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.

  1. Note that Veritasium has sometimes shaped its educational content to reflect well on its sponsors. ↩︎

Frustrating Days

Today was frustrating. I had more to do than usual, but I was also tired and, late in the day, fairly depressed. I pushed myself to do a small amount of light math reading, and I am hoping for better conditions tomorrow.

My review project hasn’t been moving forward very quickly over the past few weeks. That, too, is frustrating. Early spring is frequently very rocky for me, though. At the moment, I’m just trying to make it through without completely losing my momentum.

Discontinuous

Things are a bit disrupted today. I am out of one of my medications and am feeling the withdrawal symptoms more and more. Nevertheless, I read more of Infinite Powers. The section I read contained several interesting ideas. One was that, while calculus is very useful in modeling the natural world, the assumptions that space and time are continuous that are fundamental to calculus may not be accurate. At the smallest scales, space and time may be broken into indivisible segments that span units called Planck length and Planck time. I will have to think more about this and how it relates to the idea that mathematics is inherent in nature.

Infinitesimal

Today I had a lot of things to do that were not math, and they wore me out so that I spent most of the rest of the available time sleeping. I did manage to read a section of Infinite Powers, though. It concerned Archimedes’ early use of calculus concepts to find the area of a circle. I would name my owl after him, if I had an owl.

Another Quiet Day

Today I was tired and lay low. It’s been a lot of effort to keep at least some of my plates spinning over the past several weeks. For study time, I made more electronic flashcards based on the review I have been doing. It’s useful to make the flashcards a while after I finish reviewing a topic. That way I can tell which pieces of information have taken root in my mind and which ones will need more cultivation.

Flash News

Today I spent a long time testing flash card apps and making electronic flash cards about various math information I’ve encountered during my review, such as details about conics and their significant points. I’m still not convinced I’d rather have these than physical flash cards, though they are more portable and easier to keep organized. Perhaps tomorrow I’ll make some physical flash cards as well and compare them.

(The app I eventually settled on is called Mochi.)

Proofs That Go Poof

Today I read another section of Analysis with an Introduction to Proof. This one was dedicated to proof techniques, and one of those made use of the tautology $[p\implies (q\lor r)]\equiv [(p\land \lnot q)\implies r]$. This is intuitive enough, but proofs that use it feel like cheating. For instance, to prove that if $x^2=2x$, then $x=0$ or $x=2$, you can assume that $x^2=2x$ and $x\neq0$, then prove that $x=2$. That’s it. Poof! Done! No showing that $0^2=2(0)$ required. I do not find this intuitive.

A Wheel on a Wheel

For study today, I watched the three-video series The Wonderful World of Weird Wheels by Morphocular, which is a fun look at the mathematics of curves rolling on other curves. I think it would probably appeal to any readers who have a basic understanding of what a derivative is, and maybe to others as well, since there are a lot of engaging visuals.

The video series motivated me to read a bit about hyperbolic functions, which I never really encountered in my mathematical education. $\sinh$ and $\cosh$ were always just mysterious markings on certain calculators. I can’t say they are that much less mysterious now, but I do know their definitions and have some sense of their relationship to hyperbolas.

Shiny New Week

Today I spent two hours reading a section of Analysis with an Introduction to Proof and doing the associated exercises. I’m feeling better than I was for most of last week, though the mental weather is still not entirely clear.

Part of the section I read explained the inadequacy of examples, even quite a few examples, in proving general statements. As an illustration, it offered the statement that $n^2+n+17$ is prime for every natural number $n$. This statement is true for values of $n$ up to 15, but is not true for 16 or 17. The latter, in particular, could be predicted without trying each value of $n$, since $17^2$, $17$, and $17$ are clearly all divisible by a common factor (meaning that their sum must also be divisible by that factor).

Later, an exercise asked me to find a counterexample to the statement that $3^n+2$ is prime for every natural number $n$. This had me wondering whether there was a way to predict a counterexample, as there had been in the earlier case.1 I could not think of one, however, so I answered the question by trying each $n$. It turns out that I didn’t have to go far, as $n=5$ breaks the pattern. I still wonder if there is a better way, though.

Thank you to all of you who have commented on my posts recently. It really helps me stay committed to my project. I will try to reply soon.

  1. To do. ↩︎