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Tag Archives: math
The Shroud of Turin and an Infinite Series
Having a somewhat skeptical bent and a long interest in pseudo-science, an article on Medium about the Shroud of Turin caught my eye. For some background see Unraveling the Myths Surrounding the Shroud of Turin. The Shroud first appeared about 1355 A.D.:
A couple notes on Ancient Greek Astronomy
Following up on these posts:
Today I read Beyond the Book (and What the Greeks Knew About the Earth) in which Professor Matt Strassler explains one of the ways the ancient Greeks knew that the Earth was round and how one of them (Eratosthenes of Cyrene} was able to make a reasonably accurate calculation of its circumference. This calculation assumes that the Sun is much further away from the Earth than the Moon, and hence much bigger, than the Earth of the Moon. Which is Bigger, the Sun or the Earth? Check it Yourself! explains how the Greeks knew this.
Wheels that are not circles
Triangle-Wheeled Bike Gives New Meaning to ‘Tricycle’. The inventors
… went back to the drawing board to see if they could come up with a bike design featuring triangular wheels. They succeeded, and unlike the square-wheeled bike, these triangular wheels actually roll like round ones.
This is known as the Reuleaux triangle. I first ran across it in Poul Anderson’s 1963 SF story The Three Cornered Wheel, which I read sometime in high school (1964-68). A stranded spaceship crew needs to transport a heavy object over land. Unfortunately, the use of anything circular for mundane purposes is forbidden by the religion of the natives. However, the use of a curve of up to 1/3 of a circle is allowed for a sufficiently important cause. The young hero figures out that using such a “three cornered wheel” will solve the crew’s problem without offending the religious authorities.
Another look at the “Flat Earth” Myth
As I wrote in Columbus and the Flat Earth: “No educated European in 1492 believed that the Earth was flat. They all knew it was round.”
A recent Twitter thread (Original tweets here) covers much of the same ground, with the same conclusions, but with a current example:
First steps with LaTeX
Working through Create Beautiful Documents (better than Word) using TexStudio on Ubuntu Linux.
Continue reading
My First Peek at Renormalization
I have vaguely known about renormalization since the 1970’s, but had never seriously studied it. Out of curiosity I watched Renormalization and envelopes on YouTube Thursday evening. This was the final lecture of the Asymptotics and perturbation methods course by Prof. Steven Strogatz of Cornell University. I had watched the first two lectures of the course, but none of the others until this one. Fortunately, there were relatively few explicit dependencies on them, so I was able to follow this quite well. Here is the description:
Continue readingVector and Matrix Algebra Tutorial
Vector and Matrix Algebra is a short tutorial from Complexity Explorer. It is really quite elementary. I took a formal course in Linear Algebra long ago, and have reviewed the basics a couple times since then. I wanted to see if I still understand those basics. The answer is that I do: I had no trouble at all with the material.
Continue readingSignificant Figures
I found this post on Facebook: Why is it important to know so many digits of pi?.
My comment:
As someone who started computing with log tables and slide rules, the first question I ask is how many significant digits do the other variables in your calculation have? The smallest such number tells you how many digits of pi you need. With electronic devices there is no harm in using more in your calculation, as many as your device has, but do not let that give you a false idea of the precision of your result.
I learned about significant figures in my high school chemistry in 1967-68. (Thank you, Mr. Wheeler!). Use of appropriate significant figures, also from a chemistry class, clearly explains the concept and its use in practice.
I only first saw Star Trek (TOS) after high school, in reruns. Thanks to that chemistry class I gag every time I hear Mr. Spock reporting some calculation to an absurd number of decimal places. His input data could not possibly be that precise!
Calculating a limit
Today I saw this problem on Medium: Compute the limit
Followed by “Pause the article and attempt a solution now.” (Don’t cheat and look ahead)
So I did. I do not really like factorials, so I immediately thought of Stirling’s approximation:
ln n! ≃ n ln n – n as n → ∞
and all of the n‘s cancelled, leaving the result 1/e. I then looked at the author’s solution. My answer was correct, but he used a completely different approach, as you can see. I posted my solution, and got a nice complement from him.
This is Thanksgiving day in the USA. I am thankful that my calculus skills are still pretty good decades after my last formal course in that or any related field .
