On Day 7, we moved on to the symmetries of a square. Using the same methods as for the equilateral triangle, we got the structure of D_{4}, though we left some derivations for a homework assignment. We then made an educated guess about the structure of the symmetry group of a pentagon, D_{5}, and then generalized to the symmetries of an ndimensional regular polygon, which led us to the formal definition of D_{n}.
Month: February 2017
Discovering Group Theory, Day 6
On Day 6, we dove into dihedral group D_{3}, which students have been exploring independently as part of their homework and in the context of the symmetries of an equilateral triangle.
 We tackled the group using three methods:
 physically manipulating a cutout of an equilateral triangle,
 drawing diagrams of various operations, simple (single rotation or reflection) and combined (rotation followed by a reflection, etc.), and
 working out the results of combined operations algebraically.This was an attempt to unify the physical experience of manipulating an object, visually representing the effects of those manipulations on the object, and expressing such operations symbolically.
Symmetry transformations of an equilateral triangle.
 We also revisited the already familiar concepts of identity, inverse, generators, group order, and element order, and constructed the Cayley table for D_{3}.
 We noticed that the order in which rotations and reflections are carried out does matter, which motivated the introduction of the commutativity as a property of two elements. This led us to the definitions of Abelian and a nonAbelian groups.
Discovering Group Theory, Day 5
On Day 5, was devoted to the discussion of “Mathematics for Human Flourishing,” a farewell address given by Francis Su at at the end of his term as the President of the Mathematical Association of America.
Students shared their own experiences with mathematics and how those experiences shaped their attitude toward the subject.
Color Physics @Hampshire, Day 7
On Day 7, we finally got to understand what electromagnetic waves are.
 Students used a wonderful HTMLbased simulation of a radiating charge to explore how motion of an electrically charged particle affects its electric field lines. They wrote their observations, the key ones which we discussed as a class.
 After briefly talking about the creation of magnetic field by a changing electric field and vice versa (mostly in a handwavy way), we moved on to the more familiar representation of the electromagnetic wave far away from wiggling charged particle. For this we used a very nice Java simulation developed by Juan M. Aguirregabiria, the screenshot of which is shown below. The shift inspired a good discussion of the pros and cons of the two visual representations of the electric field (arrows and field lines) and the relation between the depictions of electromagnetic radiation in the two simulations.
A screenshot of a simulation of a plane electromagnetic wave.
 After discussing different types of electromagnetic radiation (UV, visible light, WiFi, microwave) and its effects and uses, we began discussing what it takes to make a wave in general.
Color Physics @Hampshire, Day 6
On Day 6, we began to unpack the statement that light is an electromagnetic wave, leaving the particle nature of light for a later discussion.
 We discussed electric charge and electric force.
 We introduced the concept of electric field!
 We learned about two ways of visually representing the strength and the direction of the electric field, one with arrows at various points, the other with field lines. To get a better sense of the spatial distribution of the field, students built with styrofoam balls and toothpicks 3D models of the electric field of a spherically symmetric charge.
 We worked on conceptual problems testing students’ understanding of the above concepts.
Color Physics @Hampshire, Day 5
On Day 5, we were familiarizing ourselves with notation and problemsolving strategies.
 We completed more examples of units conversions and scientific notation.
 We practiced problemsolving with examples relating speed, distance, and time
We ended the class by deflecting a stream of water with an electrically charged balloon as a segue to the discussion of light as an electromagnetic wave!
Color Physics @Hampshire, Day 4
Day 4 consisted of wrapping up the discussion of HSB, introducing the RGB model, and setting the groundwork for the upcoming discussion of physics.
 Review of of Hue, Saturation, and Brightness
 HSB cylinder
 Additive color mixing: We organized our observations of mixing of colored light from the previous class and introduced the RGB color model.
 Primary colors in additive mixing
 Complementary colors in the additive system
 RBG color model
 RGB cube
 Measurement in physics:
 What is science?
 Units and unit conversions
 Scientific notation and orders of magnitude. We watched Powers of Ten to get a better sense of what different powers of ten mean in terms of different scales, so that we can assess our answers to numerical problems.
Other Resources: A student pointed me to an interactive catalog of objects of a vast range of sizes, organized by powers of ten http://htwins.net/scale2/
Discovering Group Theory, Day 4
Realizing that there was much to be gained from using now familiar cyclic groups to introduce more group properties, we postponed the exploration of dihedral groups and instead spent the day exploring.
 The review of group properties
 After reviewing the structure of particular examples of cyclic groups (C_{1}, C_{2}, C_{3}, C_{4}, and C_{6}) students extrapolated the properties of C_{n}, which led us to the formal definition of cyclic groups.
 Modular Addition and (Z_{n},+_{n})
 We built on our existing knowledge of how to calculate the hour reading of a clock to motivate modular addition. Students were asked to come up with a general mathematical formula which can be used to do the calculations.
 Working in pairs, students then investigated whether (Z_{4},+4) forms a group by checking if it satisfies the four axioms and constructing the Cayley table. This exercises broadened our understanding of group “multiplication,” which was originally introduced as a way to represent compounding effects of subsequent rotations of a polygon. It became clear why we have been using quotes around the word multiplication.

 After comparing C_{4} and Z_{4,} we introduced the idea of a group isomorphism.
 What is Not a Group? Students wanted to know more about when a set and a “product” do not form a group, so we went through a couple of examples in which one or more of the four axioms is violated (especially associativity).
Discovering Group Theory, Day 3
On Day 3, we did two different sets of activities:
 We revisited the definition of a group and its properties in the context of cyclic groups, namely C_{1}, C_{2}, C_{3}, C_{4}, and C_{6},_{ } encoding the rotational symmetry of polygons we played with on Day 2. Particular features of the cyclic groups prompted a natural introduction to a few new concepts and new terminology:
 Order of a group
 Order of an element of a group
 Group generators
 Group multiplication (Cayley) table, which students worked out for C_{3}, C_{4}, and C_{6} .
 We began exploring the symmetries of a rectangle, but this time including the reflections. This allowed us study the abovelisted concepts in a different context and practice carrying out group multiplication and finding inverses.
The plan was to use 2 to segue into dihedral groups, but it seems more pedagogical to spend another class or two on exploring the cyclic groups in more depth.
Color Physics @Hampshire, Day 3
Topics
 Saturation, hue, and brightness
 Additive Color Mixing
We started Day 3 by playing with the HSB sliders on a computer software to determine which properties of perceived color are referred to as hue, saturation, and brightness.
After using the handheld spectrometers to investigate the spectrum of a computer screen, cellphone screens, and the video projector, students made predictions about how brightness, saturation, and hue may be affected by mixing different amounts of red, green, and blue light and tested their predictions using primary color light sticks.