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.:
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.
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.
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:
Working through Create Beautiful Documents (better than Word) using TexStudio on Ubuntu Linux.
Continue reading
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 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 readingI 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!
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 .
Inventing the Flat Earth: Columbus and Modern Historians, by Jeffrey Burton Russell, is the book for the day. Columbus did not show the world that the Earth was round. No educated European in 1492 believed that the Earth was flat. They all knew it was round. As all math geeks know, Eratosthenes of Cyrene had made a good calculation of the circumference of the Earth about 200 BCE.
Catholic church authorities did not say that the plan of Columbus to reach the orient by sailing westward was impossible because the Earth was flat. Their scholastic theology was based on the philosophy of Aristotle, who understood perfectly well that the Earth was round.
There are passages in the Bible that suggest a flat Earth, but almost all theologians of ancient and medieval times knew the evidence for a round Earth was overwhelming, and understood the Bible was not to be taken literally in this and similar cases.
The objection to the plans of Columbus was that, thanks to Eratosthenes, people had a good idea of the distance from the west coast of Europe to the east coast of China, and could easily calculate that no ship of the day could possibly carry enough supplies for the voyage.
Columbus, acting like a 21st century Republican, rejected the best science of the day and chose a smaller alternative value for the circumference that suited his purposes. He was just lucky that the Americas happened to be there. As a result their inhabitants were then horribly unlucky.
The story about Columbus and the flat Earth is a 19th century invention, not history.
Also posted on Facebook.
From Hilbert Space to Dilbert Space, and beyond 