# Polyform Curiosities

A polyform is a geometric shape formed by joining copies of a single shape. The best known polyforms are polyominoes, made from squares. Solomon W. Golomb coined the term, studied polyominoes extensively, and published a book about them. Other plane polyforms include polyiamonds, made from equilateral triangles; polyaboloes or polytans, made from isosceles right triangles; polyhexes, made from regular hexagons; and polylines or polysticks, made from straight lines. Solid polyforms include polycubes, polyominoids, and polyrhons.

• Remembrance of Software Past
• Exclusion
• Tiling
• Compatibility
• Pairwise Compatibility
• Multiple Compatibility
• Galvagni Compatibility
• Baiocchi Figures
• Cell Shifting
• Oddities
• Catalogues
• Printable Grids
• Wallpaper

• ## Remembrance of Software Past

It isn't trivial!  —Livio Zucca

From 1999 to 2005 Livio Zucca maintained a vast website of mathematical recreations called Remembrance of Software Past. It was full of new ideas, attracting contributions from many talented mathematicians. Unfortunately, it was hosted by GeoCities, which was shut down in 2009. Fortunately Giovanni Resta reconstructed Remembrance in full on his host Iread.it.

Deteriorating eyesight left Livio unable to maintain his site. With Livio's permission I have revived his Remembrance pages about polyomino compatibility. I have updated the links and many of the solutions. Livio and I welcome new contributions. If you would like to submit an improved solution, or a new solution, or a new idea, please write me at colonel@monmouth.com.

 Pento-Tro-Dominoes. Find a figure that can be tiled with any of a given pentomino, tromino, and domino. Pento-Tetro-Trominoes. Find a figure that can be tiled with any of a given pentomino, tetromino, and tromino. Tetrominoes Challenge. Find a figure that can be tiled with each of a given set of tetrominoes and no others. Triple Pentominoes. Find a figure that can be tiled with each of three given pentominoes. Pentomino Odd Pairs. Find a figure that can be tiled with an odd number of either of two pentominoes.

## Exclusion

No room! No room! they cried out when they saw Alice coming.

—Lewis Carroll, Alice's Adventures in Wonderland

The exclusion problem is to remove as few cells as possible from a given figure or the plane so as to exclude a given polyform.

 Hexomino Exclusion. Exclude a hexomino from a checkerboard. Polyking Exclusion. Exclude a polyking from a checkerboard. Polyiamond Exclusion. Exclude a polyiamond from the plane. Polyhex Exclusion. Exclude a polyhex from the plane.

## Tiling

The slabs made a most intricate and fascinating design, but a thoroughly unobtrusive one, unless one paid deliberate attention to it.

—Carlos Castaneda, The Second Ring of Power

The tiling problem is to join copies of one or more polyforms to make a given polyform.

 Two-Pentomino Balanced Rectangles. Tile a rectangle with two pentominoes in equal quantities. Three-Pentomino Balanced Rectangles. Tile a rectangle with three pentominoes in equal quantities. Two-Hexiamond Balanced Hexagons. Tile a regular hexagon with two hexiamonds in equal quantities. Three-Hexiamond Balanced Hexagons. Tile a regular hexagon with three hexiamonds in equal quantities. Polyiamond Hexagon Tiling. Tile a straight or ragged hexagon with various polyiamonds. Bireptiles. Dissect two joined copies of a polyform into equal smaller copies.

Then clap four slices of pilaster on 't,
That laced with bits of rustic makes a front.

—Alexander Pope, “Epistle to Richard Boyle”
 Tetrads. A polyform tetrad is four copies of a polyform, joined so every copy shares a boundary with every other.

## Compatibility

A maddening identity of the big picture is arrived at without using any similar pieces.

—Arpad Arutinov, The Back Door of History

The Compatibility Problem is to construct a figure that can be tiled with each of a set of polyforms.

### Pairwise Compatibility

#### Polyominoes

 Pentomino Compatibility. Given two pentominoes, construct a figure that can be tiled with either. Holeless Hexomino Compatibility. Given two hexominoes, construct a holeless figure that can be tiled with either. Tetromino-Pentomino Compatibility. Given a tetromino and a pentomino, construct a figure that can be tiled with either. Holeless Tetromino-Hexomino Compatibility. Given a tetromino and a hexomino, construct a holeless figure that can be tiled with either. Holeless Pentomino-Hexomino Compatibility. Given a pentomino and a hexomino, construct a holeless figure that can be tiled with either. Holey Heptomino Compatibility. Which polyominoes are compatible with the holey heptomino? Voided Square Octomino Compatibility. Which polyominoes are compatible with the square octomino with a hole in it? Square Enneiomino Compatibility. Which polyominoes are compatible with the square enneiomino? Holeless Pentomino Odd Pairs. Holeless solutions for Livio Zucca's Pentomino Odd Pairs.

#### Polyiamonds

 Hexiamond Compatibility. Given two hexiamonds, construct a figure that can be tiled with either. Heptiamond Compatibility. Given two heptiamonds, construct a figure that can be tiled with either. Octiamond Compatibility. Given two octiamonds, construct a figure that can be tiled with either. Mixed Polyiamond Compatibility. Given two polyiamonds of different orders, construct a figure that can be tiled with either. Triangle Enneiamond Compatibility. Find a figure that can be tiled with a given polyiamond and with the triangular enneiamond.

#### Polyhexes

 Pentahex Compatibility. Given two pentahexes, construct a figure that can be tiled with either. Pentahex Odd Pairs. Given two pentahexes, construct a figure that can be tiled with an odd number of either. Mixed Polyhex Compatibility. Given two polyhexes of different orders, construct a figure that can be tiled with either. Seven Euphoric Pentahexes. Each of these pentahexes is compatible with all 82 hexahexes. Three Euphoric Hexahexes. Each of these hexahexes is compatible with all 82 hexahexes. Ring Hexahex Compatibility. Which polyhexes are compatible with the ring hexahex? Disk Heptahex Compatibility. Which polyhexes are compatible with the disk heptahex? Yin/Yang Pentahexes. Given two pentahexes, find a polyhex that can be tiled with either, leaving a hole shaped like the other.

#### Other Plane Polyforms

 Pentapent Compatibility. Figures that can be tiled by two different pentapents. Mixed Polypent Compatibility. Figures that can be tiled by polypents of different orders. Ring Hexapent Compatibility. Which polypents are compatible with the ring hexapent? Pentahept Compatibility. Given two pentahepts, construct a figure that can be tiled with either. Tetrabolo Compatibility. Given two tetraboloes, construct a figure that can be tiled with either. Tetrabolo-Pentabolo Compatibility. Given a tetrabolo and a pentabolo, construct a figure that can be tiled with either. Tetrahop Compatibility. Given two tetrahops, construct a figure that can be tiled with either. Didrafter Compatibility. Given two didrafters, construct a figure that can be tiled with either.

#### Solid Polyforms

 Tetracube Compatibility. Given two tetracubes, construct a figure that can be tiled with either. Pentacube Compatibility. Given two pentacubes, construct a figure that can be tiled with either.

### Multiple Compatibility

 Holeless Triple Pentominoes. Holeless solutions for Livio Zucca's Triple Pentominoes. Pentomino Odd Triples. Like Livio Zucca's Triple Pentominoes, with an odd number of tiles. Pento-Tetro-Tetrominoes. Given a pentomino and two tetrominoes, find a polyomino that each can tile. Pento-Pento-Tetrominoes. Given two pentominoes and a tetromino, find a polyomino that each can tile. Multiple Polyomino Compatibility. Figures that can be tiled by many polyominoes of the same order. Triple Hexiamonds. Given three hexiamonds, construct a figure that can be tiled with each. Hexa-Penta-Tetriamonds. Given a hexiamond, a pentiamond, and a tetriamond, construct a figure that can be tiled with each. Multiple Polyiamond Compatibility. Figures that can be tiled by many polyiamonds of the same order. Zucca's Challenge Problem for Polyiamonds. Given a set of polyiamonds of the same order, construct a figure that can be tiled with any member of the set and no other. Multiple Polyhex Compatibility. Figures that can be tiled by many polyhexes of the same order. Zucca's Challenge Problem for Tetrahexes. Given a set of tetrahexes, construct a figure that can be tiled with any member of the set and no other. Zucca's Challenge Problem for Extrominoes. Given a set of extrominoes, or extended trominoes, construct a figure that can be tiled with any member of the set and no other. Multiple Polypent Compatibility. Figures that can be tiled by many polypents of the same order. Zucca's Challenge Problem for Polypents. Given a set of polypents, construct a figure that can be tiled with any member of the set and no other.

### Galvagni Compatibility

Galvagni's problem is to construct a figure that can be tiled with a polyform in more than one way. A Reid Figure is a Galvagni Figure without holes. A Plover Figure is a Galvagni Figure made without reflecting the polyform.

### Baiocchi Figures

A Baiocchi figure has full symmetry and is formed by joining copies of a single polyform. Most Baiocchi figures involve Galvagni figures.

### Cell Shifting

The Cell Shifting Problem is to join copies of a polyform to construct two figures that differ in just one cell, as near as possible.

 Cell Shifts for Polyominoes of order up through 6. Cell Shifts for Pentomino Pairs. Cell Shifts for Polyhexes of order up through 5. Cell Shifts for Pentahex Pairs. Cell Shifts for Heptominoes. Cell Shifts for Hexahexes. Cell Shifts for Polyiamonds of order up through 7. Cell Shifts for Heptahexes. Cell Shifts for Octiamonds. Knight's-Move Cell Shifts for Polyominoes.

## Oddities

That is another of your odd notions, said the Prefect, who had the fashion of calling everything odd that was beyond his comprehension, and thus lived amid an absolute legion of oddities.

—Poe, The Purloined Letter

An oddity is a figure with binary symmetry made by joining an odd number of copies of a polyform.

### Polyominoes

 Polyomino Oddities. Oddities for polyominoes of order up to 7. Pentomino Oddities. Pentomino oddities with specific symmetries. Hexomino Oddities. Hexomino oddities with specific symmetries. Polyomino Semi-Oddities. Semi-oddities for polyominoes. A semi-oddity is a figure with quaternary symmetry and an even number of tiles that is not a multiple of 4. Heptomino Oddities. Heptomino oddities with specific symmetries.

### Polykings

 Pentaking Oddities. Oddities with specific symmetries for pseudopolyominoes of order 5.

### Polyiamonds

 Pentiamond, Heptiamond, and Enneiamond Oddities. Oddities for pentiamonds, heptiamonds, and enneiamonds. Oddities for polyiamonds of odd order can have only bilateral symmetry. Hexiamond Oddities. Hexiamond oddities with specific symmetries. Octiamond Oddities. Octiamond oddities with specific symmetries. Polyiamond Tri-Oddities. These are like oddities but with ternary symmetry.

### Polyhexes

 Polyhex Oddities. Oddities for polyhexes of order up to 5, with specific symmetries. Hexahex Oddities. Hexahex oddities with specific symmetries. Heptahex Oddities. Heptahex oddities with specific symmetries. Polyhex Tri-Oddities. Tri-oddities for polyhexes of order up to 5. Hexahex Tri-Oddities. Tri-oddities for hexahexes. Heptahex Tri-Oddities. Tri-oddities for heptahexes.

### Polypents

 Tetrapent Oddities. Oddities for tetrapents. Pentapent Oddities. Oddities for pentapents. Hexapent Oddities. Oddities for hexapents. Heptapent Oddities. Oddities for heptapents.

### Polyhepts

 Tetrahept Oddities. Oddities for tetrahepts.
Pentahept Oddities. Oddities for pentahepts.

### Polyocts

 Polyoct Oddities. Oddities for polyocts of order 1 through 5.

### Polyenns

 Pentenn Oddities. Oddities for pentenns.

### Polydecs

 Polydec Oddities. Oddities for polydecs.

## Catalogues

It was a strange collection, . . . but so much larger and so much more varied that I think I never had more pleasure than in sorting them.

—Robert Louis Stevenson, Treasure Island

 Catalogue of Polypents. Enumerations and pictures of these neglected polyforms. Catalogue of Polyhepts. Some more neglected polyforms. Catalogue of Polyocts. Still more neglected polyforms. Catalogue of Polyhops. Thomas Atkinson's hopscotch-style polyominoes. Catalogue of Polyjogs. Polyforms formed of squares joined by half edges. Catalogue of Polypentagrams. Polyforms formed of pentagrams joined edge to edge. Catalogue of Polypennies. Polyforms formed by joining equal disks tangentially. Catalogue of Polyrhons. Polyforms formed by joining rhombic dodecahedrons. Polycube Symmetries. In how many ways can a polycube be symmetrical?

## Printable Grids

Something was out to get him, something here, among the paper.

—Thomas Pynchon, Gravity's Rainbow

 Printable Grid Files. Square, triangular, and hexagonal grids, for those who like to play with polyominoes, polyiamonds, and polyhexes.

## Wallpaper

My wallpaper and I are fighting a duel to the death. One or the other of us has to go.

—Oscar Wilde

 Tetromino Wallpaper. Pentiamond Wallpaper. Tetrahex Wallpaper. Dragomino Wallpaper. A Dragomino is the polyomino equivalent of a Dragon Curve.