Sheet D (Geometric Structures)

**Polyominoes II**

prepared by:

Joseph Malkevitch

Department of Mathematics and Computing

York College (CUNY)

Jamaica, New York 11451

Email: malkevitch@york.cuny.edu

web page:

http://www.york.cuny.edu/~malk

1. Given a polyomino, a graph H can be obtained from this polyomino by putting a dot into each of the squares of the polyomino and joining two such dots if the squares they represent have an edge in common.

a. Draw all the graphs H associated with the triominoes

b. Draw all the graphs H associated with the tetrominoes

c. Draw all the graphs H associated with the pentominoes

Do you notice any patterns?

2. a. Write down the perimeter for all 12 of the pentominoes.

b. Which pentomino(s) solves the isoperimetric problem? (Which pentominoe has the smallest perimeter for fixed area?)

3. Do all the pentominoes that fold to a cubical box without a lid have the same perimeter?

4. Is any tree the H graph for some polyomino? If not, what conditions must a tree obey to be the H graph of some polyomino? (See Problem 1.)

5. What restrictions, if any, must there by on an arbitrary graph to be the H graph for some polyomino? (See Problem 1.)