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.

Color Physics @Hampshire, Day 11

Day 11 was more or less a rehash of the second half of Day 10, but in more depth and with more contact points with the visual arts. 

The topics covered were:

  • Infrared and ultraviolet photography.
  • Metamers.
  • Missing complementary pairs of spectral colors.
  • Three types of additive color mixing and its applications:
    • Direct addition of lights (TV monitors and projections on white surfaces in stage design).
    • Partitive mixing and pointillism in painting (Seurat, Signac, and van Gogh).

      A self portrait of van Gogh's as an example of the pointillist painting technique which relies on the principle of additive color mixing.
      A self portrait of van Gogh’s as an example of the pointillist painting technique which relies on the principle of additive color mixing.
    • Mixing light “in time” with Newton’s disks, which students made for a homework assignment a few weeks ago.

Finally, we formally introduced subtractive mixing with filters and the transmittance graphs. Students then used diffraction gratings to break up the light from a tungsten lamp, observed which colors were absorbed by various filters, and sketched the transmittance graphs for those filters.

Student observing the color spectrum after the light has passed through a filter.
A student observing the color spectrum generated by a diffraction grating after the light has passed through a filter.

Other interesting resources: 

 

 

Color Physics @Hampshire, Day 10

After reviewing the concepts of wavelength, amplitude, and frequency in general, we discussed the special case of electromagnetic waves and the electromagnetic spectrum. The key ideas covered were the

  • Inverse relation between the wavelength and frequency of an electromagnetic wave.
  • Intensity of a wave and its relation to the amplitude of the wave.

In a lecture-heavy portion of the class, we went over how different properties of an EM wave in the visible spectrum correlate with the hue, saturation, and brightness of the perceived light.  I used very good slides developed by Geoff Boynton at the University of Washington.

Geoff Boynton's slide on the relation of the width of the spectral intensity-distribution of light and the perceived saturation.
Geoff Boynton’s slide on the relation of the width of the spectral intensity-distribution of light and the perceived saturation.

We introduced the spectral intensity-distribution graphs for light sources and the distinction between spectral colors and non-spectral colors, but also reviewed additive color mixing in the context of the intensity-distribution graphs. We touched on the relevance of additive mixing to the pointillist painting movement developed by Seurat.

Students then used hand-held spectrometers to determine the intensity-distribution for the red, green, and blue LED’s we used in additive mixing exercises earlier in the semester.  We also tried to determine the spectrum of the mixture various pairs of lights. This was tricky because of the light pollution in the room and the fact that the light from two LED’s was coming from different angles and scattering within the spectrometer.  


Students measured and recorded the intensity distributions of light from LEDs.Students measured and recorded the intensity distributions of light from LEDs.

Some of the students got to start playing with color filters and explore their absorption spectrum, but more formal introduction was left for the following class.

Discovering Group Theory, Day 11

On Day 11, we began with all the grammatically correct ways to say “I love physics” in Bosnian/Croatian/Serbian language. That happens to be a total of 3!=6 ways, each a permutation of “JA VOLIM FIZIKU,” which is pronounced roughly as |ya volim feezikoo|. I couldn’t resist the temptation to bring in my native tongue(s) into the lecture; the students mostly rolled with it. 🙂

  • The class  started with students constructing the sentences by arranging thee pieces of paper with individual words in all possible orders.

All the ways “I love Physics.”

  • Once we figured out all the possible combinations of the three words, we gave a symbol to each permutation taking the original sentence to a different one, as well as the identity.

A whiteboard with all the permutations of "Ja volim fiziku"Introducing notation for permutations.

  • After we clarified what it means to apply permutations successively, students wrote out the multiplication table for the permutations. This was particularly interesting to me because each student seemed to have a different method of tracking the changes after each permutation. Some moved the pieces of paper, some sketched the process by writing the string of words after each permutation, and one kept track of the permutations by creating a “dictionary ” pictured below which he used to track the changes. A subtle question arose of whether the first subsequent permutation was to be applied to the specific words used in the definition of the permutation, or on whichever words were in the original locations of the words from the definition.
    Working on the multiplication table for permutations.

 

  • We inspected the multiplication table to find that these permutations form a group, which led to a formal introduction of the symmetric group S3, and the symmetric groups of permutations in general.
  • We concluded the class by drawing parallels to D3, which is isomorphic to S3.

 

Pieces of paper reading "Ja"
A student noted that this is a very funny pile. 

Discovering Group Theory, Day 10

On Day 10, we continued to talk about subgroups and generators. After going over the basic definition more formally than the last time and cleaning up our notation, we worked out more examples of subgroups of cyclic groups and practiced applying Lagrange’s theorem to find the orders of possible subgroups. We concluded with finding subgroups of D3, from which the takeaway was that there could be multiple instances of a subgroup of the same order. It was not particularly exciting day, but it solidified students’ understanding of the basic concepts. And we also had tea.

A worksheet on subgroups of D3 and some tea.
A worksheet on subgroups of D3 and some tea.

Color Physics @Hampshire, Day 9

Day 9 was a continuation of our exploration of waves. The class time was divided into two main activities:

  • Working in groups on conceptual exercises related to waves (drawing waves, ranking them based on various properties, etc.), led by me.
  • Carrying out a more systematic study of waves on a slinky, which included quantitative measurements of the speed of a wave pulse, led by my colleague Fred Wirth who has offered to lend a helping hand this semester and who designed and coordinated this portion of the class.
Conceptual Exercises on Waves
Conceptual Exercises on Waves

At the end, we discussed as a class the solutions to some of the conceptual exercises.