Polyominoes

Presented by GeometersPresented by GeometersMick Raney & Sunny MallMick Raney & Sunny Mall

Our Task

How does the particular mathematics discussed fit into the tapestry of geometry as a whole?What are some aspects of its historical development?When does the particular mathematics appear in the K-16 curriculum, and how is it unfolded throughout the curriculum?What websites, software, etc., can assist in visualizing, representing, and understanding the mathematics?What are some good additional references, either physical or online?

History of Polyominoes

First mentioned by Solomon Golomb in a 1953 paperInitially, appeal was primarily in puzzles and games such as TetrisMultiple games have been spawned since the inception of the conceptNumerous sites offer on-line and downloadable playSchool projects have resulted in sponsored websites and groupsDevelopment led to discussion of numbers and types of polyominoesApplications include packing problems in 2D and 3DCurrent areas of study include Combinatorial Geometry

“Founding Father”

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A Quick Description

Solomon Golomb, mathematician and inventor of pentominoes. If two squares side by side is a "domino", then n squares joined side by side to make a shape is a "polyomino", an idea invented by mathematician Solomon Golomb of USC. There are two distinct "triominoes" (three squares): a straight line and an L. There are five distinct "tetrominoes" (four squares), popularized in the computer game Tetris, which was inspired by Golomb's polyominoes.

Some other applications

Convex Polyominoes - with perimeter equal to “bounding box”

Possible use to estimate size of irregular shapes as follows: (does this problem look familiar?)

P = 26P = 34

Enumeration

Most enumeration schemes use computer programs

We define free, one-sided and fixed polyominoes

Free can be picked up, moved or flipped

One-sided

Fixed can be rotated, translated, flipped

For example, there are 12, 18, and 63 pentominoes respectivelyThe claim is that the ratio of fixed to one-sided is <=4 and fixed to free is <=2 *D. H. Redelmeier, W. F. Lunnon, Kevin Gong, Uwe Schult, Tomas Oliveira e Silva, and Tony Guttmann, Iwan Jensen and Ling Heng Wong (2000)

Kevin Gong used Parallel Programming to enumerate polyominoes with the “rooted translation method”

Side by Side Comparison

name free one-sided

fixed with holes

Sloane   A000105 A000988 A001168 A001419

1 monomino 1 1 1 0

2 domino 1 1 2 0

3 triomino 2 2 6 0

4 tetromino 5 7 19 0

5 pentomino 12 18 63 0

6 hexomino 35 60 216 0

7 heptomino 108 196 760 1

8 octomino 369 704 2725 6

9   1285 2500 9910 37

10   4655 9189 36446 195

Pentominoes Online

A five square polyomino is a pentomino. There are a multitude of applications for pentominoes from games to tilings to packing problems.

http://www.kevingong.com/Polyominoes/

http://www.stetson.edu/~efriedma/polyomin/

http://mathnexus.wwu.edu/Archive/resources/detail.asp?ID=60

Grades Pre-K to 2

Sort, classify, and order polyominoes by number of squares needed to form the shapes.

Sort polyominoes that have seven or more squares by “ones with holes” and “ones without holes.”

Extend patterns such as a sequence of polyomino shapes.

Classify each pentomino according to the letter that is most closely resembles.

Pre-K to 2 Example

Sort the shapes below. Explain how you sorted them.

Pre-K to 2 Example

Match each pentomino with the letter that it most closely resembles:

F I L N P T U V W X Y Z

Grades 3 to 5

Identify, compare, analyze and describe attributes of two-dimensional polyominoes and the three-dimensional open and closed boxes that pentominoes and hexominoes form.

Classify nets of pentominoes and hexominoes based on whether or not they will fold into boxes.

Investigate, describe and reason about the results of transforming pentominoes and hexominoes into boxes.

Build and draw all the pentominoes. How many are there? Determine the area and perimeter of each pentomino. Create and describe mental images of polyominoes. Identify and build a three-dimensional object from two-dimensional

representations of that object.

Grades 3 to 5 Example

Which pentominoes do you think will make a box (open cube)? Make a prediction. Then cut out the shapes and try to form a box.

A

B

C D

E F

G

L

K

JIH

Grades 3 to 5 Example

Using all 12 3-D pentominoes, make the following:

• 6 x 10 rectangle• 5 x 12 rectangle• 4 x 15 rectangle• 3 x 20 rectangle• 8 x 8 square with 4 pieces missing in the middle• 8 x 8 square with 4 pieces missing in the corners• 8 x 8 square with 4 pieces missing almost anywhere• 3 x 4 x 5 cube• 2 x 5 x 6 cube• 2 x 3 x 10 cube• 2D replica of each piece, only three times larger• 5 x 13 rectangle with the shape of 1 pentomino piece missing in the middle• Tessellations using a pentomino• Hundreds of other shapes!

Grades 6 to 8

Use two-dimensional polyomino nets that form three-dimensional boxes to visualize and solve problems such as those involving surface area and volume.

Describe sizes, positions, and orientations of polyominoes under informal transformations such as flips, turns, slides and scaling.

Grades 6 to 8 Example

Which hexominoes do you think will make a cube? Make a prediction. Then cut out the shapes and try to form a cube.

Determine the surface area and volume of each cube that you form.

Figure 1Figure 2Figure 3Figure 4Figure 5Figure 6

Grades 6 to 8 Example

Given the original hexomino below, classify each transformation as either a flip, slide, turn, or scaling.

Original FigureTransformation 1Transformation 2Transformation 3Transformation 4Transformation 5

“Chasing Vermeer is a novel about a group of middle school students who tackle the mystery behind the disappearance of A Lady Writing, a famous painting by Joahnnes Vermeer. Students employ pentominoes to create secret messages to communicate as they use their problem-solving skills and powers of intuition to solve the mystery. They explore art, history, science, and mathematics throughout their adventure.”

Mathematics Teaching in the Middle School

October 2007

Grades 9 to 12

Using a variety of tools, draw and construct representations of two-dimensional polyominoes and the three-dimensional boxes formed by pentominoes and hexominoes.

Understand and represent translations, reflections, rotations, and dilations of polyominoes in the plane by using sketches and coordinates.

Grades 9 to 12 Example

Draw a pentomino by connecting, in order, the coordinates below.

(0, 0), (0, 1), (2, 1), (2, 0), (1, 0), (1, -1), (-2, -1), (-2, 0), (0, 0)

Find the new set of coordinates to connect after applying the following transformations:

Translate the pentomino 5 units left and 2 units down.Reflect the pentomino over the y-axis.Rotate the pentomino 90° about the point (3, 2).Quadruple the area of the pentomino.

Process Standards Pre-K to 12

Make and investigate mathematical conjectures surrounding polyominoes. (Reasoning and Proof)

Organize their mathematical thinking about polyominoes through communication. (Communication)

Create and use representations to organize, record, and communicate their knowledge of polyominoes. (Representation)

Grades 13 to 16

Explore free and fixed polyominoes and the relationship between them;

Explore one-sided polyominoes; Explore polyominoes with holes; Define the bounds on the number of n-polyominoes; Derive an algebraic formula to determine the number of n-

polyominoes . . . Currently there is not a formula for calculating the number of different polyominoes. There are only smaller result for n, obtained by empirical derivation through the use of computer technology; and

Explore other polyforms (polyabolos, polyares, polycubes, polydrafters, polydudes, polyiamonds, and so on) and the relationships between them.

Grades 13 to 16 Example

Polyiamonds HexiamondsBarCrookCrownSphinxSnakeYachtChevronSignpostLobsterHookHexagonButterfly

The Tapestry

What else?

Tiling problems like: given a rectangular shape, determine the optimum number of polyominoes which will fill the rectangle

http://www.users.bigpond.com/themichells/packing_pentominoes.htm (Mark’s packing pentominoes page)

Combinatorial Geometry: involves many different problems including “Decomposition” problems, covering problems.

The Heesch Problem: seeks a number which describes the maximum number of times that shape can be completely surrounded by copies of itself in the plane. What possible values can this number take if the figure is a polyomino and not a regular polygon?

We now welcome your . . .

Questions?

Comments.

Heckling!