Pentacube Compatibility

A pentacube is a solid made of five cubes joined face to face. There are 23 pentacubes, not distinguishing reflections and rotations:

The six blue tiles have left- and right-handed forms. Kate Jones's systematic names are shown in green. The mirror forms of V1, S1–S2, and L1–L4 are called V2, N1–N2, and J1–J4. But L3 and J3 are identical because they have mirror symmetry through a plane diagonal.

Here are the least known numbers of tiles needed to construct a solid that can be tiled with either of two pentacubes:

  A B C E F H I J K L M N P Q R S T U V W X Y Z
A*22222102224222224242222
B2*222242422222242224224
C22*222102224222224232622
E222*2242222222222224222
F2222*252222222222442222
H22222*42224222224242822
I10410454*3102202241064545525
J2222223*224222222224422
K242222102*22222222222822
L222222222*4222222222822
M42422420424*422242884224
N22222222224*222222221222
P222222222222*2222222422
Q2222224222222*224232622
R222222102222222*24242422
S242222622242222*2222622
T4242244222222442*326422
U22224252228222223*22224
V423244422282234222*63224
W2424225422422222626*624
X2262285488212464642326*220
Y222222222222222222222*2
Z24222252224222222444202*

6 Tiles

8 Tiles

10 Tiles

12 or More Tiles

One-Sided Pentacubes

Here are the values when tile reflection is not allowed. The blue figures differ from the values where reflection is allowed.

  A B C C′ E E′ F H H′ I J J′ K L M N P Q R R′ S S′ T U V W X Y Z
A*23322222102222422233224242222
B2*222222242242222222442224224
C32*222224102222422222244442622
C′322*22242102222422222424442622
E2222*222242222222232222224222
E′22222*22242222222223222224222
F222222*2252222222222222442222
H2224222*242222422222224242822
H′22422222*42222422222224242822
I104101044544*33102202241010664545525
J2222222223*222422222222224422
J′22222222232*22422222222224422
K2422222221022*2222222222222822
L2222222222222*422222222222822
M424422244204424*42222442884224
N222222222222224*22223322221222
P2222222222222222*222222222422
Q22222222242222222*22224232622
R3222322221022222222*2234442422
R′32222322210222222222*324442422
S24242222262222432223*23222622
S′244222222622224322322*3222622
T4244222444222222244433*326422
U22442242252222822244223*22224
V424422444422228223442222*63224
W2422442225442242222222626*624
X2266222885448821246446642326*220
Y222222222222222222222222222*2
Z24222222252222422222222444202*

Last revised 2012-03-19.

Back to Polyform Curiosities.
Col. George Sicherman [ HOME | MAIL ]