Searching for a Mathematical Definition
This exercise is designed for liberal arts students, teachers, and anyone who
believes that mathematics is about numbers.
Introduction: The purpose of this discussion is to understand the nature
and role of definitions in mathematics. For example, suppose we say that
the value of an investment portfolio is increasing over time. What is our
definition of increasing? Do we mean that the value only rises as we move
forward in time? Or do we mean that over long periods of time the value rises
even though there may be unpredictable fluctuations during shorter periods
of time? The answer certainly depends on the types of investments in the
portfolio. If only stocks are present (as opposed to bonds), then short term
fluctuations, including losses, should be expected. So our definition, in this
case, of the word increasing can be inferred from context.
Although the word “definition” can have a broad meaning to most people, mathematicians maintain a very rigid interpretation of this term. The
reason is that meaning cannot be easily inferred from context in most mathematical and scientific applications. The next few pages are designed to help
you understand the mathematical notion of a definition by developing one
particular example. The concept that we are about to discuss is a familiar
one. However, we will learn that defining this concept can be a difficult
process. Along the way, I also hope to dispel some of the stereotypes about
the nature of mathematics.
Exercise 1: Compare and contrast the snowflakes shown in slide 1. Are any
two the same? Do the snowflakes have any qualities or properties that make
them the ”same” in some sense?
(Photo from [1])
1
“The beauty of a snow crystal depends on its mathematical regularity and
symmetry; but somehow the association of many variants of a single type, all
related but not two the same, vastly increases our pleasure and admiration.”
D’Arcy Thompson
(On Growth and Form, Cambridge, 1917)
Exercise 2: Describe what the word “symmetry” means to you. Feel free to
use examples in your description. It may be helpful to compare an object
with lots of symmetry to another object with little or no symmetry.
So What Is Symmetry in Mathematics?
Now let us attempt to define symmetry. The following definition from
Encarta 96 describes symmetry as “orderly, mutually corresponding arrangement of various parts of a body producing a proportionate, balanced form.”
Although this characterization provides an intuitive notion of symmetry, it
is much too imprecise for mathematical usage. There would be many mathematical objects that may or may not have symmetry depending on who you
ask. A rigorous mathematical definition is the following:
Definition: A symmetry f of a geometric object O is a mathematical transformation that maps O onto itself. We say that O is left invariant by the
symmetry f .
In other words, we say that a symmetry of O is an imperceptible motion.
Suppose that two people have a square card on a rectangular table so that the
sides of the card are parallel to the sides of the table. While the first person is
looking the other way, the second person rotates the card 90 degrees. When
the first person looks at the card again, they will not realize it has been
moved. On the other hand, a 45 degree rotation would be perceivable. So
we say that a 90 degree rotation is a symmetry of the square whereas a 45
degree rotation is not. At this point you may wonder how many symmetries
there are for a square. In addition to a 90 degree rotation, motions of 180
and 270 degrees are also imperceptible. So we now have 3 symmetries of a
2
square. There are also 4 reflections about four different axes. This brings the
total to 7. For mathematical reasons we add the motion that does nothing
to the list. This brings us to the final total of 8 symmetries for a square.
You may wonder why we did not include a rotation of -90 degrees. This is
an imperceptible motion. However, the net effect of this motion is the same
as a rotation of 270 degrees. We consider these to be the same rotation.
6
-
?
3
Activity: Comparing Symmetries (Physical models will be provided in
class. Students will try to determine how many symmetries each object has).
The Price We Have to Pay ........
By now you may have realized that our definition of symmetry is very
restrictive. Defining the concept in terms of a transformation rules out many
objects that we may feel posses symmetry. For example, people exhibit
bilateral symmetry. This means the left side of a human body looks basically
the same as the right side. However, there are some small variations, such
as freckles, that “break” the symmetry. If we consider the interior of a
person, then the off-center heart is a major asymmetry. Still, it is clear that
humans posses a symmetric quality. So our perceptions of patterns in nature
sometimes necessitate a more general usage of this concept.
Exercise 3: Has your understanding of the word “definition” changed at all
during this discussion?
Exercise 4: Has your understanding of the nature and role of mathematics
changed at all during this discussion?
Exercise 5: Examine the 23 definitions in Book 1 of Euclid’s Elements (see
[2] for an online version). Do all of these meet the modern mathematical
criterion for a definition? Why are some of these concepts now taken to be
undefined terms?
Exercise 6: Compare and contrast the meaning of symmetry and chaos. You
do not need to think of these concepts mathematically. It may help to give
examples of objects that possess one or both of these characteristics.
4
References
[1] “The Snowflake: Winter’s Secret Beauty”, Kenneth Libbrecht, with photographer Patricia Rasmussen, Voyageur Press, October 2003.
[2] http://aleph0.clarku.edu/ djoyce/java/elements/elements.html
[3] “Symmetry, Shape, and Space: An Introduction to Mathematics Through
Geometry. L. Christine Kinsey and Teresa E. Moore. Key College Publishing, 1999.
[4] http://www.its.caltech.edu/ atomic/snowcrystals/
[5] http://snowflakes.lookandfeel.com/
5