Today I watched this fun video from Numberphile. I watched it earlier in the month—it’s the one I stopped so I could investigate its topic a bit myself—but I had since forgotten the details. I’d recommend it to all of my readers. It is very accessible but has connections to more advanced topics. When I am feeling better, I would like to work on proving the theorem presented at the end of the video.
Tag: video
Back to School
Well, I’m back. I ended up taking a somewhat longer break than planned. I was just too discouraged to study last Thursday, and yesterday I was still getting settled again after being at camp. Today I watched YouTube videos in an attempt to ease into my project again. One was about the recent discovery of a new formula for approximating $\pi$. (It is not superior to existing formulae, just new.) The other was a fun video about some properties of prime numbers. It should be accessible to most people, and I have embedded it below.
Topical Math
Today was a sleepy day. (I wonder what causes those.) I didn’t do any exercises, but ended up watching six videos on voting systems. It seemed an appropriate topic as the US presidential election begins in earnest. I’ve long thought about making a study of this topic, becoming a voting system wonk and joining one of the organizations that advocates for reform. I do think adoption of better systems might benefit the country, especially in the primaries, where it is already common to have more than two candidates. (Elections with more than two candidates being where our current system is most likely to produce perverse results.)
Below is the video I watched that I think is most likely to be interesting to non-wonks. It does a good job explaining some alternative voting systems, though it does not say much about their pros and cons:
Back to Calculus
Today was dedicated to reading in my calculus textbook. I read the first two sections of chapter two, which introduce the idea of limits, as well as an appendix that covers its rigorous definition. I didn’t have any particularly interesting thoughts about them, unless you count “Hey, look at me! I’m doing calculus again.” And that put me in mind of this fun video I’ve shared before:
Rolling Along
Today I worked on exercises in my calculus book concerning parametric curves. I also talked with my father a bit about cycloids, the curves traced by a point on the perimeter of a circle as the circle rolls along a line. (These are simplest to define parametrically.)
As part of our discussion, we imagined a related class of curves that turn out to be called called cyclogons, traced by a point at the vertex of a rolling regular polygon. Just as a circle is the limit of a sequence of polygons as the number of sides approaches infinity, so a cycloid is the limit of a sequence of cyclogons as the number of sides of the polygon approaches infinity. Visualizing this can make it more intuitive that the point that traces a cycloid never moves backward as the circle rolls. Check out this demonstration and notice how the point always traces part of the top half of a circle, passing through each horizontal position only once.
(I’d also like to draw readers’ attention to the series of videos on rolling curves that I shared a while ago: The Wonderful World of Weird Wheels by Morphocular)
Next Week
I had a difficult day and spent my study time watching videos, including a nice documentary-style one about the math of relativity. I admit that I’m looking forward to my day of rest tomorrow.
I am actually contemplating taking all of next week off study and blogging. I will be traveling again, settling in as a live-in sitter for two cats, visiting with family, and attending Pride events. It will be a lot for me.
Fuzzbrain
My concentration today was terrible, though I don’t know why. I tried to read in my calculus text, but it was hopeless. I did watch a couple of videos from 3blue1brown’s Essence of Calculus series. I didn’t absorb them very well, however. I have decided not to feel bad about all this, and to try again tomorrow.
Ptolemy’s Theorem and Others
Things continued difficult today, but I felt a bit better after dinner. I watched four videos from the Numberphile channel, notably one about proving a geometric relationship called Ptolemy’s Theorem by a technique called inversion of the plane. It was a great video, but quite involved. I would not recommend it to those without a strong interest in math.
When I am feeling better, I would like to have a go at proving Ptolemy’s Theorem by another method.1 The lecturer in the video said that it can be done using plane geometry, provided you are clever, and can also be done by crunching trig ratios, something I have some experience in. I’d also like to prove, by some other method, a fact that can be proven using Ptolemy’s Theorem: that in the figure shown below, the distance from $D$ to $B$ is the sum of the distances from $D$ to $A$ and $C$.
A cool thing about Ptolemy’s Theorem is that the Pythagorean Theorem falls right out of it. In fact, you could say that the Pythagorean Theorem is a special case of Ptolemy’s. This means that Ptolemy’s Theorem must only be true if you assume the Parallel Postulate. I wonder if that is the case for the proposition above.
- To do. ↩︎
Tilings Again
Today I watched an enjoyable interview with one of the researchers involved in the discovery of the first aperiodic monotile, as described in the second video from my post last Wednesday. (The channel it comes from, Numberphile, has a wealth of other interesting math videos, as well, most featuring mathematical guests explaining their interests.)
Shapes on a Plane
Today I watched three videos about tilings, specifically non-periodic tilings. These are arrangements of tiles that cover an infinite plane in a pattern that never repeats.
Two of the videos were quite accessible. The first was from the popular science channel Veritasium1 and covered the state of this subject three years ago:
The second was from a smaller channel called Up and Atom and explained recent developments that have caused excitement among math enthusiasts.
The third video was more technical and probably of less interest to nonspecialist readers. It included a little more detail about why some collections of shapes form non-periodic tilings.
I’m still very tired from yesterday and don’t have much bright to say about all this. I feel like there is something to understand, though, about the relationship between rotational symmetry and periodicity of tilings. That is, between the ability of a tile pattern to match its original configuration when rotated versus when translated.
- Note that Veritasium has sometimes shaped its educational content to reflect well on its sponsors. ↩︎