Symmetric Pentacube Triples

A pentacube is a geometric solid formed by joining five equal cubes face to face. There are 23 pentacubes, six with distinct mirror images, making 29:

The number of pentacube triples, or sets of three different pentacubes, is C(29, 3), or 3654. Most of these triples can be joined to form symmetric 15-cubes. These 39 cannot: AFI, AIS/AIS′, AIT, ASX/AS′X, ATX, CFI/C′FI, CIK/C′IK, CKX/C′KX, HIX/H′IX, HIZ/H′IZ, IEV/IE′V, IJW/IJ′W, IJX/IJ′X, IKS/IKS′, IKU, IQT, IQZ, IRX/IR′X, JXZ/J′XZ, KLX, QTX, QXZ, RTX/R′TX, RXZ/R′XZ.

At the other extreme, the triple BLN can form 27086 different symmetric polycubes!

Just 23 triples have unique solutions. These triples are shown below. To see a solution, click on the triple. Cross-sections are shown from top to bottom.

BIKBIS (BIS′)CFX (C′FX)CFZ (C′FZ)
HIT (H′IT)HXZ (H′XZ) IJS (IJ′S′)IJZ (IJ′Z)
IRT (IR′T)IRZ (IR′Z)KUWQUX
RSX (R′S′X)

I am indebted to Gál Péter for his investigation of symmetric pentacube pairs.


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Col. George Sicherman [ HOME | MAIL ]