An Introduction to Polycubes

Author: David Goodger <goodger@python.org>
Date: 2016-02-20
Revision: 627
Web site:http://puzzler.sourceforge.net/
Copyright: © 1998-2015 by David J. Goodger
License:GPL 2
images/puzzler.png

Contents

Polycubes are polyforms constructed from unit cubes joined face-to-face in regular three-dimensional Cartesian space. The polycubes that are flat (only one cube thick) are known as "planar polycubes" or "solid polyominoes".

Here is a puzzle containing all the polycubes of order 1 through 5:

images/cubes/polycubes-12345-overlapping-blocks-1.png

See Polycubes: Puzzles & Solutions, Pentacubes: Puzzles & Solutions, and Solid Pentominoes: Puzzles & Solutions for many more puzzles.

Polyforms

The number and names of the various orders of polycubes are as follows:

Order
Polyform
Name
Polycubes
Planar
Polycubes
1 monocube 1 1
2 dicube 1 1
3 tricubes 2 2
4 tetracubes 8 5
5 pentacubes 29 12
6 hexacubes 166 35

"*" above means that forms with enclosed holes exist.

The numbers of polycubes can also be found in the following sequences from The On-Line Encyclopedia of Integer Sequences: A000162 (3-D) and A000105 (planar polycubes / solid polyominoes).

Examples of the polycubes from order 1 (monocube) to order 6 (hexacubes) are given in the tables below.

The polycubes are named with letters (like the monocube "M") or a letter-number scheme (like the "I3" tricube, and the "F5" and "V15" pentacubes). The names used by Polyform Puzzler are based on the traditional names for pentominoes, Kadon's names for their Superquints, hexacubes, and Poly-4 Supplement sets, as well as Thorleif Bundgård's names for Soma cubes.

The initial letter of each name is the letter of the alphabet that the polycube most closely resembles, or an initial. In some cases, that resemblance is weak, and the letters are arbitrary. The final digit of the number represents the polyform order (how many unit cubes are in the polycube). There are more pentacubes than letters in the alphabet, so the names of many of the non-planar pieces have an extra middle digit to differentiate the variations. All 166 of the hexacubes have 3-character names.

In the tables below, "Aspects" refers to the number of unique orientations that a polyform may take (different rotations, flipped or not). This varies with the symmetry of the polyform.

The "Notes" column records which pieces are planar and non-planar, as well as chiral pairs (matching left-hand and right-hand mirror images). The 3-D polycubes cannot be "flipped" into their mirror images (that would require access to a fourth physical dimension).

Alternate names and name origins are noted in the "Name" column.

Monocube

There is only one monocube (order-1 polycube):

Name Image Aspects Notes
M
(from "Monocube")
images/pieces/polycubes/M.png 1 planar

Dicube

There is only one dicube (order-2 polycube), also planar (a solid monomino):

Name Image Aspects Notes
D
(from "Dicube")
images/pieces/polycubes/D.png 3 planar

Tricubes

There are 2 tricubes (order-3 polycubes), also planar (a solid domino):

Name Image Aspects Notes
I3 images/pieces/polycubes/I3.png 3 planar
V3
(Soma "V")
images/pieces/polycubes/V3.png 12 planar

Tetracubes

There are 8 tetracubes (order-4 polycubes), 5 of which are planar (solid tetrominoes):

Name Image Aspects Notes
A4
(Soma "a"; Kadon "V3")
images/pieces/polycubes/A4.png 12 non-planar; chiral pair with "B4"
B4
(Soma "b"; Kadon "V1")
images/pieces/polycubes/B4.png 12 non-planar; chiral pair with "A4"
I4 images/pieces/polycubes/I4.png 3 planar
L4 images/pieces/polycubes/L4.png 24 planar
O4 images/pieces/polycubes/O4.png 3 planar
P4
(Soma "p"; Kadon "V2")
images/pieces/polycubes/P4.png 8 non-planar
S4
(Soma "Z")
images/pieces/polycubes/S4.png 12 planar
T4 images/pieces/polycubes/T4.png 12 planar

Soma Cubes

The Soma Cubes, invented by Piet Hein in 1933, consist of 7 pieces: all the non-convex polycubes of order 3 (1) and 4 (6).

Name Image Aspects Notes
V
(polycube "V3")
images/pieces/polycubes/V3.png 12 planar; order 3
L
(polycube "L4")
images/pieces/polycubes/L4.png 24 planar; order 4
T
(polycube "T4")
images/pieces/polycubes/T4.png 12 planar; order 4
Z
(polycube "S4"; Kadon "S")
images/pieces/polycubes/S4.png 12 planar; order 4
a
(polycube "A4"; Kadon "V3")
images/pieces/polycubes/A4.png 12 non-planar; order 4; chiral pair with "b"
b
(polycube "B4"; Kadon "V1")
images/pieces/polycubes/B4.png 12 non-planar; order 4; chiral pair with "a"
p
(polycube "P4"; Kadon "V2")
images/pieces/polycubes/P4.png 8 non-planar; order 4

Pentacubes

There are 29 pentacubes (order-5 polycubes), 12 of which are planar (solid pentominoes):

Name Image Aspects Notes
A5 images/pieces/polycubes/A5.png 24 non-planar
F5 images/pieces/polycubes/F5.png 24 planar
I5 images/pieces/polycubes/I5.png 3 planar
J15 images/pieces/polycubes/J15.png 12 non-planar; chiral pair with "L15"
J25 images/pieces/polycubes/J25.png 24 non-planar; chiral pair with "L25"
J45 images/pieces/polycubes/J45.png 24 non-planar; chiral pair with "L45"
L5 images/pieces/polycubes/L5.png 24 planar
L15 images/pieces/polycubes/L15.png 12 non-planar; chiral pair with "J15"
L25 images/pieces/polycubes/L25.png 24 non-planar; chiral pair with "J25"
L35 images/pieces/polycubes/L35.png 24 non-planar; Kadon's Super Deluxe Quintillions set includes a second copy called "J35" (also adopted by Polyform Puzzler's Pentacubes Plus)
L45 images/pieces/polycubes/L45.png 24 non-planar; chiral pair with "J45"
N5 images/pieces/polycubes/N5.png 24 planar
N15 images/pieces/polycubes/N15.png 24 non-planar; chiral pair with "S15"
N25 images/pieces/polycubes/N25.png 24 non-planar; chiral pair with "S25"
P5 images/pieces/polycubes/P5.png 24 planar
Q5 images/pieces/polycubes/Q5.png 24 non-planar
S15 images/pieces/polycubes/S15.png 24 non-planar; chiral pair with "N15"
S25 images/pieces/polycubes/S25.png 24 non-planar; chiral pair with "N25"
T5 images/pieces/polycubes/T5.png 12 planar
T15 images/pieces/polycubes/T15.png 12 non-planar
T25 images/pieces/polycubes/T25.png 24 non-planar
U5 images/pieces/polycubes/U5.png 12 planar
V5 images/pieces/polycubes/V5.png 12 planar
V15 images/pieces/polycubes/V15.png 12 non-planar; chiral pair with "V25"
V25 images/pieces/polycubes/V25.png 12 non-planar; chiral pair with "V15"
W5 images/pieces/polycubes/W5.png 12 planar
X5 images/pieces/polycubes/X5.png 3 planar
Y5 images/pieces/polycubes/Y5.png 24 planar
Z5 images/pieces/polycubes/Z5.png 12 planar

Hexacubes

There are 166 hexacubes (order-6 polycubes), 35 of which are planar (solid hexominoes):

Name Image Aspects Notes
A06
(Kadon's "A")
images/pieces/polycubes/A06.png 12 planar
Aa6
("Fat A")
images/pieces/polycubes/Aa6.png 12 non-planar
Ba6
("B")
images/pieces/polycubes/Ba6.png 4 non-planar
C06 images/pieces/polycubes/C06.png 12 planar
D06 images/pieces/polycubes/D06.png 12 planar
E06 images/pieces/polycubes/E06.png 12 planar
F06
("hi F")
images/pieces/polycubes/F06.png 24 planar
F16
("low F")
images/pieces/polycubes/F16.png 24 planar
F26
("hi 4")
images/pieces/polycubes/F26.png 24 planar
F36
("low 4")
images/pieces/polycubes/F36.png 24 planar
Fa6
("F1")
images/pieces/polycubes/Fa6.png 24 non-planar
Fb6
("F2")
images/pieces/polycubes/Fb6.png 24 non-planar
Fc6
("F3")
images/pieces/polycubes/Fc6.png 24 non-planar
Fd6
("F4")
images/pieces/polycubes/Fd6.png 24 non-planar
Fe6
("F5")
images/pieces/polycubes/Fe6.png 24 non-planar
Ff6
("Fb1")
images/pieces/polycubes/Ff6.png 24 non-planar
Fg6
("Fb2")
images/pieces/polycubes/Fg6.png 24 non-planar
Fh6
("Fb3")
images/pieces/polycubes/Fh6.png 24 non-planar
Fi6
("Fb4")
images/pieces/polycubes/Fi6.png 24 non-planar
Fj6
("Fb5")
images/pieces/polycubes/Fj6.png 24 non-planar
G06 images/pieces/polycubes/G06.png 24 planar
H06 images/pieces/polycubes/H06.png 24 planar
I06 images/pieces/polycubes/I06.png 3 planar
J06 images/pieces/polycubes/J06.png 24 planar
Ja6
("J1l")
images/pieces/polycubes/Ja6.png 24 non-planar
Jb6
("J1r")
images/pieces/polycubes/Jb6.png 24 non-planar
Jc6
("J2l")
images/pieces/polycubes/Jc6.png 24 non-planar
Jd6
("J2r")
images/pieces/polycubes/Jd6.png 24 non-planar
Je6
("J3r")
images/pieces/polycubes/Je6.png 24 non-planar
Jf6
("J4l")
images/pieces/polycubes/Jf6.png 24 non-planar
Jg6
("J4u")
images/pieces/polycubes/Jg6.png 24 non-planar
Jh6
("J4d")
images/pieces/polycubes/Jh6.png 24 non-planar
Ji6
("J4v")
images/pieces/polycubes/Ji6.png 12 non-planar
K06 images/pieces/polycubes/K06.png 12 planar
L06 images/pieces/polycubes/L06.png 24 planar
La6
("L1")
images/pieces/polycubes/La6.png 12 non-planar
Lb6
("L4")
images/pieces/polycubes/Lb6.png 24 non-planar
Lc6
("L5")
images/pieces/polycubes/Lc6.png 24 non-planar
Ld6
("Lb1")
images/pieces/polycubes/Ld6.png 12 non-planar
Le6
("Lb5")
images/pieces/polycubes/Le6.png 24 non-planar
Lf6
("L1l")
images/pieces/polycubes/Lf6.png 24 non-planar
Lg6
("L1r")
images/pieces/polycubes/Lg6.png 24 non-planar
Lh6
("L2l")
images/pieces/polycubes/Lh6.png 24 non-planar
Li6
("L2r")
images/pieces/polycubes/Li6.png 24 non-planar
Lj6
("L3l")
images/pieces/polycubes/Lj6.png 24 non-planar
Lk6
("L4r")
images/pieces/polycubes/Lk6.png 24 non-planar
Ll6
("L4u")
images/pieces/polycubes/Ll6.png 24 non-planar
Lm6
("L4d")
images/pieces/polycubes/Lm6.png 24 non-planar
Ln6
("L4v")
images/pieces/polycubes/Ln6.png 12 non-planar
M06 images/pieces/polycubes/M06.png 24 planar
N06
("short N")
images/pieces/polycubes/N06.png 12 planar
N16
("long N")
images/pieces/polycubes/N16.png 24 planar
Na6
("N1")
images/pieces/polycubes/Na6.png 24 non-planar
Nb6
("N2")
images/pieces/polycubes/Nb6.png 24 non-planar
Nc6
("N3")
images/pieces/polycubes/Nc6.png 24 non-planar
Nd6
("N4")
images/pieces/polycubes/Nd6.png 24 non-planar
Ne6
("N5")
images/pieces/polycubes/Ne6.png 24 non-planar
Nf6
("Nb1")
images/pieces/polycubes/Nf6.png 24 non-planar
Ng6
("Nb2")
images/pieces/polycubes/Ng6.png 24 non-planar
Nh6
("Nb3")
images/pieces/polycubes/Nh6.png 24 non-planar
Ni6
("Nb4")
images/pieces/polycubes/Ni6.png 24 non-planar
Nj6
("Nb5")
images/pieces/polycubes/Nj6.png 24 non-planar
Nk6
("N13")
images/pieces/polycubes/Nk6.png 24 non-planar
Nl6
("N14")
images/pieces/polycubes/Nl6.png 12 non-planar
Nm6
("N1l")
images/pieces/polycubes/Nm6.png 24 non-planar
Nn6
("N1r")
images/pieces/polycubes/Nn6.png 24 non-planar
No6
("N1u")
images/pieces/polycubes/No6.png 12 non-planar
Np6
("N1b3")
images/pieces/polycubes/Np6.png 24 non-planar
Nq6
("N1b4")
images/pieces/polycubes/Nq6.png 24 non-planar
Nr6
("N23")
images/pieces/polycubes/Nr6.png 12 non-planar
Ns6
("N2r")
images/pieces/polycubes/Ns6.png 24 non-planar
Nt6
("N2b3")
images/pieces/polycubes/Nt6.png 12 non-planar
Nu6
("N2b4")
images/pieces/polycubes/Nu6.png 24 non-planar
O06 images/pieces/polycubes/O06.png 6 planar
P06 images/pieces/polycubes/P06.png 24 planar
Pa6
("P1")
images/pieces/polycubes/Pa6.png 24 non-planar
Pb6
("P2")
images/pieces/polycubes/Pb6.png 24 non-planar
Pc6
("P3")
images/pieces/polycubes/Pc6.png 24 non-planar
Pd6
("P4")
images/pieces/polycubes/Pd6.png 24 non-planar
Pe6
("P5")
images/pieces/polycubes/Pe6.png 24 non-planar
Pf6
("Pb1")
images/pieces/polycubes/Pf6.png 24 non-planar
Pg6
("Pb2")
images/pieces/polycubes/Pg6.png 24 non-planar
Ph6
("Pb3")
images/pieces/polycubes/Ph6.png 24 non-planar
Pi6
("Pb4")
images/pieces/polycubes/Pi6.png 24 non-planar
Pj6
("Pb5")
images/pieces/polycubes/Pj6.png 24 non-planar
Q06 images/pieces/polycubes/Q06.png 24 planar
Qa6
("Q14")
images/pieces/polycubes/Qa6.png 12 non-planar
Qb6
("Q4r")
images/pieces/polycubes/Qb6.png 24 non-planar
Qc6
("Q4d")
images/pieces/polycubes/Qc6.png 24 non-planar
Qd6
("Q4v")
images/pieces/polycubes/Qd6.png 24 non-planar
Qe6
("Q4b1")
images/pieces/polycubes/Qe6.png 12 non-planar
R06 images/pieces/polycubes/R06.png 24 planar
S06
("long S")
images/pieces/polycubes/S06.png 12 planar
Sa6
("S13")
images/pieces/polycubes/Sa6.png 24 non-planar
Sb6
("S14")
images/pieces/polycubes/Sb6.png 12 non-planar
Sc6
("S1l")
images/pieces/polycubes/Sc6.png 24 non-planar
Sd6
("S1r")
images/pieces/polycubes/Sd6.png 24 non-planar
Se6
("S1u")
images/pieces/polycubes/Se6.png 12 non-planar
Sf6
("S23")
images/pieces/polycubes/Sf6.png 12 non-planar
Sg6
("S2l")
images/pieces/polycubes/Sg6.png 24 non-planar
Sh6
("S2d")
images/pieces/polycubes/Sh6.png 24 non-planar
T06
("long T")
images/pieces/polycubes/T06.png 12 planar
T16
("short T")
images/pieces/polycubes/T16.png 24 planar
Ta6
("T1")
images/pieces/polycubes/Ta6.png 24 non-planar
Tb6
("T2")
images/pieces/polycubes/Tb6.png 24 non-planar
Tc6
("T3")
images/pieces/polycubes/Tc6.png 24 non-planar
Td6
("T4")
images/pieces/polycubes/Td6.png 24 non-planar
Te6
("T5")
images/pieces/polycubes/Te6.png 24 non-planar
Tf6
("T1u")
images/pieces/polycubes/Tf6.png 24 non-planar
Tg6
("T1d")
images/pieces/polycubes/Tg6.png 24 non-planar
Th6
("T24")
images/pieces/polycubes/Th6.png 12 non-planar
Ti6
("T2u")
images/pieces/polycubes/Ti6.png 24 non-planar
Tj6
("T3u")
images/pieces/polycubes/Tj6.png 24 non-planar
Tk6
("T3d")
images/pieces/polycubes/Tk6.png 24 non-planar
Tl6
("T4l")
images/pieces/polycubes/Tl6.png 24 non-planar
Tm6
("T4r")
images/pieces/polycubes/Tm6.png 24 non-planar
Tn6
("T4d")
images/pieces/polycubes/Tn6.png 24 non-planar
To6
("T4v")
images/pieces/polycubes/To6.png 24 non-planar
Tp6
("T4b4")
images/pieces/polycubes/Tp6.png 6 non-planar
U06 images/pieces/polycubes/U06.png 24 planar
Ua6
("U1")
images/pieces/polycubes/Ua6.png 24 non-planar
Ub6
("U2")
images/pieces/polycubes/Ub6.png 24 non-planar
Uc6
("U3")
images/pieces/polycubes/Uc6.png 24 non-planar
Ud6
("U4")
images/pieces/polycubes/Ud6.png 24 non-planar
Ue6
("U5")
images/pieces/polycubes/Ue6.png 24 non-planar
V06 images/pieces/polycubes/V06.png 24 planar
Va6
("V1")
images/pieces/polycubes/Va6.png 24 non-planar
Vb6
("V2")
images/pieces/polycubes/Vb6.png 24 non-planar
Vc6
("V3")
images/pieces/polycubes/Vc6.png 24 non-planar
Vd6
("V4")
images/pieces/polycubes/Vd6.png 24 non-planar
Ve6
("V5")
images/pieces/polycubes/Ve6.png 24 non-planar
Vf6
("V1l")
images/pieces/polycubes/Vf6.png 12 non-planar
Vg6
("V1d")
images/pieces/polycubes/Vg6.png 24 non-planar
Vh6
("V3r")
images/pieces/polycubes/Vh6.png 12 non-planar
Vi6
("V3d")
images/pieces/polycubes/Vi6.png 24 non-planar
W06
("Wa")
images/pieces/polycubes/W06.png 24 planar
W16
("Wb")
images/pieces/polycubes/W16.png 12 planar
W26
("Wc")
images/pieces/polycubes/W26.png 24 planar
Wa6
("W1")
images/pieces/polycubes/Wa6.png 24 non-planar
Wb6
("W2")
images/pieces/polycubes/Wb6.png 24 non-planar
Wc6
("W3")
images/pieces/polycubes/Wc6.png 24 non-planar
Wd6
("W4")
images/pieces/polycubes/Wd6.png 24 non-planar
We6
("W5")
images/pieces/polycubes/We6.png 24 non-planar
X06 images/pieces/polycubes/X06.png 12 planar
X16
("italic X")
images/pieces/polycubes/X16.png 12 planar
Xa6
("X1")
images/pieces/polycubes/Xa6.png 24 non-planar
Xb6
("X3")
images/pieces/polycubes/Xb6.png 6 non-planar
Y06
("hi Y")
images/pieces/polycubes/Y06.png 24 planar
Y16
("low Y")
images/pieces/polycubes/Y16.png 12 planar
Ya6
("Y1")
images/pieces/polycubes/Ya6.png 24 non-planar
Yb6
("Y2")
images/pieces/polycubes/Yb6.png 24 non-planar
Yc6
("Y3")
images/pieces/polycubes/Yc6.png 24 non-planar
Yd6
("Y4")
images/pieces/polycubes/Yd6.png 12 non-planar
Ye6
("Y5")
images/pieces/polycubes/Ye6.png 24 non-planar
Yf6
("Yb1")
images/pieces/polycubes/Yf6.png 24 non-planar
Yg6
("Yb2")
images/pieces/polycubes/Yg6.png 24 non-planar
Yh6
("Yb4")
images/pieces/polycubes/Yh6.png 12 non-planar
Yi6
("Yb5")
images/pieces/polycubes/Yi6.png 24 non-planar
Z06
("long Z")
images/pieces/polycubes/Z06.png 12 planar
Z16
("short Z")
images/pieces/polycubes/Z16.png 24 planar
Za6
("Z1")
images/pieces/polycubes/Za6.png 24 non-planar
Zb6
("Z2")
images/pieces/polycubes/Zb6.png 24 non-planar
Zc6
("Z3")
images/pieces/polycubes/Zc6.png 12 non-planar
Zd6
("Zb1")
images/pieces/polycubes/Zd6.png 24 non-planar
Ze6
("Zb2")
images/pieces/polycubes/Ze6.png 24 non-planar
Zf6
("Zb3")
images/pieces/polycubes/Zf6.png 12 non-planar

Coordinate System

Polycubes (including solid pentominoes and Soma cubes) use a 3-dimensional (x,y,z) Cartesian coordinate system.

Each unit cube has 6 immediate neighbors. The neighbors of the cube at coordinates (x, y, z) are:

{(x+1, y, z), (x-1, y, z), (x, y+1, z), (x, y-1, z), (x, y, z+1), (x, y, z-1)}

See the FAQ for more details.