Activity 7
Polyominoes and Polyiamonds

Polyominoes and polyiamonds are connected arrangements of squares and triangles, respectively. They can be used to explore both reflectional and rotational symmetry, they can be arranged to form other shapes, and they can be extended into three-dimensional shapes to form solid objects.

Polyominoes

A polyomino is a connected arrangement of squares that satisfies the following conditions:

1. If two squares touch, they must have a side in common,
   or both must share a side with another square.
2. All squares must be in the same plane.
3. Two polyominoes are the same if one can be flipped,
   turned, or moved so as to coincide with the other.

Dominoes

There is only one domino.

Trominoes

There are two trominoes.

Tetrominoes

There are five tetrominoes. Use cut-out squares or cubes to make all five of the different tetrominoes and sketch the shapes on the grid paper below. Label the shapes a, b, c, d, e and find the number of lines of reflectional symmetry and the order of rotational symmetry for each tetromino.

Pentominoes

There are twelve pentominoes. Use cut-out squares or cubes to make all twelve of the different pentominoes and sketch the shapes on the grid paper below. Label the shapes with a letter that looks like the shape and find the number of lines of reflectional symmetry and the order of rotational symmetry for each pentomino.

Polyiamonds

A polyiamond is a connected arrangement of equilateral triangles that satisfies the following conditions:

1. If two triangles touch, they must have a side in common, or both must share a side with another triangle.
2. All triangles must be in the same plane.
3. Two polyiamonds are the same if one can be flipped, turned, or moved so as to coincide with the other.

Diamonds

There is only one diamond.

Triamonds

There is only one triamond.

Tetriamonds

There are three tetriamonds. Use cut-out triangles to find all three and sketch them on the triangular grid on the next page. Label each shape with a, b, c and examine each tetriamond for reflectional and rotational symmetry and record your results in the table below the grid.

Pentiamonds

There are four pentiamonds. Use cut-out triangles to find all four and sketch them on the triangular grid below. Label each shape with a, b, c, d and examine each pentiamond for reflectional and rotational symmetry and record your results in the table below the grid.

Hexiamonds

Puzzles

The most famous of the polyomino puzzles is one which uses the twelve pentomino shapes. The commercial name used for this puzzle has varied over the years. The most recent name was Hexed. The object of the game is to arrange the twelve shapes into a rectangle. Since each shape is made from five squares, there will be 60 squares among the 12 shapes. The rectangular arrangement would have to be either 3 by 20, 4 by 15, 5 by 12, or 6 by 10. The existence of the "X" shape prohibits a 2 by 30 rectangle. In all, there are 2339 solutions to this puzzle.

Use either a set of plastic pentomino shapes, or cut out a set of shapes, and arrange them into a rectangle. It is easier to do this if a rectangular frame is used and if the shapes are of different colors. Record your solution on one of the grids below by shading in each piece with a different colored pencil. Try to get a solution for each grid.

A similar puzzle exists with the twelve hexiamonds. These twelve shapes can be placed into a rhombus arrangement. Use cut-out hexiamond shapes and arrange them into a rhombus. Record your solution on the rhombus grid below by shading in each piece with a different colored pencil.

     

One puzzle that is more interesting uses the pentomino shapes constructed out of colored cubes. Each shape consists of five cubes that are either glued or locked together. The puzzle is to place the twelve solid pentomino shapes into a rectangular solid arrangement. Since there are 12 shapes with 5 cubes in each shape, there will be a total of 60 cubes. The only possible rectangular solids would have to have dimensions of 2 by 5 by 6, 3 by 4 by 5, or 2 by 3 by 10. All of these are possible.

Construct a set of solid pentomino shapes using interlocking cubes. Fit them together into each of the rectangular solids listed above.