Group Theory

In 2017, I taught an introductory course titled “Symmetries of Nature: An Introduction to Group Theory” at Hampshire College, which I designed as a hands-on, interactive course accessible to all students, even those without much math background. In an effort to keep a record of what we did and share the pedagogy with other interested educators, I posted our daily activities, along with the pictures, and links to resources.

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Discovering Group Theory @Hampshire, Days 24-27

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For the last two weeks of the semester students worked independently on projects which involved the material we learned in class.The class time was spent on research and discussion of their individual projects.

A first-year art student explored and identified the approximate symmetries of geometric shapes and symbols of ancient South-Western petroglyphs and rock paintings, which included dihedral groups and the translation group.
A first-year art student explored and identified the approximate symmetries of geometric shapes and symbols of ancient South-Western petroglyphs and rock paintings, which included dihedral groups and the translation group.
A third-year student with a concentration in math researched the application of group theory to Rubik's cube.(Photo by By Lars Karlsson (Keqs), Own work, via Wikimedia Commons)
A third-year student with a concentration in math researched the application of group theory to Rubik’s cube.(Photo by By Lars Karlsson (Keqs), Own work, via Wikimedia Commons)
A third-year art student researched the connections between group theory and the tree branching patterns.
A third-year art student researched the connections between group theory and the tree branching patterns.
A first-year student, still undecided on her concentration, dove into a paper on applications of group theory to molecular systems biology and used what we learned in class to understand the key points made in the article (Rietman EA, Karp RL, Tuszynski JA. Review and application of group theory to molecular systems biology. Theoretical Biology & Medical Modelling. 2011;8:21. doi:10.1186/1742-4682-8-21.)
A first-year student, still undecided on her concentration, dove into a paper on applications of group theory to molecular systems biology and used what we learned in class to understand the key points made in the article (Rietman EA, Karp RL, Tuszynski JA. Review and application of group theory to molecular systems biology. Theoretical Biology & Medical Modelling. 2011;8:21. doi:10.1186/1742-4682-8-21.)
A second-year pre-med/math student researched the use of group theory in chemistry.
A second-year pre-med/math student researched the use of group theory in chemistry.
  • A second-year pre-med/math student researched the use of group theory in chemistry.
  • A first-year student, still undecided on her concentration, dove into a paper on applications of group theory to molecular systems biology and used what we learned in class to understand the key points made in the article (Rietman EA, Karp RL, Tuszynski JA. Review and application of group theory to molecular systems biology. Theoretical Biology & Medical Modelling. 2011;8:21. doi:10.1186/1742-4682-8-21.)
  • A first-year art student explored and identified the approximate symmetries of geometric shapes and symbols of ancient South-Western petroglyphs and rock paintings, which included dihedral groups and the translation group.
  • A third-year art student researched the connections between group theory and the tree branching patterns.
  • A third-year student with a concentration in math researched the application of group theory to Rubik’s cube.

The course concluded with class presentations.

After I complete narrative evaluations for the students in the course, I will write a post reflecting on my own experience teaching this course.

Discovering Group Theory, Days 21-23

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We spent three days discussing the conceptual framework of special relativity.

We reviewed the expressions for time dilation and length contraction (we didn’t derive the latter) and discussed what they tell us about the nature of space and time.  We introduced the Lorentz boosts as a replacement for the Galilean boosts, and formally introduced the Lorentz group, SO(1,3). New concepts and ideas included

  • Spacetime events
  • Spacetime diagrams and null cones
  • Causally connected events 
  • Relativity of simultaneity

The key takeaway was that Lorentz transformations leave the causal structure of spacetime the same, just like rotations of a polygon leave the shape and the orientation of the polygon the same.

 

 

Discovering Group Theory, Day 20

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On Day 20, after reviewing the Galilean group and the classic formula for the addition of velocities, we went on to talk about their incompatibility with the results of the Michelson–Morley experiment. We begun to resolve the issue by deriving the expression for time dilation using the example of a moving light clock.

                                                                Light clock

 

Discovering Group Theory, Day 19

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For this portion of the course, my key goal is that students understand  how group theory is used in physics, so that they can pursue the subject further if it catches their interest.

In preparation for day 19, students were asked to work through Khan Academy videos on 2×2 matrices outside of class

For most students, this was a review, but some of them have not seen matrices before. Rather than introducing everything about matrices, we only discussed those operations needed to prove that rotation matrices form a group SO(2) under matrix multiplication.

While proving that the set of 2×2 rotation matrices is closed under matrix multiplication, we used our recently acquired knowledge of complex numbers to derive the equations for cosine and sine of the sum of the angles.

We introduced SO(3) by analogy to SO(2), but without going into the technical details.

After wrapping up rotations, we then moved on to moving frames of reference and the Galilean principle of relativity. We showed that acceleration is the same in all frames of reference moving at constant velocity relative to one another. This calculation required the understanding of a derivative, which was familiar to most students but not all, so I emphasized the conceptual understanding. We then defined Galilean group.

Galilean relativity

To create excitement for what is to come, I briefly reiterated the physical interpretation of the law of addition of velocities, which we derived in the above-discussed calculation, and introduced the contradiction between it and the observed constant speed of light in vacuum.

 

Discovering Group Theory, Day 18

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On Day 18, we returned to the spacetime symmetries we discussed last time, but treated them with a bit more rigor and tighter notation.  In the context of the Newton’s 2nd law, we worked out the math for

  • Translation group as applied to three-dimensional space and to time.
  • Rotations in space and their matrix representation in 2D.
  • We formally defined the orthogonal group O(2) and the special orthogonal group SO(2).

Discovering Group Theory, Day 17

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Realizing that space-time symmetries provide more intuitive introduction to the abstract notion of a symmetry used in physics,  on day 17 we switched gears by discussing Richard Feynman’s lecture “Symmetry in Physical Law,” which students watched outside the class.

Feynman introduces the notion of a symmetry of a physical law, which we disected in the context of Coulomb’s law.

  • We defined a coordinate system and discussed the difference between passive and active transformations.
  • We worked out the translational and rotational spatial symmetries in 2D and showed that translations form a group.
  • We also introduced translations in time and showed how they leave the kinematic equations unchanged.

As a transition to the discussion of moving observers, we watched the first half of a wonderful video from the 60’s titled Frames of Reference. 

 

 

Discovering Group Theory, Day 16

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On Day 16, we finished up complex numbers in polar form and introduced several new concepts:

  • The circle group T and unitary group U(1)
  • Infinite vs finite groups
  • Continuous vs discrete groups

We also started going through t’Hooft’s Scientific American article “Gauge Theories of the Forces between Elementary Particles“, which proved to be challenging but exciting. We discussed the Standard Model of particle physics and the distinction between spacetime symmetries and internal symmetries.

Discovering Group Theory, Day 15

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On Day 15, we continued working with complex numbers, but now in exponential or polar form. After an introduction to complex numbers exponential form and its relation to the rectangular form, students worked on exercises covering:

  • Complex numbers in the polar form and their relation to the rectangular form
  • Complex conjugation and modulus
  • Complex numbers as points in a complex plane

About half of the class managed to get to the proof that the complex numbers form a group under multiplication, which was easier to do in the polar form.

We ended the class by watching a TED talk by Murray Gell-Mann,  “Beauty, truth, and …, physics?”.

Discovering Group Theory, Day 14

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In preparation for the discussion of continuous groups and their applications in physics, we began the class with a short and a bit of a handwaving introduction of the more abstract notion of symmetry, namely that of a symmetry of a physical law. We then moved on to complex numbers, which most students have seen before.  After a brief introduction, students practiced working with:

  • Complex numbers in the rectangular form
  • Operations with complex numbers
  • Complex conjugation and modulus
  • Complex numbers as points in a complex plane

Finally, students showed that complex numbers form a group under addition, (C,+).

Discovering Group Theory, Day 13

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On day 13, we reviewed symmetrical groups and introduced Cayley’s theorem. Students then worked out the mapping between the elements of D3 and  S3.

Students explored isomorphism between D3 and S3.