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PROFESSOR: So now that we've talked about the different dynamics of droplets in the air, we can think about how different types of pathogen can leverage those droplets to transmit from one person to another through respiration through the air. So let's begin with bacteria. So there are many different kinds of bacteria. The typical size of a single bacterium is on the order of several microns. So let's just say, 1 to 10 microns. On the other hand, bacteria can also exist in colonies or larger structures. And that does determine, to some extent, what kinds of droplets can transmit those bacteria. So let's begin with an example of a large-drop bacteria, which is typically going to be found in large drops. And that would be the bacteria that causes strep throat, the streptococcus. So the streptococcus bacteria, which is shown here, has a typical size that's around two microns. So it's on the smaller end. And it forms chains and even larger colonies the...

PROFESSOR: We've talked a lot about de Broglie saying that the wavelength is given by h over p. But we have not said much yet about the frequency of the waves. So what is the frequency of those matter waves? So what is the frequency-- frequency-- of the matter waves. So de Broglie did answer that same question. And the answer was obtained by analogy. We have p equal h bar k. And he said, well, just like the wavelength is determined by the momentum, we'll have e equal h bar omega. So the frequency-- so this equation is the one that now completes the story. Omega is equal to e over h bar. Fixes omega in terms of the energy. And we're going to say a few things. In fact, this will be an interesting digression into an important subject about waves that illustrates why this answer makes a lot of sense. And that's, really, all you can do at this moment. This is a postulate of quantum mechanics. That you do this thing, and with this, you get quantum mechanics. So ...

PROFESSOR: All right, we've been slogging our way through derivation of the plane groups. And I think I'll do a few more, because we'll stumble across some major tricks in deriving a subfamily of them. But to not get lost in the forest because of all the trees, I have a set of notes. They are handwritten because my secretary would resign if she had to fit in all these figures and subscripts and strange symbols. So, they are as neat as I could make them. Sorry to say that, in running them through the Xerox machine in an attempt to get everything on one sheet, some of the last lines got clipped. So I'll run these through again and give you a copy that's minus those truncations. All right. What we've been doing so far, to have a brief reprise, was to take the symmetries, the 10 two-dimensional plane group symmetries. And they were one-, two-, three-, four-, or sixfold axes, a mirror plane, 2mm, 3m, 4mm, 6mm. So there are 10 of them. And these are the ...

PROFESSOR: --questions about what we've done. I think it's been fast, but hopefully if you understand the principles, we didn't go too fast. But any questions on what we've done? I guess you haven't had a chance to think of questions yet. Let me take care of our oddball symmetry at the end of the chart that I handed out-- and this 4 bar-- and ask what we can do there. Having discovered the four bar operation in 2, 2, 2 with diagonal mirror planes, we can consider that as a new type of symmetry element. And this is the symbol for it. And this would take a pair of objects and repeat them by a 180 degree rotation. And then another pair of opposite handedness, opposite chirality would be rotated 90 degrees and inverted, rotated 90 degrees and inverted. And I mentioned last time that a solid that has this shape is something called a sphenoid. And the 4 bar axis takes a pair of faces that are up and a pair of faces that are down and skewed by 90 degrees. Now...

PROFESSOR: All right, well, we've seen how the query language works. Now, let's talk about how it's implemented. You already pretty much can guess what's going on there. At the bottom of it, there's a pattern matcher. And we looked at a pattern matcher when we did the rule-based control language. Just to remind you, here are some sample patterns. This is a pattern that will match any list of three things of which the first is a and the second is c and the middle one can be anything. So in this little pattern-matching syntax, there's only one distinction you make. There's either literal things or variables, and variables begin with question mark. So this matches any list of three things of which the first is a and the second is c. This one matches any list of three things of which the first is the symbol job. The second can be anything. And the third is a list of two things of which the first is the symbol computer and the second can be anything...