Symmetry Lab
Note: This assignment will not be graded. Students should complete on this word document. Use Track Changes to make your entries.
Purpose:
· To investigate the symmetry operations of an equilateral triangle.
· To gain experience with symbolic operations.
· To open your mind to new types of learning.
Introduction and Theory:
Symmetry appears in nature and is used as an element in art and literature. Symmetry in the physical universe and physical law is more than just aesthetically pleasing or elegant, however. As proved by Emmy Noether (1882-1935), the renowned mathematician, there is a fundamental relationship between symmetry in nature and the conservation laws which are the cornerstones of physics. Noether’s Theorem states that for every continuous symmetry in nature there is a conservation law and vice-versa. Thus conservation of energy, conservation of linear momentum, conservation of angular momentum, and conservation of electric charge are manifestations of four basic symmetries.
Each type of symmetry can be described by behavior of the system under changes (operations) which transform the system from one position (state) to a symmetric one. The symmetry operations which “change states” in a system are results of the symmetry of the system.
For symmetries of geometric figures (triangles, squares, etc.) - discrete symmetries - there is a limited number of operations to “change states”. These operations form a group with specific mathematical properties. Working with geometric figures helps to illustrate the mathematical operations in the group. This laboratory exercise will investigate the symmetry group of the equilateral triangle, the simplest group with operations for which the order is important (operations do not commute like ordinary numbers).
Symmetry is an invariance of an object or system to a set of changes.
In simpler language, a thing (a system) is said to possess symmetry if we can make a change (a transformation) in the system such that, after the change, the thing appears exactly the same (is invariant) as before.
More information is available at:
www.emmynoether.com
www-ed.fnal.gov/samplers/hsphys/people/hill.html
look for more information on Emmy Noether -
http://turnbull.mcs.st-and.ac.uk/history/Mathematicians/Noether_Emmy.html
www.awm-math.org/noetherbrochure/AboutNoether.html
Procedures:
Use the diagram of an equilateral triangle along with a transparent overlay to perform the operations described below. After each operation or set of operations, return to the original position (ABC) described below.
A. Elements of the group.
1. Rotations: The triangle may be rotated. Rotations in multiples of 120° will bring the triangle to a position symmetric to its original one. The original position is
which can be represented as ABC. The rotation of the overlay 120°
clockwise can be represented as R120°.
Rotate the overlay 120° clockwise and see which letters now appear
where the original A, B, and C were. The overlay C should be where
the A was. The overlay A should be where the B was. The overlay B
should be where the C was. This operation can be represented by the following equation:
R120° (ABC) = CAB
What is R240° (ABC) where R240° represents a 240° clockwise rotation?
R240° (ABC) = ___BCA___
What is R360°?
R360° (ABC) = ____ABC_____
What is R-120° (counterclockwise rotation)?
R-120° (ABC) = ____BCA_____
What is R-240°?
R-240° (ABC) = ____CAB_____
How many unique (different) answers did you get in total? ____3_____
Note that since there are only three unique rotated states (ABC, CAB, BCA) there are only three unique operations: R120°, R240° and a rotation which returns the triangle to its natural state (ABC). This third operation R360° is referred to as the “do nothing” or unity operation and is symbolize by the number 1.
R360° = R-360° = 1
The order of symmetry describes how many unique, discrete (different) symmetries there are.
What is the order of symmetry for rotations of an equilateral triangle? ______3______
Note that all objects have an order of symmetry of at least 1 (the do nothing).
2. Reflections: The overlay triangle may be folded over an axis (shown below).
If one turns the over the overlay keeping a symmetry axis
(RI, RII, or RIII) fixed, the configuration or state of the triangle
changes. For example, a reflection with fixed axis RI is:
RI (ABC) = ACB
What do reflections RII and RIII yield?
RII (ABC) = ______
RIII (ABC) = ______
3. Operations of the group: If you think of the operations as numbers it is possible to set up a table of single operations like the one below. Fill in the results.
Operation on ABC / Result1 / ABC
R120°
R240°
RI
RII
RIII
Are there operations which can produce another “final state”? If not, the group can be considered complete with only the six operations above. (Order of symmetry of 6)
B. Multiple Operations
What happens if one operation is followed by another? Symbolically we represent the first operation to the left of the (ABC), the second operation to left of that, and so on. For example, a rotation clockwise of 120° followed by a reflection about RI would be represented as
RI R120° (ABC)
2nd operation 1st operation
Show that RI R120° (ABC) = CBA.
If you look back to the table from the previous section you will note that CBA = RII. Therefore,
RI R120° (ABC) = RII (ABC) = CBA
or
RI R120° = RII
Now try R120° RI (ABC). Is the final state the same? In general, it will not be. The order of operation is important because operations do not commute under symbolic multiplication like real numbers do when multiplied.
Although 2 x 3 = 3 x 2 RI R120° does not equal R120° RI.
Complete the “multiplication table” for 2 successive operations below:
1st operation is: Þ / 1 / R120° / R240° / RI / RII / RIII2nd operation is: ß / X / X / X / X / X / X
1 / 1
R120° / R120°
R240° / R240°
RI / RI / RIII
RII / RII
RIII / RIII
What occurs if a third operation is added? Show by example that three successive operations are equivalent to a single operation:
______(ABC) = ______
operation 3 operation 2 operation 1
= ______(ABC)
equivalent operation
WHAT IS THE POINT?
Why are we calculating multiple operations? The goal is to attempt to use known operations to create new operations. Have we created any new operations?
C. Questions to test understanding:
1. How many unique discrete rotations are there for a square?
2. How many unique discrete rotations are there for a rectangle?
3. How many unique discrete rotations are there for an isosceles triangle?
4. How many unique discrete rotations are there for a regular pentagon?
5. How many unique discrete rotations are there for a regular octagon?
6. How many unique discrete rotations are there for a circle?
7. How many axes of reflection are there for a square?
8. How many axes of reflection are there for a regular pentagon?
9. How many axes of reflection are there for a regular octagon?
10. How many axes of reflection are there for a circle?