## Challenge Solution #1

I've posted the solution for the First Challenge Question in the comments section to that post.  St. Jack Spell's comment here is the winner, and my answer key is here, along with some secret knowledge which I promised to "Andrew2" (St. Andrew II?)

Congratulations to Jack---As a prize, he may suggest the topic for a post.  (But no promises about when I would do that post.)

Posted in Challenge Questions | 8 Comments

## Random Santa Fact

Oh, and I meant to include this random piece of information in the previous post but it slipped my mind:

♦  I just got back on Tuesday from a 3 week trip to Europe to go to some physics workshops in Brussels, Belgium and in Benasque, Spain.  While we were in Brussels, Nicole and I saw among many other things the Église St-Nicolas, a strange church near the Grand Place, which actually has privately owned shops built into the walls, completely surrounding the church!  According to our guidebook,

"A market church was built on this site at the end of the 13th century, but, like much of the Lower Town, it was damaged in the 1695 French bombardment.  A cannon ball lodged itself into an interior pillar and in 1714, the bell tower finally collapsed.  Several restoration projects were planned but none came to fruition until 1956, when the west side of the building was given a new Gothic style façade.

Dedicated to St. Nicolas, the patron saint of merchants, this low-lit atmospheric church is known for its choir stalls, dating from 1381, which depict the story of St. Nicolas on medallions..."

So, now you know which side Santa's bread is buttered on!  Explains a lot, doesn't it?

It is now 145 days until Christmas, so you have plenty of time to forget this fact if it is going to ruin your holidays.  Ho ho ho!

## Lots More Random Stuff

♦  A Roman cooking blog: Pass the Garum.

♦  An amazing carved tree trunk.

♦  I have a mild case of synethesia: I associate colors with letters and numbers.  Words tend to be associated primarily with the color of the first letter.  It's not really an actual perception of color, just a really strong association.

One day I made a table of which letters correspond to which colors.  But then I made a table of which numbers were associated with which letters:

## 0123456789

... and at around the time I got to E, I said, hey wait a minute, that's in rainbow order!  The culprit is here, and apparently I am not alone.

However, there appear to have been some mutations which have overcome various aspects of the original letter blocks.  First of all, the purples all seem to have washed out, mostly to browns, though L is a light bamboo. Each letter has differentiated to some degree from the others that were the same color, e.g. A is maroon while S is a lighter red while M is pink; these were all the same pinkish-red color in the magnet set.

I was mathematically inclined from an early age, and the association of 1/I with True/Something and 0/O with False/Nothing seems to have caused them to become white and black respectively.  The whiteness of I is probably also related to Ice.  P became Pink.  4-8 are related numbers, and so are 3-6-9.

My one regret is that, having become hopelessly corrupted by a mass-produced commercial product, my preferences aren't good evidence for what the REAL colors of each letter are.

♦  Having broached the subject of math and mysticism— here is a letter from André Weil (famous mathematician), to his sister St. Simone Weil (mystical writer, activist, and math teacher), explaining the role of analogy in mathematics.  (Not an easy read for non-math people!)

♦  The spoon theory of disease.  Of course, the interesting thing is that it has nothing per se to do with spoons, it just needs to be something concrete with positive associations which you can count.  Counting is one of the most fundamental mathematical analogies: you can use any kind of object to represent any other kind of object, in fact you can even use nonsense words like we were taught to do in kindergarten.  John Baez:

I like to think of it in terms of the following fairy tale. Long ago, if you were a shepherd and wanted to see if two finite sets of sheep were isomorphic, the most obvious way would be to look for an isomorphism. In other words, you would try to match each sheep in herd A with a sheep in herd B. But one day, along came a shepherd who invented decategorification. This person realized you could take each set and "count" it, setting up an isomorphism between it and some set of "numbers", which were nonsense words like "one, two, three, four,..." specially designed for this purpose. By comparing the resulting numbers, you could see if two herds were isomorphic without explicitly establishing an isomorphism!

According to this fairy tale, decategorification started out as the ultimate stroke of mathematical genius. Only later did it become a matter of dumb habit, which we are now struggling to overcome through the process of "categorification".

♦  Which is more important?  Random, low-quality mummy masks, or learning more about early manuscripts of the Gospels and other 1st-3rd century literary documents, by disassembling them into the papayrus fragments from which they were made?

Possible early fragment of St. Mark's Gospel from before 90 AD, but we'll have to wait for it to be published to assess the credibility of this.  (Some claim the earliest fragment of Mark's Gospel is 7Q5 (mid-1st century) from the Dead Sea Scrolls, but in my opinion the reconstruction of that text fragment is far too speculative to be convincing.

♦  Longtime commenter St. Jack Spell is currently writing a series on the historical evidence for the Resurrection: parts 1 2 3 4, with I think more to come 5, 6, 7, 8.  I found particularly noteworthy his argument that certain critical facts surrounding the Resurrection (the burial of Jesus, that the tomb was found empty, dating the earliest claims to have seen Jesus to very early on) are accepted even by most skeptical New Testament scholars.

♦  From the Wikipedia article on the origins of the University of Bologna.  The first university was run by the students:

The University arose around mutual aid societies of foreign students called "nations" (as they were grouped by nationality) for protection against city laws which imposed collective punishment on foreigners for the crimes and debts of their countrymen. These students then hired scholars from the city to teach them. In time the various "nations" decided to form a larger association, or universitas—thus, the university. The university grew to have a strong position of collective bargaining with the city, since by then it derived significant revenue through visiting foreign students, who would depart if they were not well treated. The foreign students in Bologna received greater rights, and collective punishment was ended. There was also collective bargaining with the scholars who served as professors at the university. By the initiation or threat of a student strike, the students could enforce their demands as to the content of courses and the pay professors would receive. University professors were hired, fired, and had their pay determined by an elected council of two representatives from every student "nation" which governed the institution, with the most important decisions requiring a majority vote from all the students to ratify. The professors could also be fined if they failed to finish classes on time, or complete course material by the end of the semester. A student committee, the "Denouncers of Professors", kept tabs on them and reported any misbehavior. Professors themselves were not powerless, however, forming a College of Teachers, and securing the rights to set examination fees and degree requirements. Eventually, the city ended this arrangement, paying professors from tax revenues and making it a chartered public university.

In some ways it makes a lot of sense that the people paying for the product should set the terms for what they would get in exchange.

♦  But maybe college kids shouldn't be allowed to run universities... apparently about half of college students believe that we see because of rays that come out of our eyes.  The extramission theory of vision strikes back!  Original article here (behind paywall).

♦  Then again, maybe we shouldn't let the people running them now be in charge either...

♦  The educational philosophy of mistakes.

♦  Why we should Radically Simplify Law.  On the Cato Institute website, but really both conservatives and liberals should be able to go along with this.  No one wins when the law is a complicated mess.

♦  Don't let fear stop you from travelling, a charming comic.

♦  An erie piece about an unusual wedding.  The testimony of this saint is also well worth reading, and also her apocalyptic experience.

## Physics Challenge #1: Nonrelativistic Metric

This is a new feature which I'd like to try out.  I will ask a question about concepts in theoretical physics, and the readers will try to answer it.  It may be a straightforward question, or it may be a trick question.  I figure this will give me a better idea of what my readership does and does not understand about what I've written, and give readers a chance to show off their skills.

Challenge #0 happened kind of accidentally in the comment section of this post, and there it was suggested that this might be a cool regular feature.  So let's give Challenge #1 a whirl, and see what happens.

The metric of special relativity in Cartesian coordinates, in units where the speed of light $c = 1$, is

(The d's are just a calculus notation to indicate that you can use this metric to measure infinitesimal distances between nearby points, something which is very useful in general relativity where the metric is a function of position.  Here, however the metric is constant in space and time, so you could replace the d's with $\Delta$'s if you like.)

It has a ten dimensional group of symmetries, called the Poincaré group, which preserve the metric.  These are the set of transformations acting on the t, x, y, and z coordinates which preserve the metric.  (See the link for details.)

Suppose that instead we want to do nonrelativistic Newtonian mechanics.  These are the laws of physics which people believed were true before St. Maxwell and Einstein came along, and which are still valid for describing objects travelling much slower than the speed of light.

1. What is the appropriate metric to use when describing the geometry of spacetime in non-relativistic physics?

2. What is the symmetry group of this metric?  How many dimensions does it have?

3. Are these the same as the symmetries of Newtonian physics, which this metric is supposed to describe?  Why or why not?

The correct answer to these questions reveals something surprising about the way in which relativity is an improvement on nonrelativistic physics.

You need not answer all of these questions, but the answers to one may help confirm that the answers to the others are correct.  Experts (e.g. those with graduate education in physics) are requested to wait a while before attempting an answer, in order to give others a chance to respond.

Posted in Challenge Questions | 31 Comments