Transcript Polyominoes - Mathematical sciences

Polyominoes
Presented by Geometers
Mick 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 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
“Founding Father”
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 = 26
P = 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 respectively
The 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,
Wong (2000)
and Tony Guttmann, Iwan Jensen and Ling Heng
Kevin Gong used Parallel Programming to enumerate
polyominoes with the “rooted translation method”
Side by Side Comparison
name
Sloane
free
onesided
fixed
with holes
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.
B
C
D
A
E
F
G
K
H
I
J
L
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.
Figure 1
Figure 2
Figure 3
Figure 6
Figure 4
Figure 5
Determine the surface area and volume of each cube that you form.
Grades 6 to 8 Example
Given the original hexomino below, classify each transformation as
either a flip, slide, turn, or scaling.
Original Figure
Transformation 3
Transformation 1
Transformation 4
Transformation 5
Transformation 2
“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:
1.
2.
3.
4.
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 npolyominoes . . . 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
Hexiamonds
Bar
Crook
Crown
Sphinx
Snake
Yacht
Chevron
Signpost
Lobster
Hook
Hexagon
Butterfly
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_pentominoe
s.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!