Polyominoes

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Polyominoes

• Presented by Geometers
• Mick Raney Sunny Mall

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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?

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History of Polyominoes

• First mentioned by Solomon Golomb in a 1953 paper
• Initially, appeal was primarily in puzzles and
  games such as Tetris
• Multiple games have been spawned since the
  inception of the concept
• Numerous sites offer on-line and downloadable
  play
• School projects have resulted in sponsored
  websites and groups
• Development led to discussion of numbers and
  types of polyominoes
• Applications include packing problems in 2D and
  3D
• Current areas of study include Combinatorial
  Geometry

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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.

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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 26
P 34
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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
  respectively
• The claim is that the ratio of fixed to one-sided
  is lt4 and fixed to free is lt2 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

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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
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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?ID60

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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.

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Pre-K to 2 Example

• Sort the shapes below. Explain how you sorted
  them.

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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

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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.

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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.

B
C
D
A
E
F
G
K
L
J
I
H
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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!

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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.

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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.

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Grades 6 to 8 Example

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

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• 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

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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.

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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.

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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)

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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.

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Grades 13 to 16 Example

• Polyiamonds

Hexiamonds Bar Crook Crown Sphinx Snake Yacht Chev
ron Signpost Lobster Hook Hexagon Butterfly
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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_p
  entominoes.htm (Marks 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?

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We now welcome your . . .

• Questions?
• Comments.
• Heckling!