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.)
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Today I watched part of an interesting video by 3blue1brown about the surface area of a sphere. I had a lot of trouble concentrating on it, though. Near the end it began to require more participation from the viewer, and I decided to save the rest for later. There’s some hope underlying that, I guess.
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I wasn’t up to much today and slept for most of the day. I did read some of Infinite Powers, but I had enough trouble focusing and understanding that I will probably read the same section again next time. Nothing interesting to share today, I’m afraid.
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Today I’ve been playing with tilings to try to understand the relationship between radial symmetry and periodicity. I’ve come up with the following to show that each can exist independently of the other.
I’m still wondering, though, if a tiling can be radially symmetrical about more than one point and be non-periodic. I suspect not.1
- To do. ↩︎
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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. ↩︎
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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.
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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.
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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.
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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.