Mathematical Symmetries

1
Mathematical Symmetries and the Soma
Cube Westerly Drong April 13, 2001
2
Introduction
What is a Soma? What are symmetries? How do
symmetries help solve Soma puzzles?
3
Seven Pieces of the Soma Puzzle
1 V
2 L
3 T
4 Z
5 A
6 B
7 P
4
Soma Puzzles
Cube
Crystal
Tower
Airplane
Dog
5
Notation
Give each piece a number and write the figure
in text
Soma007 7..6 / 7733366 5..1 / 7443261 5..1 /
5544222
6
What Is a Symmetry?
A symmetrical transformation must, by
definition retain distance, angle, shape, and
size attributes
A transformation maps a set of points from an
original figure, called the domain, onto another
called the range
I look at symmetries that also imply
self-similarity. This means that symmetrical
transformation exists where every element of the
range is an element of the domain
7
A B D C
Symmetries of a Square
A B D C
B C A D
C D B A
D A C B
Rotational Symmetries
B A C D
D C A B
A D B C
C B D A
Reflectional Symmetries
8
Matrix Notation
Any rotational or mirror transformation can be
expressed in the form of 3x3 matrices which
follow these rules
1. The only possible values are -1, 0, and 1
2. Each row and each column of the matrix must
have one and only one non-zero value
0 1 0 -1 0 0 0 0 1 90º rotation about z axis
1 0 0 0 1 0 0 0 1 identity matrix
-1 0 0 0 1 0 0 0 1 planer reflection
9
Matrix Notation
These matrices are applied to each A, B, C point
in a Soma figure to give it a new X, Y, Z point
X Y Z
X Y Z
X Y Z
A B C
A B C
A B C
0 1 0 -1 0 0 0 0 1 90º rotation about z axis
1 0 0 0 1 0 0 0 1 identity matrix
-1 0 0 0 1 0 0 0 1 planer reflection
10
Matrix Multiplication
It should be noted that matrix multiplication can
be used to show the progression of
transformations
0 1 0 -1 0 0 0 0 1 90º clockwise rotation
about z axis
0 -1 0 1 0 0 0 0 1 270º clockwise rotation
about z axis

-1 0 0 0 -1 0 0 0 1 180º clockwise
rotation about z axis
X
11
Symmetries of the Pieces
Each one of the seven pieces of the Soma have
symmetries. Most have a reflectional symmetry
with themselves Pieces 5 and 6 are the only
ones which do not, and they are symmetrical
with each other
12
Reducing Possible Solutions

• If a symmetry applies to a figure then the first
  copy of
• piece 1 (referred to as 1a) does not need to be
  placed in
• any location that is symmetrical to the previous
  placement

• By controlling the symmetries of piece 7,
  planer mirror
• pairs of solutions can be eliminated

13
Piece 7
There are exactly 8 orientation variants of piece
7
. 77 7 . . . 7
. . 7 . 7 . 7 7
. 7 . . 7 7 . 7
7 77 . 7 . . .
7 . 7 7 . . 7 .
7 7 . 7 . 7 . .
7 . . . 7 77 .
. . . 7 . 77 7
If solutions are generated containing piece 7
variants from the top half of the chart above,
then no solutions will be produced that are
their planer mirror pair.
14
Symmetries of the Puzzles
There are only two solutions to the Gorilla.
The two solutions are actually mirror images of
each other
. .5 . . . .5 . . 6 . . . 765577 . . .
. . 66447 . . . . . 33344 . . . . . .
322 . . . . . . . 112 . . . . . . . 1. 2
.
. .6 . . . .6 . . 7 . . . 577665 . . .
. . 74455 . . . . . 44333 . . . . . .
223 . . . . . . . 211 . . . . . . . 2. 1
.
The Gorilla
By limiting the use of piece 7, these
symmetrical solutions can be eliminated
15
Symmetries of the Puzzles
This puzzle has two planar mirror reflections
which implies a rotational symmetry in addition
to the reflectional symmetry. Puzzles in this
family can be tamed by using even fewer variants
of orientation for piece 7.
The Bathtub
16
Next . . .
Working in the fourth dimension
Symmetry Matrix
Puzzle Piece
1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
X . . . X . . . X X X . . . . .
17
Acknowledgements
Mrs. Shilepsky Mr Stiadle Bob Nungester Thorleif
Bundgård all of my fellow MPS majors
18
Problems worthy of attack Prove their worth by
hitting back -Piet Hein
Any Questions?