Form: 022/22
8 Points: 2,1,1|2,1,2|3,1,1|3,1,2|1,2,1|1,2,2|2,2,1|2,2,2
Bounds: 3x2x2 (1-3, 1-2, 1-2)
Has overhangs: false
Footprint area: 4
Used by Angel Cube

Elevation

3D view

P

Key: P
Compact Form: 11/11/1
5 Points: 1,1,1|2,1,1|1,2,1|2,2,1|1,3,1
Bounds: 2x3x1 (1-2, 1-3, 1-1)
Has overhangs: false
Footprint area: 5
Used by P-Pentominos, Clones Tile A Rectangle - Restricted Neighbors

Elevation

3D view

Q

Key: Q
Compact Form: 11/11/01
5 Points: 1,1,1|2,1,1|1,2,1|2,2,1|2,3,1
Bounds: 2x3x1 (1-2, 1-3, 1-1)
Has overhangs: false
Footprint area: 5
Used by P-Pentominos

Elevation

3D view

A

Key: A
Compact Form: 111/11/1
6 Points: 1,1,1|2,1,1|3,1,1|1,2,1|2,2,1|1,3,1
Bounds: 3x3x1 (1-3, 1-3, 1-1)
Has overhangs: false
Footprint area: 6

Elevation

3D view

PLR

Key: PLR
Compact Form: 221/1
6 Points: 1,1,1|1,1,2|2,1,1|2,1,2|3,1,1|1,2,1
Bounds: 3x2x2 (1-3, 1-2, 1-2)
Has overhangs: false
Footprint area: 4
Used by Clones Tile A Rectangle - Restricted Neighbors

Elevation

3D view

L+3

Key: L+3
Compact Form: 411/1
7 Points: 1,1,1|1,1,2|1,1,3|1,1,4|2,1,1|3,1,1|1,2,1
Bounds: 3x2x4 (1-3, 1-2, 1-4)
Has overhangs: false
Footprint area: 4
Used by Clones Tile A Rectangle - Restricted Neighbors
Note: L plus 3-spike

Elevation

3D view

PQR

Key: PQR
Compact Form: 221/11
7 Points: 1,1,1|1,1,2|2,1,1|2,1,2|3,1,1|1,2,1|2,2,1
Bounds: 3x2x2 (1-3, 1-2, 1-2)
Has overhangs: false
Footprint area: 5
Used by Clones Tile A Rectangle - Restricted Neighbors

Elevation

3D view

PLR+

Key: PLR+
Compact Form: 43/2
9 Points: 1,1,1|1,1,2|1,1,3|1,1,4|2,1,1|2,1,2|2,1,3|1,2,1|1,2,2
Bounds: 2x2x4 (1-2, 1-2, 1-4)
Has overhangs: false
Footprint area: 3
Note: High footprint variation (3 to 7) may be useful for some puzzles.

Elevation

3D view

APL

Key: APL
Compact Form: 321/2
8 Points: 1,1,1|1,1,2|1,1,3|2,1,1|2,1,2|3,1,1|1,2,1|1,2,2
Bounds: 3x2x3 (1-3, 1-2, 1-3)
Has overhangs: false
Footprint area: 4
Used by Clones Tile A Rectangle - Restricted Neighbors

Elevation

3D view

PQA

Key: PQA
Compact Form: 321/21
9 Points: 1,1,1|1,1,2|1,1,3|2,1,1|2,1,2|3,1,1|1,2,1|1,2,2|2,2,1
Bounds: 3x2x3 (1-3, 1-2, 1-3)
Has overhangs: false
Footprint area: 5
Used by Clones Tile A Rectangle - Restricted Neighbors

Elevation

3D view

VPL

Key: VPL
Compact Form: 311/11
7 Points: 1,1,1|1,1,2|1,1,3|2,1,1|3,1,1|1,2,1|2,2,1
Bounds: 3x2x3 (1-3, 1-2, 1-3)
Has overhangs: false
Footprint area: 5
Used by Clones Tile A Rectangle - Restricted Neighbors

Elevation

3D view

1x2x3

Key: 1x2x3
Compact Form: 33
6 Points: 1,1,1|1,1,2|1,1,3|2,1,1|2,1,2|2,1,3
Bounds: 2x1x3 (1-2, 1-1, 1-3)
Has overhangs: false
Footprint area: 2
Used by Clones Tile A Rectangle - Restricted Neighbors

Elevation

3D view

2x4x4

Key: 2x4x4
Compact Form: 2222/2222/2222/2222
32 Points: 1,1,1|1,1,2|2,1,1|2,1,2|3,1,1|3,1,2|4,1,1|4,1,2|1,2,1|1,2,2|2,2,1|2,2,2|3,2,1|3,2,2|4,2,1|4,2,2|1,3,1|1,3,2|2,3,1|2,3,2|3,3,1|3,3,2|4,3,1|4,3,2|1,4,1|1,4,2|2,4,1|2,4,2|3,4,1|3,4,2|4,4,1|4,4,2
Bounds: 4x4x2 (1-4, 1-4, 1-2)
Has overhangs: false
Footprint area: 16

Elevation

3D view

2x3x5

Key: 2x3x5
Compact Form: 555/555
30 Points: 1,1,1|1,1,2|1,1,3|1,1,4|1,1,5|2,1,1|2,1,2|2,1,3|2,1,4|2,1,5|3,1,1|3,1,2|3,1,3|3,1,4|3,1,5|1,2,1|1,2,2|1,2,3|1,2,4|1,2,5|2,2,1|2,2,2|2,2,3|2,2,4|2,2,5|3,2,1|3,2,2|3,2,3|3,2,4|3,2,5
Bounds: 3x2x5 (1-3, 1-2, 1-5)
Has overhangs: false
Footprint area: 6
Used by Clones Tile A Rectangle - Restricted Neighbors

Elevation

3D view

3x4x5

Key: 3x4x5
Compact Form: 5555/5555/5555
60 Points: 1,1,1|1,1,2|1,1,3|1,1,4|1,1,5|2,1,1|2,1,2|2,1,3|2,1,4|2,1,5|3,1,1|3,1,2|3,1,3|3,1,4|3,1,5|4,1,1|4,1,2|4,1,3|4,1,4|4,1,5|1,2,1|1,2,2|1,2,3|1,2,4|1,2,5|2,2,1|2,2,2|2,2,3|2,2,4|2,2,5|3,2,1|3,2,2|3,2,3|3,2,4|3,2,5|4,2,1|4,2,2|4,2,3|4,2,4|4,2,5|1,3,1|1,3,2|1,3,3|1,3,4|1,3,5|2,3,1|2,3,2|2,3,3|2,3,4|2,3,5|3,3,1|3,3,2|3,3,3|3,3,4|3,3,5|4,3,1|4,3,2|4,3,3|4,3,4|4,3,5
Bounds: 4x3x5 (1-4, 1-3, 1-5)
Has overhangs: false
Footprint area: 12

Elevation

3D view

PQO

Key: PQO
Compact Form: 221/21
8 Points: 1,1,1|1,1,2|2,1,1|2,1,2|3,1,1|1,2,1|1,2,2|2,2,1
Bounds: 3x2x2 (1-3, 1-2, 1-2)
Has overhangs: false
Footprint area: 5
Used by Clones Tile A Rectangle - Restricted Neighbors

Elevation

3D view

knot

Key: knot
Compact Form: 322/31/11
13 Points: 1,1,1|1,1,2|1,1,3|2,1,1|2,1,2|3,1,1|3,1,2|1,2,1|1,2,2|1,2,3|2,2,1|1,3,1|2,3,1
Bounds: 3x3x3 (1-3, 1-3, 1-3)
Has overhangs: false
Footprint area: 7
Note: Identical footprint on 3 sides.

Elevation

3D view

knot+1

Key: knot+1
Compact Form: 422/31/11
14 Points: 1,1,1|1,1,2|1,1,3|1,1,4|2,1,1|2,1,2|3,1,1|3,1,2|1,2,1|1,2,2|1,2,3|2,2,1|1,3,1|2,3,1
Bounds: 3x3x4 (1-3, 1-3, 1-4)
Has overhangs: false
Footprint area: 7

Elevation

3D view

hollowCube

Key: hollowCube
Compact Form: 333/313/333//0//0/01
26 Points: 1,1,1|1,1,2|1,1,3|2,1,1|2,1,2|2,1,3|3,1,1|3,1,2|3,1,3|1,2,1|1,2,2|1,2,3|2,2,1|3,2,1|3,2,2|3,2,3|1,3,1|1,3,2|1,3,3|2,3,1|2,3,2|2,3,3|3,3,1|3,3,2|3,3,3|2,2,3
Bounds: 3x3x3 (1-3, 1-3, 1-3)
Has overhangs: true
Footprint area: 9
Note: 3x3x3 cube, with middle spot empty.

Elevation

3D view

helix

Key: helix
Compact Form: 02/011//1
5 Points: 2,1,1|2,1,2|2,2,1|3,2,1|1,1,2
Bounds: 3x2x2 (1-3, 1-2, 1-2)
Has overhangs: true
Footprint area: 3
Note: Overhang in every direction.

Elevation

3D view

kian

Key: kian
Compact Form: 403/02/151
16 Points: 1,1,1|1,1,2|1,1,3|1,1,4|3,1,1|3,1,2|3,1,3|2,2,1|2,2,2|1,3,1|2,3,1|2,3,2|2,3,3|2,3,4|2,3,5|3,3,1
Bounds: 3x3x5 (1-3, 1-3, 1-5)
Has overhangs: false
Footprint area: 6
Note: Kian shape

Elevation

3D view

OL

Key: OL
Compact Form: 131/101/111//0//001
11 Points: 1,1,1|2,1,1|2,1,2|2,1,3|3,1,1|1,2,1|3,2,1|1,3,1|2,3,1|3,3,1|3,1,3
Bounds: 3x3x3 (1-3, 1-3, 1-3)
Has overhangs: true
Footprint area: 8
Note: Used to show orientations

Elevation

3D view