Discovering Group Theory, Day 17

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

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.

Color Physics @Hampshire, Days 15-17

Days 15, 16, and 17 were spent on hands-on activities in which students investigated subtractive mixing by reflection, and explored the use of additive and subtractive mixing in print and painting. The activities included:

  • Creating partitive mixtures in the pointillist style

  • Investigating and replicating the use of halftone in print
        
    Students used magnifying glasses to look at newsprint and determined how the CMYK system is used for printing. Then the made their own halftone designs.

  • Investigating subtractive mixing by reflection.This portion involved three different activities:
    • In the first, students placed pieces of paper of various colors in a light-tight box,  illuminated them by one of the lights we have (red, green, or blue), and recorded their observations. The papers we used were to an OK approximation, red, blue, green, cyan, magenta, and yellow.
    • In the second activity, students used their observations from the first to predict what colors the same pieces of paper would appear to have when illuminated by white light, but viewed through filters of different colors. They then tested their predictions and commented on any discrepancies.
    • Finally, using what they learned from the first two activities, students devised a method by which they can determine how “pure” the cyan, magenta, and yellow pigments in their watercolor sets were.

 

A student-made hologram of a thread spool

Discovering Group Theory, Day 15

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

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

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.

Color Physics @Hampshire, Day 14

On day 14, in preparation for an activity of making holograms, Prof. Wirth gave a mini lecture on interference of light, its use in early color photography, and its application to holography.

After that, we continued our exploration of subtractive mixing with filters.  Students worked through a worksheet in which  they had to answer quantitative and qualitative questions. Among other things, students

  • Made predictions about the intensity distribution of transmitted light given a wide-spectrum white light incident on ideal filters with a variety of transmittance curves, and compared them with observations.
  • Made predictions about the intensity distribution of transmitted light when colored light is incident on ideal filters with a variety of transmittance curves, and compared them with observations.
  • Sketched the transmittance curves for a combination filters.
    We worked on transmittance curves for multiple filters.
    We worked on transmittance curves for multiple filters.


  • Looked at the real transmittance curves of our filters to understand why not all of the light is blocked by overlapping cyan, magenta, and yellow filters.

At the end, we went over all of the answers as a class.

Light passing through overlapping filters
Subtractive mixing with filters.

Materials:

 

Color Physics @Hampshire, Day 13

On day 13, a guest speaker, Prof. John Castorino gave a lecture on the biology of color vision.  His presentation included an introduction to the anatomy of the eye, the placement of rods and cones, their response to the sensory input, the mechanism behind trichromacy of color vision, the biological basis for color blindness, and many more (I am certainly missing a number of topics). It was great for students to be able to ask biology-related questions and get a straight answer.


 

Color Physics @Hampshire, Day 12

On day 12 we continued to talk about subtractive mixing with filters.  We covered the following topics

  • Subtractive mixing with filters
  • Transmittance curves
  • Ideal vs real fiters
  • Cyan, Magenta, and Yellow (CMY) color system
  • We also touched on color photography (and in particular the work of the physicist J. C. Maxwell)  the early color films, including the two-color system and the three-color Technicolor system. We saw a nice example of early film created using the two-color system, for which the camera had rapidly changing red and green filters.

Other resources:

 

Discovering Group Theory, Day 12

On Day 12, we continued to explore the group of permutations of finite sets using S3 as an example, but we also practiced calculations with elements of symmetric groups of higher orders.

A worksheet on permutations
On day 12, more on permutations.

In addition to reviewing the concepts of order of an element, inverse, commutativity, we introduced some new concepts:

  • Cycles and their properties
  • Parity/ Even and odd permutations

Students practiced switching between different notations used to represent permutations, while also checking our answers by physically moving pieces of paper labeled with numbers 1, 2, and 3.

Students carried out permutations of pieces of paper
to either find answers or check them.