# Polycube Tiling

A polycube is a geometric solid formed by joining equal cubes face to face. The tiling problem is to join copies of one or more polyforms to make a given polyform.

## Boxes

 Pentacubes in a Box. Join copies of a pentacube to make a rectangular prism. Pentacubes in a Box Without Corners. Join copies of a pentacube to make a rectangular prism with its corner cells removed. Pentacubes in a Box With Four Edges Removed. Join copies of a pentacube to make a rectangular prism from which the cells along four parallel edges have been removed. Pentacubes in a Box With All Edges Removed. Join copies of a pentacube to make a rectangular prism from which the cells along the edges have been removed. Pentacube Pair Boxes. Join copies of two pentacubes, using one of each, to make a rectangular prism. Pentacube Pair Odd Boxes. Join copies of two pentacubes, using one of each, to make a rectangular prism with odd dimensions.

## Other Prisms

 Polycube Prisms. Join copies of a polycube to make a prism. Tiling L Shapes with a Pentacube. Join copies of a pentacube to make an L-shaped prism. L Shapes from Pentacube Pairs. Join copies of two pentacubes to make an L-shaped prism. L Shapes from the 29 Pentacubes. Join the 29 pentacubes to make an L-shaped prism. Tiling a Triangular Prism Polycube with a Tetracube. Find a triangular prism polycube that can be tiled with copies of a given tetracube. Tiling a Pyramid Prism Polycube with a Tetracube. Find a pyramid prism polycube that can be tiled with copies of a given tetracube. Tiling a Diamond Prism Polycube with a Tetracube. Find a diamond prism polycube that can be tiled with copies of a given tetracube. Tiling a Triangular Prism Polycube with Two Pentacubes. Find a triangular prism polycube that can be tiled with two given pentacubes, using at least one of each. Tiling a Pyramid Prism Polycube with Two Pentacubes. Find a pyramid prism polycube that can be tiled with two given pentacubes, using at least one of each. Tiling a Diamond Prism Polycube with Two Pentacubes. Find a diamond prism polycube that can be tiled with two given pentacubes, using at least one of each.

## Other Tilings

 Polycube Reptiles. Join copies of a polycube to make a larger copy of itself. Polycube Bireptiles. Join two copies of a polycube, then dissect the result into equal smaller copies of it. Proper Minimal Polycube Irreptiles. Join variously sized copies of a polycube to make a larger copy of itself, using fewer copies than would be needed if they were all the same size. Prime Boxes for the Clip Pentakedge. Identify the irreducible boxes that can be tiled by the Clip Pentakedge. 33 + 43 + 53 = 63. Dissect a cube of side 6 to make cubes of sides 3, 4, and 5. Tiling a Solid Diamond Polycube With Right Tricubes. Dissect an octahedron-shaped polycube into L-shaped tricubes. Symmetric Pentacube Triples. Join three different pentacubes to form a symmetric polycube. Pentacube Pair Pyramids. Join copies of two pentacubes to make a pyramid. Tiling a Cubic Polycube with Two Pentacubes. Join copies of two pentacubes to make a cube with side 5. Tiling a Rhonic Polycube with Two Pentacubes. Join copies of two pentacubes to make a polycube shaped like a rhombic dodecahedron. Tiling a Cuboctal Polycube with Two Pentacubes. Join copies of two pentacubes to make a polycube shaped like a cuboctahedron. Tiling a Heavy Cuboctal Polycube Shell with Pentacubes. Excise a heavy cuboctal polycube from a bigger one and tile the residue with copies of two or three pentacubes. Filling Space with the Pansymmetric Heptacube. Use the 3D analogue of an X pentomino to fill space. Tiling a Scaled Pentacube with a Pentacube. Use a pentacube to tile various pentacubes scaled up by 2 or 3. Prisms with Square-Symmetric Bases from the 29 Pentacubes. Use all 29 pentacubes to form a prism whose base is a fully-symmetric 29-omino. Twisted Polycube Rings. Construct a re-entrant polycube with a twist in its loop. Entangled Polycubes. Make a rectangular box out of polycubes that cannot be separated.

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Col. George Sicherman [ HOME | MAIL ]