Neuschwanstein and Hohenschwangau

Multitudes of more photos of the Reynolds clan on their expedition to Neuschwanstein and Hohenschwangau can be found on facebook (specifically Kathy's and Megan's pages), but here are a couple of my favorites.

Here is the Alpsee, the lake closest to the castles. There are many lakes in the region, but on the day we were there, there was not much visibility - since it was snowing - so we could only see a few. At this particular moment, it does not appear to be snowing, so I got a decent view.

Here is the Hohenschwangau castle, the birthplace of King Ludwig II, the "crazy king,"  and when he grew up he built ...

... Neuschwanstein castle (among others - in my German class we discussed how Bavaria was not very happy with Ludwig II spending so much money on castles, but it seems to have been an excellent long-term investment, since yesterday Megan and I estimated that on a typical day they receive about 100,000 EUR just in entrance fees, and that doesn't include all the food and souvenirs and entrance to Hohenschwangau and train tickets and bus tickets and food and souvenirs, which means that they certainly net several hundred millions per year, Boy wouldn't Ludwig be happy now).

Now, the question is, was Neuschwanstein really the inspiration for Sleeping Beauty's castle at Disneyland/world? On this gray day, it did appear so much to me. The stone is not white, but more a subdued grayish brown. Although the turret looks pretty good, the shape is a little different. Judge for yourself here.

Schnee in Pullach!!

Finally, the day after Christmas, it snowed in München. There was about 4 cm on the ground when we awoke. In fact, a friend who is up in Hochfügen (see this post), a popular skiing destination, said that 20 cm fell there. This is very late, as often the first snow is in October, and there is usually snow on the ground by the first of December. In any case, the skiers are loving it, because the local Alps now have snow for some downhilling.

So, we decided to head to Neuschwanstein and Hohenschwangau for the day (along with all the other hordes of tourists - we heard lots of languages, Japanese, Italian, Russian...), and this was our walk to the S-bahn. Of course, the Reynolds girls are walking in the middle of the street. But it's only suburban Pullach (technically really Großhesselohe).

And here is the small Platz at the S-bahn station. There's a restaurant and a cafe, and a Kinderplatz (children's playground) and some other things (like a fruit stand) that are only open in the summer. But a gorgeous scene. Certainly not Florida!

Frohe Weihnachten und schöne neue Jahr!

Glyptothek (1)

Last week I was sick (Ich war krank) so I didn't feel like posting. I'm back in the world of the living again, so here goes.

Last Sunday I visited several museums - and Sunday is a good day since admission is only 1 EUR - and had a good time. There are some very nice collections. I accidentally took the wrong U-bahn train (the first time!) but this ended up being serendipitous, since I went to the Glyptothek first, the museum that I planned to visit last. It turned out to be my favorite. It's essentially a museum of ancient scupltures, with pieces like the "Barberini faun",

the Medusa (luckily I didn't turn into stone),

and statues from the West Pediment of the Aegina Temple. (Actually, the statues from both the West and East Pediments are on display.)

The Temple pays homage to the Greek goddess Aphaia, and was built in about 500 BCE on the island of Aegina. It's too bad that they are not back on the Greek island, because that is probably where they belong. But the display is very nice, and it has quite a bit of explanatory text (in German and English) that is nice. The way that the artist depicted the Trojan war, confined by the triangular space, is very cool.

semester@HM

Das Magazin der Hochschule München.

In the Dezember 2014 issue there is an article about the Fellowship-Programm where they interviewed me and the other Gastprofessor, Chih-Kuang (Jack) Lin.  If you can read German, here it is. If you can't read German, ask my sister or daughter to translate for you. Or, if they want practice, they could place a translation in the comments (wink, wink, nudge, nudge).

Update: Here is a link to the pdf of the full magazine.

Nordfriedhof

The Mach number, almost as good as the Reynolds number, is one of the famous non dimensional quantities in physics. It is especially useful in fluid dynamics (as is the Reynolds number) because it allows you to perform experiments that are of a different size, different flow speed, using a different viscosity fluid, than the theoretical situation you are trying to replicate. However, even though these physical quantities are different, as long as certain dimensionless quantities match your desired situation, the experiment will be an accurate predictor.

Instead of having a gravestone among other gravestones, Mach has a pillar set apart from other gravestones. I assume that he's buried underneath. There are words on 3 of the four sides of the pillar. First, an explanation of who he was (as if that was needed!).

Second, the usual information. Interestingly, on German gravestones it often states what the person did, i.e., architect, artz (doctor), ingenieur (engineer), etc. Mach, of course, was a physicist and a philosopher.

The third side, not shown, was information about his son, who presumably is also buried here.

Arnold Sommerfeld was the other grave I wished to visit in Nordfriedhof. He was one of the founders of quantum theory, working with Niels Bohr to develop the Bohr-Sommerfeld quantization condition (one of the attempt's of "old quantum theory" to explain atomic spectra that didn't really work). His doctoral students include Werner Heisenberg, Hans Bethe, Wolfgang Pauli, Rudolf Peierls, and Peter Debye, many of whom won Nobel Prizes.

As I mentioned here, he my academic ancestor. You can my family tree here. You'll see that I currently have no PhD students, but my advisor, Burton Fried, has several. In addition, his advisor, Gregor Wenztel, also has several. And of course, his advisor, Arnold Sommerfeld, has many. And if you go back even further, you'll see that I am "descended" from great mathematicians like Gauss, Poisson, and Fourier. I don't presume to have anywhere their intellect, but it is nice to know that perhaps some of my skill comes from them. Certainly much of the way I approach problems is due to Fried, and I wouldn't be surprised if it is often like that.

Education wisdom

Sometimes the internet just sucks you in. Really. In. Yesterday evening I attended a colloquium at LMU on galaxy formation (like, how did the structure of the universe come to look like how it does?). Unforutnately, the speaker gave a talk for experts, of which I am not one, and spent a lot of time on the details, which I'm sure were interesting if I understood the issues, and not very much time on the big picture (for us novices). But I did take some notes, and I was searching on the web trying to understand AGN feedback and star formation in the early universe and why dark matter stays in the halos of galaxies but does not collapse into the core. (Not in a detailed, mathematical, way, but just an overview for a person with some physics background, which I have. I believe it was Fermi who, when asked by a speaker how they should pitch their talk, replied, "as if to a person with infinite intelligence, but zero knowledge." I like this very much, because one of the most common faults of scientific talks, if not THE most common fault, is that they assume too much knowledge of the audience. Even though they use big words and sophisticated language, if you watch closely they are not usually assuming that the audience is smart - although if the audience were infinitely intelligent they could follow anything - but usually that the audience has knowledge that they don't have. Or at least they don't have knowledge that the speaker is assuming that they have - and the speaker should know better.) And I ran across this blog called Rate Your Students. It discusses issues that professors have with their students in a funny way. Now, I've read several of the "best of" posts, which can also be found under "The Ring of DistinKtion" or "Things We Can Do to Improve..." but my favorites (so far) are On Grading and On Evaluations. Both of these (and probably more) should be read out loud for maximum effect. And do you notice what I've done? Since I really have nothing to say, I've pointed you to other sites so that you have hours of reading pleasure while thinking about how much fun you're having on Plasmaphile. I've multiplied your enjoyment with only a little work of my own. Ahhh, the internet.

PS. I've done a little research (i.e., Google, which amazingly, is what students think research is these days - I wonder if RYS has a post on that topic?), and John Baez (excellent math and science website, by the way, claims that it was Leo Szilard who made the statement about intelligence and knowledge. Well, I won't quibble. Both Szilard and Fermi had way more intelligence AND knowledge than I do!

Update: David J. Linden, in The Accidental Mind, claims it was Max Delbrück. I've found many more (unattributed) examples. I'm beginning to think that this quote has entered our collective consciousness and it may well be impossible to determine its origin.

Cemeteries of Munich

One of my goals is to visit the graves of famous physicists that happen to be buried in Munich. Usually because they lived in Munich (or died in Munich) and taught at the University of Munich (now called the Ludwig-Maximilians-Universität, LMU).

The first one I visited, two weeks ago, was the Alter Südfriedhof, just outside the old city walls near Sendlinger Tor. Buried there is Joseph Fraunhofer.

And also Georg Ohm.

Fraunhofer, of course, is one of the fathers of spectroscopy, and he was the first to discover the "Fraunhofer lines," absorption lines in the solar spectrum that allows us to determine the chemical makeup of the solar atmosphere.

Ohm is more famous. His law, Ohm's Law, is a description of the proportionality between current and voltage, and he was a professor at the Universität from 1852-1854. There is a nice statue of Ohm on Theresienstraße at the Technische Universität, TU.

Finally, last week, I visited the Waldfriedhof - the Forest Cemetery. Werner Heisenberg is buried there, the father of the uncertainty principle.

While he was never a professor here, he was a student. His advisor was another famous physicist, Arnold Sommerfeld, and Sommerfeld is buried in another Munich cemetery, the Nordfriedhof. I plan to visit that next, given that Ernst Mach (of Mach number fame) is buried in Nordfriedhof. Those pictures will be coming soon.

I have a special connection to Sommerfeld, as he is my academic great-grandfather. My PhD advisor, Burt Fried, did his graduate school at the University of Chicago under Gregor Wenzel. And Wenzel was also a student of Sommerfeld's. So I'm especially interested in Sommerfeld's grave.

Wank

My third trip to the Alps was to Garmisch-Partenkirchen, two towns that were combined into one by Hitler before the 1936 winter Olympics were held there. The towns are at about 720 m altitude, and nearby rise the sharp peaks of Zugspitze, Germany's highest mountain, on the border with Austria, and Alpspitze, a picturesque peak with a huge slab of an east face, sloping down at about 45 degrees.

I went with a former student of Hochschule München and some of his friends to 'climb' Wank, a smaller hill with no snow - currently the path up Zugspitze is closed because of snow. Here is a photo looking out over Garmisch from a few hundred meters up the trail. The Matterhorn-like peak in the background is Daniel, and is in Austria.

And here, looking slightly to the left of the previous photo, is the massif that contains Zugspitze (on the right) and Alpspitze (on the left).

And here they are together.

Wank is about 1800 m high, so the trail gained about 3500 feet in a very short distance. It was extremely steep. But I was with youngsters only half my age, so I had to keep up! After about 3 hours we made it to the top. Of course, since this is Bavaria, at the false summit just west of the peak, there was a large crucifix. And also, of course, since this is Bavaria, at the summit there was a restaurant and beer garden. Unfortunately, these were closed because of the season. They are open both in summer and winter, when the visitor population increases, but on a November weekend that wasn't supposed to be warm, but turned out to be about 55 degrees, they didn't think there would be enough hikers to support their sales.

Here is another view from the top, of a small town named Farchant.

Finally, after we hiked down, one of the young hikers had actually grown up in Garmisch, and took us to visit his grandmother. We were able to look around her house - which was 350 years old - and also have some homemade cookies - which were delicious. Then we went with his father, sister, and brother-in-law to have dinner at a local restaurant. On the advice of his father, who has been a chef for 35 years (a few in Washington, DC, and a few at the famous Hofbräuhaus) I ordered Wildgulash, which was a goulash with deer meat, a local specialty. Very good. Here's the restaurant.

Dachau

I went to Dachau on Monday.  It is very strange, because it is a nice little town who's name has such a bad connotation. To get there, you take the S2 train to Petershausen, although sometimes it doesn't go all the way to Petershausen, but only goes to Dachau. For these trains, it is somewhat uncomfortable boarding a train that says "Dachau." Of course, the people who live there probably wish that they weren't so infamous to the rest of the world.

Then, you take a bus to the KZ-Gedenkstätte (concentration camp memorial site). You can also walk the distance, following the path that the prisoners had to walk when they were discharged from the train, and there are signs along the way. In Dachau, as with many other German cities, there is a J-F Kennedy Platz, which is situated on the Sudetenlandstraße. (In Munich, there's a bridge over the Isar river named the JF Kennedybrücke.)

The memorial site itself is quite powerful. It was the first camp in Hitler's regime, started in 1933. It was intended for German politicians in order to solidify his control over Germany. Some things that I hadn't known before include the fact that for most of the war there were no Jews there. Hitler expressly did not want Jews in camps that were in Germany. This is why they went to camps in the occupied countries, like Auschwitz. Also, it's one of the few camps where some original building survive - although some have been reconstructed also. Many of the German camps were demolished and there is only a memorial site. In Bavaria, all school children are required to visit one of the camps, and many go to Dachau. Also, in Germany, all military officers are required to visit one of the camps.

The day I went it was very cold and gray. I suppose it gave me more of a feeling of what it was like, not that anything really could. I did think about the book by Louis Zamperini, and his description of the camps in Japan that he was in. The description of my guide was very similar.

Here are a few photos:

A "back alley" between buildings where - at the wall at the end - some firing squad executions took place.

The main square where "roll call" was taken twice a day. The prisoners had to stand for up to 3 hours as they were counted. It took so long because by the end of the war there were 40,000 prisoners.

The rows of foundations of the bunkhouses. The foundations are original, but the two bunkhouses at the far end are reproductions. The buildings had been demolished in the 1960s.

One of the poster in their extensive history exhibition. Translation: "Our last hope: Hitler." It is certainly true that, from what I've read, Hitler did seem like one who would lead Germany out of it's terrible economic crisis precipitated by the end of WWI and the world wide depression of the thirties.

Overall, quite an experience.

S-bahn/Olympics

To get to work in the morning - and back in the evening, I take the S7 line of the S-bahn (Suburban or Surface). There is also the U-bahn (Underground) lines, which are more the typical subway. However, while the U-bahn is always underground, the S-bahn is mostly above ground in the suburbs, but underground in the city center. The U-bahn dates from the 1972 Olympics, and the S-bahn is about 20 years old.

Starting from a little south of my house, the stops are, heading north

Höllriegelskreuth
Pullack
Großhesselohe Isartalbahnhof
Solln
Siemenswerke
Mittersendling
Harras
Heimeranplatz
Donnersbergerbrücke
Hackerbrücke
Hauptbahnhof
Karlsplatz (Stachus)
Marienplatz
Isartor
Rosenheimerplatz
Ostbahnhof
...

I get on at Großhesslohe and off at Hauptbahnhof (central train station). Then I take the tram (either line 20, 21, or 22) north on Dachauerstrasse toward Hochschule München. The stops are

Hauptbahnhof Nord
Karlsstrasse
Stiglmeierplatz
Sandstrasse
Hochschule München (Lothstrasse)

I you keep going north on Dachauerstrasse, you will reach Olympiapark, the area where the Olympics were held: swim stadium, track and field stadium, and basically a very nice park. Here's a view of the swim stadium, where Mark Spitz won his 7 gold medals!

Hochfügen, Österreich

This past weekend I made it to Austria!! Friends of a friend - who are now my friends - invited me to join them on their weekend trip to their hütte (cabin) in the mountains. Until we were in the Alps, I did not realize that we were going to Austria. But now I've got another country under my belt. I don't think I'll make it to Vienna, Megan, but the Tirol region is close enough. Here was our route. What was really nice about it is that we climbed up the northern range of the Alps, and then dropped into the Inntal, which translates into 'Inn valley'. {This is just like the lakes, which are 'see', such as Königssee, where I went last weekend. Although, the English maps usually write 'Lake Königssee', which means 'Lake King's Lake.'  It's kind of like 'The La Brea Tar Pits' [more about that in my next post].} The Inntal is the valley of the Inn river, which flows into the Danube (or the Donau in Germany). There's a very nice map on the Wikipedia page. Of course, up river is the city of Innsbruck, which, since brücke means bridge, means 'Bridge over the Inn river.' It's amazing what a little language can teach you!! Here is a photo of a town on the Aachensee, which we passed just above the Inntal.

So then we crossed the Inntal and entered the Zillertal, which, as you might guess, is the 'Ziller valley.' The Ziller is a river that flows into the Inn. There is a lot of skiing up the Zillertal - really in the mountains above the Zillertal - and we headed up into the mountains at the town of Fügen. At the end of the road was the ski area, hotel, and cabins, called Hochfügen. And here's a photo from the front porch of the hütte, which overlooks the ski area.

It reminds me alot of the Big Bear/Lake Arrowhead area of LA. It's about a 2-3 hour drive from Munich, the cabins are at 1.600 m (about 5,000 feet), and the upper peaks that are visible are 1.000 m (3,000 ft) higher. The snow is at lower elevations here, I guess because it's further north. But it's very similar. And, of course, the first order of business was to decide which beer to drink!!

Here's the same view the next morning, when the sun was on the slopes. The temperature varied from about 2˚C - 10˚C. It was cold but pleasant because of the sun.

At the local tavern, I posed with a young friend next to a sign that must be wrong? The circumference of the earth is close to 40.000 km. Even though the Earth is an oblate spheroid, with an equatorial radius of 6378 km and a polar radius of 6357 km, this makes any circumference within 100 km of 40.000. This sign implies a polar circumference of 41.390 km, which is definitely too large. The equatorial circumference is 40.074 km, and the polar distance must be less that that. I didn't notice this until I was posting this photo. I don't know if I'll make it back there, but if I do, I'll let the proprietor know!

Königssee

This past Samstag I went to Berchtesgaden, the city, which is just outside of the Berchtesgadener National Park, located in the Alps in the southeast corner of Germany near Salzburg. It was a four hour train ride to Berchtesgaden, then a short bus up to Königssee, a deep glacial lake. Finally, a boat ride across the lake to St Bartholomä, which is where I'm standing below.

The snow on the ground was fresh - last week was cold and the Alps got their first snowfall. I'm sure the snow in the mountains was not fresh. In fact, in the notch in the mountains behind me, is a glacier, which you can't see.  The peak behind me to my left is Watzmann (2713 m), and the glacier is called the Eiskapelle. But we went for a hike up that valley - didn't get to the glacier, but saw some pretty scenery. "We" is Jack and I. Jack is another engineering professor in the same fellowship program as I am. He's teaching in the Mechanical Engineering department, and is from Taiwan.

It was very nice to be hiking on hills, and the smell in the air was wonderful - trees, snow.  It reminded me quite a bit of Glacier National Park. As far as I could tell from the German brochures, much of the geology was similar, too.

The only bad part was the long train ride, which, on the way there was made even longer. Just outside of Munich the train stopped because of some blockage on the track ahead. We had to go back to Munich and wait for the next train - two hours later. So a 4-hour trip took 6 hours. But, the did give everyone a coupon for a free coffee and "Butterbreze", which is a pretzel sliced in half and spread with lots of butter. Pretty decadent.

Munich impressions (3)

Tipping is not as prevalent as it is in the US. And when tipping does occur, it is at lower levels and is much more relaxed. For example, if you eat a meal at a restaurant, you would tip only 1-2 EUR rather than a fixed percentage (and a rather high percentage) of the bill, as in the US. Also, the way you tip is more casual. This evening I ate with a friend and the bill (which was simply written on a piece of paper) was 13,80 EUR. I gave the waitress a 50 EUR note and said "fünfzehn" (fifteen) indicating that I wanted to pay 15 EUR (i.e., a 1,20 EUR tip). She gave me back 35 EUR in change. It's all very above board, rather than the somewhat secretive way we tip in the US, leaving an amount on the table  and then leaving.

They put mayonnaise on French Fries (or Pommes). Just like John Travolta learned in Pulp Fiction. And there are many things that are 'finger foods' in the US, but here they use a fork and knife. French Fries, for example. With mayonnaise. And burritos. In order to not stick our like a sore thumb, I have to eat my burritos with a fork and knife. It's important for me to watch people around me to make sure I'm not committing a blunder.

One thing that's not so different. As I rode my bike around central Munich today, I saw quite a few people who I assume are homeless. I guess it's a problem everywhere.

Around Munich

On Saturday, I went for a bike ride around Munich, led by my knowledgeable host. We visited the famous Chinese Tower in the Englischer Garten, and of course, saw the surfers on the stationary wave in the tributary of the Isar river.

Of course, we had to stop at the Biergarten near the tower. The weather is so beautiful - unusually warm for October, that it's almost sacrilegious to not stop. I had a Schweinschnitzel and pommes, with, of course, ein Maß.

Then we moved on and found a piece of the Berlin wall by the US Consulate building...

... and a piece of the Universität München (now called LMU, or the Ludwig Maximilian University) whose wall still showed signs of the fighting in WWII. Or so my host claimed.

All in all, a pleasant day.

Pullach im Isartal

This is my current residence, and it means something like "Pullach-on-Isar" because it's situated next to the Isar river. 'Tal' means valley, so more correctly it would be "Pullach on the Isar valley". Here's a nice shot of the Isar river.

If you look closely, you can see the Alps in the distance (the picture is taken toward the south). Also if you look closely, the river is on the left, and there's a canal (a diverted part of the river) on the right. The canal appears to be used to run some power stations. In between these two is a raised berm with a running path on it. That's one of the places where I (and many others) run.

And here is my house. Well, it's Karl's house.  We'll, 1/4 of it is Karl's house. It's a 'quadplex' in which four different 'apartments' have been made. We are in the front left.

I actually live in the Großhesselohe section of Pullach. Großhesselohe and Pullach used to be two separate cities, but recently they merged and have one Mayor. In addition, it's a separate city, not a part of Munich. This meant that to get my 'residence permit', I had to go to the office specifically for the outlying regions, not for those who wish to be a resident in Munich. But the good news is that I think I'm done with the German bureaucracy....for now.

Munich impressions (2)

As in my last post, here are a few things that have struck me about the differences between Germany and the US.

Pizza: what looks like a pepperoni pizza really has salami on it (the slices are larger than pepperoni usually is in the US). If you ask for a 'pepperoni pizza' you get a pizza with chopped up pepperoncinis. So you must beware!

Children travel on their own much more than in the US. For example, on the S-bahn and U-bahn trains, there are many young students going to school. Many look like they're 10 years old, or even less. Even though it's perfectly safe, it's not something that you would see any more in the US. Perhaps 30 years ago, but not any more.

When someone sneezes in public, there is no "God Bless You" or "Gesundheit" response. That just doesn't seem to be the custom here.

There is no prohibition on "open containers" in public. In fact, people drink on the S-bahn trains, especially the youth heading to Oktoberfest. Since the beer is expensive at "the Wiesn", they drink beforehand. And there are breweries everywhere. There's a small brewery and restaurant about a 5-minute walk from where I live (it's in the "platz" right next to the train station) so tonight my host Karl gave me a 2-L jug and told me to go to the brewery and get it filled. Which I did, for about 7,50 EUR. And I walked back. Now this jug just has a cork in it, so in the US it would be considered an open container, but no problem in Munich, where beer is king. Anyway, so we each had some beer with our dinner of homemade buffalo wings. Did I mention that Karl is a lover of spicy American food?

Addendum: The train drivers went on strike last night, but only for 9 hours. So, while the morning commute was disrupted (my S-bahn train into Munich was 20 minutes late) nothing serious happened. The worker-manager relationship seems to be different here, too, although I don't know enough about it to say in what way.

Munich impressions

Now that I've been here almost two weeks (!!!) I seem to have some time to write about it. I've got lots of pictures saved up (Oktoberfest and all) but I have noticed several things that are distinctly different than in the US. Not better or worse, just different, and sometimes surprisingly different. So here goes, a random list.

People are very nice. If you're introduced to them, almost all speak English, and if they don't, they'll still try (since they know more English than I know German). But, walking by someone on the street, there is almost never a greeting. I understand this in the big city, but in the tiny suburb of Pullach, where there's not too many people that you pass while walking, it is uncommon to even make eye contact.

At the end of a lecture - or class - the students show their appreciation by knocking on the tables. Instead of clapping. Not that my students usually clap at the end of each class. But here they do. I kind of like it. Now, in one of my classes, most of the students are international, so they have to learn this custom, too.

You get a lot of coins in change. This is because there are 5 € bills, but 2 € and 1 € coins. Of course, there are also 0,50 and 0,20 and 0,10 and 0,05 and 0,01 'euro cent' coins. So basically my pockets are always bulging with coins after I go into a store.

I mentioned this yesterday, that Germans love bank transfers. I'm going to become an expert in online banking in German! And another funny thing about banks. My bank is Kreissparkasse. There is another bank, with exactly the same logo (a red S with a dot over it) called Stadtsparkasse. You might think they are the same bank, but they're not. They are related, I've been told, and I can withdraw money out of either ATM system with no fees. But they're not the same.

Finally, I had Leberkäsesemmel for lunch. Leberkäse literally means 'liver' and 'cheese', but it is a meat similar to bologna that has neither liver nor cheese. In fact only products called 'Bavarian Leberkäse' are allowed to not have liver in them. All others, such as 'Stuttgarte Leberkäse', must have at least 5% liver. Go figure. And the semmel is the roll, on which you also put some sweet mustard, which I did. Very delicious.

Das Deutsche Museum

I went with my friend Wayne to the Deutsches Museum yesterday. It is a huge museum that has exhibits ranging from 'the history of calculators' to 'musical instruments'. We happened to be in the musical instrument room when a docent was playing several of the instruments, one of which was a harpsichord from the 1500s!  It sounded very cool.

My favorite spot (so far) was the roof, on which there was a sundial garden. There were about 20 different sundials, even one on the ground (look for the figure 8 - it's called an 'analemma').

However, one of the neatest was a polyhedron sundial, with a different sundial on each face.

I plan to go back because I became an annual member of the museum. Well...kind of. I haven't paid yet. In Germany, their preferred method of payment is the bank transfer. So, while you could pay cash for the 8,50 € single entry fee, you could not pay cash for the 52 € annual membership. I filled out a form, they gave me a temporary membership card, and I promised to transfer 52 € into their bank account. Very different from the U.S., where they wouldn't let you set foot in the place until you had paid.

Such are the lessons that I'm learning.

Fibonacci numbers

You've all heard of Fibonacci numbers, written down in 1202 by Leonardo of Pisa when investigating the number of rabbits in a group after successive generations. He assumed that in each cycle (about a month), each adult rabbit A would have one baby rabbit B. Also, each baby rabbit would turn into an adult rabbit. The transformations at each generation therefore are A→AB and B→A. So if we start off with one baby rabbit, the population in successive generations becomes

B
A
AB
ABA
ABAAB
ABAABABA
ABAABABAABAAB
ABAABABAABAABABAABABA
ABAABABAABAABABAABABAABAABABAABAAB
...

and the number of rabbits is the Fibonacci sequence, 1 1 2 3 5 8 13 21 34..., where each number is the sum of the preceding two numbers.

There are tons of interesting mathematical properties related to this sequence, my favorite of which is the golden ratio Φ. That is, if you take the ratio of successive entries in the sequence, you get better and better approximations to Φ as you use larger and larger entries in the Fibonacci sequence. For example, 3/2 = 1.5, 5/3 = 1.66666..., 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.61538. As you can see, these are getting closer and closer to Φ = 1.6180339887499..., which is the golden ration (or golden mean). A neat property of this number is if you take its inverse, it is 0.6180339887499..., which is the same number minus 1. For you math geeks, this simply means that Φ is the solution to the quadratic equation Φ² - Φ - 1 = 0.

But that's not the property of the Fibonacci sequence that I want to talk about. I want to show how it is related to quasicrystals, a topic that I've discussed recently. First, notice that in the AB sequences above, each sequence is simply the concatenation of the preceding two sequences. This is something you might not expect, since each is generated by making the replacements A→AB and B→A from the previous sequence, and, in principle, is not related to the sequence before that.

If you look at all the, say, four-letter sequences in the last line, you might expect that there would be equal numbers of all the 2⁴ = 16 possible sequences:

AAAA
AAAB
AABA
AABB
ABAA
ABAB
ABBA
ABBB
BAAA
BAAB
BABA
BABB
BBAA
BBAB
BBBA
BBBB

However, if you look, you will only find the following 5 sequences

1. ABAA
2. BAAB
3. AABA
4. ABAB
5. BABA

You see this in the following way. For example, there are never more than two As in a row. The reason is that there is no way, from the two rules at the top (A→AB and B→A), to create three As in a row.

Winter Semester MUAS

My courses have been finalized for the winter semester at "Hochschule Munchen" (Munich University of Applied Sciences). See here for a listing of all the course offerings in English. I'll be teaching a Modern Physics course, based on my course that I teach at Embry-Riddle. And I'll teach an Orbital Mechanics course, using Howard Curtis's excellent textbook.

Finally, I'll be helping with a project course for students interested in participating in a NASA design competition. I don't know if the design specifications have been released yet, but last year it was to simulate a mission to the moon.

I'm looking forward to having a lot of fun!

What is a quasicrystal?

My post on 01.09.2014 about quasicrystals didn't answer the question of what exactly they are. I thought I'd remedy that now.

The atoms of most metals are arranged in a regular array, often called a crystal array (or a crystalline array), that is periodic. This simply means that if you move the array a certain distance in a certain direction, it will look the same. This is called "translational symmetry." In addition, if you rotate the array by a certain angle, it will also look the same. This is called "rotational symmetry."

This is easiest to think about in only two dimensions. That is, you can completely cover a flat space (i.e., a plane) with identical, equilateral triangles.

This called a "tiling" or a "tesselation" (not to be confused with "tesseract," a four-dimensional hypercube, or the tesseract from A Wrinkle in Time). A triangle is one of three shapes than can completely tile the plane. The other two are squares and hexagons.

(The connection with crystals comes when you imagine placing an atom at each vertex. Most crystals are in some form of cubic symmetry, but a few have hexagonal symmetry.)

In addition to translational symmetry, the triangular tiling has three-fold rotational symmetry. That is, if you rotate the triangular pattern by 120, 240, or 360 degrees, you obtain the same pattern. And, as you might expect, a pattern of squares has four-fold rotational symmetry, and the hexagons has six-fold rotational symmetry. They all also exhibit translational symmetry.

You don't have to use only one shape, you can use a repeating pattern of two (or more) different shapes.

This pattern can be moved left or right, up or down, and cover itself exactly - translational symmetry. It also has rotational symmetry. Can you see how many fold?

One thing that you CAN'T do (try it) is to completely cover the plane with pentagons, or any set of shapes with five-fold rotational symmetry. Here's an attempt at that:

If you look closely, it appears to have a five-fold rotational symmetry, but it doesn't exactly. To prove this, you'd have to print out two copies and see if you could match them up after rotation. In addition, this tiling is not periodic - that is, it is not translationally symmetric. Again, it's almost periodic, but not quite. This is the loose definition of a quasi-crystal. An array (three dimensional, of course) that has symmetry and periodicity over the short range, but not over the long range.

The picture above is of a "Penrose tiling," developed by Roger Penrose, a British physicist and mathematician. They are extremely fascinating, and I'm working on a post just about Penrose tiles. They are fun to play with, just like Zometools, and I had a lot of fun doing just that at PCMI in Utah.

Baseball (3) - Orioles Park at Camden Yards

Part of my plan for my visit to DC was to go to two more baseball games, an Orioles game and an Nationals game. I knew that my good friend Bill, also a baseball fan, would help me out. So on Tuesday night, we braved the rush hour traffic and went to Baltimore. We had excellent seats, about in the 20th row. 

How did we get such good seats? StubHub! People re-sell their game tickets on this site, and the closer it gets to game time, the cheaper the tickets usually are, since if they don't sell, the sellers get nothing. In fact, one guy we met suggested waiting until a few minutes until game time, because then they get really cheap. Of course, you must have a smart phone, because you can just download the tickets to your phone, show your phone at the gate, and in you go!  It's a brave new world.

My culinary choice in Baltimore was an 'Old Bay sausage,' which was a sausage with Old Bay seasoning, famous in Maryland. It was spicy and quite good. The only down side was that it had been raining for about an hour prior to the start of the game, but they had the infield covered, and it ended up starting about 20 minutes late. The above photo is the first pitch, and if you zoom in, you'll see a white streak just to the left of first base. That's the ball.

Of course, during the second hitter's at-bat, it started raining again, and we had to suffer through a two-hour rain delay this time. Below is the grounds crew putting away the tarp and spreading more dry dirt on the infield.

You can also see beyond right field a large warehouse. Between the park and this building was a long "alley," where much of the food was sold. It gave the park a nice feel, similar to Wrigley, being in the middle of the city.

After about the 7th inning, since there were so few people at the game, and so few left, they invited all the fans to move to any seat that was empty. So we moved down to about the 8th row, just behind home plate.

During some pitch, it could have been this one, the batter hit the ball and broke his bat. The top half of the bat flew into the stands right toward us! It hit one of the guys in the orange shirts a few rows ahead of us, bounced off, and was caught by a kid a few seats to the left. Luckily nobody was hurt. The bat was sharp. 

Finally, the game ended at 12:20am, with a 5-4 win for the Orioles over the Reds, and we drove back to DC. A long (but fun) night.