It is not clear to me how this refutes the article. Can you explain a bit more? Do you have an example of a piece of data that lasts forever by virtue of these facts?
Really enjoyable read. The discovery that kicks it off made intuitive sense to me, but I loved the follow on explorations. Especially the frequency plots of some of the more structured noise functions!
From text. That connection has been investigated for a long time.
Although I must admit, it isn't entirely clear what the author is talking about. It's a bit like thoughts that appear when one is somewhat intoxicated. But extracting a mathematical model from linguistic utterances is very much what linguistics is about (and hasn't been able to achieve). Perhaps rotating the representational space of an LLM suits his/her fancy.
The issue is the opposite. Chinese has a comparitively small phonetic pallette (depending on how you view tones). Chinese written completely phoenetically can easily become incomprehensible.
The functional load of tones (that is, the importance of a pronunciation difference for distinguishing words) in Chinese is comparable to that of vowels[1]
"Depending on how you view tones" dismisses the important phonemic value of tones. Writing Chinese completely phonetically includes writing the tones.
Curious to know how you think it's possible that Chinese people are able to speak with each other if you think writing their language phonetically would render it incomprehensible.
I find hard to believe every language in the planet can be successfully expressed using some sort of alphabetic system, except for Japanese/Chinese (and local variants)
They could be written alphabetically, of course. The question is just what you lose, given that the characters are a massive part of Chinese and Japanese culture.
Chinese has evolved alongside its writing system for about 3000 years, and switching over to pinyin (the standard Latin transliteration) would be a complete revolution in Chinese culture.
I was responding to the assertion that writing Chinese using an alphabet would render the text incomprehensible, not arguing that China or Japan should switch over.
This must be true, because the diagonals are both straight lines that go through the centre and are bound by the edges, so it follows they must be equal to the diameter of the circle by definition.
> Do we know that a parallelogram with equal diagonals is a rectangle?
As another commenter points out, this is a theorem you can reach for, but proving it by itself is a bit more of a task.
Yes.
So, here when we rotate the triangle, we are essentially rotating each of the endpoints. For each endpoint, we rotate it by 180 degrees around the line segment joining the endpoint and the center. This by definition will result in a new position for each endpoint that creates a chord (as the two endpoints lie on the circle) and passes through the center (we rotated around it). A chord that passes through the center is by definition a diameter.
Thanks. Thinking about it, another way of looking at it is to construct the rotated triangle by drawing a line from each point through the center to where it intersects with the circle on the other side. This is obviously a 180' rotation, but by construction we explicitly know the diagonal goes through the center.
It maybe more clear if you visualize the reflection as a pair of perpendicular reflections, first across the diameter (which is also across the center) and then internally reflecting the diameter (which is again also across the center.)
Two reflections with a common fixed point make a rotation around that fixed point (angle of reflection is double the angle between the reflection axes.).
Two perpendicular reflections make a 180 degree rotation around the intersection of the axes of rotation.
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