“The mystery no longer lingers:
Found, at last, two missing fingers.
They both belonged, as did one tooth,
To Galileo. That’s the truth.”

So writes Digital Cuttlefish, in response to a BBC report that begins: “Two fingers and a tooth belonging to famed astronomer Galileo Galilei have been found more than 100 years after going missing, a museum in Italy says.” The image here, reportedly of another of Mr. G’s fingers, is from the BBC site.

String theory implies that black holes can come in all kinds of forms and flavours, according to a cosmologist who has catalogued all known types

String theory is physicists' best guess at a unified theory of all interactions but it comes with some strange predictions. One of these is that spacetime consists of ten dimensions rather than just the four we're familiar with. And that raises some interesting questions.

One of them is what shape singularities can form in this higher dimensional space. In four dimensions, the only solution is spherical and that's the type of black holes cosmologists have imagined all over the universe.

But in higher dimensions, there are all kinds of other solutions. We've looked at the possibility of black rings but today Maria Rodriguez at the Max Planck Institute for Gravitational Physics in Golm, Germany, compiles a catalogue of all know species of black hole.

It turns out there's a whole managerie of other black hole solutions. Here are just a few: the black saturn, the black helical ring, the di-ring, the black bowtie and the bicycling black ring as well as the more general blackfolds.

While these solutions may exist mathematically, they may or may not exist in the real Universe. In fact, Rodriguez is able to work out certain criteria that a solution must meet for it to have a hope of existing in the real world. For example, a black ring can only exist if there is enough centrifugal repulsion to prevent it from collapsing.

Rodriquez points out that the list is incomplete. "The catalogue of different species (exact solutions) of black holes shows a very rich structure but seems far from being complete."

That makes it an interesting topic for ambitious cosmologists. But be warned: there's a good reason the list is incomplete. The solutions in this higher dimensional space are fiendishly difficult to find.

Nevertheless, it would be good to either rule out the possibility of their existence or work out if and how they can be distinguished observationally from common or garden spherical black holes.

Ref: arxiv.org/abs/1003.2411: On the Black Holes Species (By Means Of Natural Selection)

Today is Einstein's birthday. If he were still alive he'd be 131. Those of you who have been reading here for a long time know that Einstein was (and is) one of my "culture heroes." When I was a kid I sent him birthday cards (yes, I'm that old) and when he died made a scrap book filled with news clippings. One of the great loves of my younger life gave me an Einstein bust as a present and it still sits on my desk, more than 40 years later (she reads the blog from across the ocean, so I hope she sees this! Mrs. R. knows and likes her so this isn't a guilty secret). I also have first editions of his second and third published works and a fairly large library of books by and about him. Unlike quantum mechanics, relativity theory is essentially the achievement of a single person, Albert Einstein. Both are beautiful theories and quantum theory may be the most successful theory in the history of science. But relativity is no slouch, either, having been confirmed again as recently as this week. Not bad.

Given all this, it seems fitting to commemorate the occasion on the blog. Enjoy:

Happy birthday, Albert, hero of my youth. Still a hero.

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Today is the much celebrated pi-day . Ok, perhaps it’s not that big a holiday – I don’t think Hallmark is selling any pi-day cards yet – but anyone who uses google today knows that something mathematical and geeky is being honored. I promise not to go into diatribes about calculations of the first few million digits of pi, or how many digits one needs to keep in order to calculate the radius of the universe to atomic accuracy. Instead, I merely want to relay a simple short story a colleague of mine recounted to me years ago.

Several years ago, before pi-day was famous, a student called the phone number associated with the digits in pi that appear after the decimal point, i.e., 1-415-926-5358. Apparently this is rather common now, and in fact, appears to be promoted as a mnemonic for the first 10 decimal places for those folks we need to have those numbers handy at all times. But this story happened in earlier times, back before the Bay Area split into several area codes. And, as the clever reader has already guessed, that student reached the SLAC main gate. How cool to phone pi and reach the main gate of a major national scientific research laboratory!

Alas, time and phone numbers march on, and nowadays phoning pi yields a “your call cannot be completed as dialed” message. (And I’m told that I cannot publish this post without noting that 3-14-15 will be a more accurate pi day.)

It was already known that we were totally doomed, but now there is a new and exciting scenario. In this one, comets rain down on us from the Ooort Cloud, said comets loosened by contact and interaction with a star called Gliese 710.

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Sine, cosine, and tangent are of course the workhorse functions of trigonometry. You learn 'em in high school, and if you go on in math and science you never stop using them. Now on many occasions you might have the sine or cosine or tangent of some angle, and you want a way to invert those functions to recover the angle from those values. Let's take a look at the inverse tangent function:

Now we'll do one of those things were we skip any motivation and just do some stuff for reasons we'll get to at the end. Let's find the derivative of the arctangent function. To start, take the tangent of both sides of the above definition. The tangent undoes the arctangent, so we get:

Differentiate both sides. The derivative of tangent is secant squared, which we could prove if we felt like it, and since f is a function we have to use the chain rule. Doing thus, and recalling that the derivative of x is 1:

Solve for the derivative:

Which isn't all that helpful except for the fact that there's a basic trig identity which connects the secant and tangent functions: sec(x)^2 - tan(x)^2 = 1. Substituting:

Ah! But tangent(f) is x, so we can swap that in:

Ok, ok. What's that good for? Well, if just write that down as our Sunday Function and graph it, we'll have something to celebrate:

That is the graph of a function, 1/(1+x^2), which has nothing at all to do with circles or geometry in any obvious way. But we know it's the derivative of the inverse tangent, and conversely the inverse tangent is the integral of that function, so we can easily plug in and see that the area under the curve is pi. Which is nice, because to day is Pi Day. Happy Pi Day!

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I'm en route to the March Meeting in Portland, which involves a three-hour layover in Chicago, between two flights on Southwest, my preferred airline. I'm always impressed by how much more efficient Southwest seems that the other major airlines.

One weird manifestation of that efficiency is the flight plans that Southwest uses. Where most flights on other airlines seem to go back and forth between two cities over and over, Southwest's routes tend to roam all over the country. This morning's flight from Albany to Chicago continued on to San Antonio, TX, Phoenix, and San Jose. Another recent trip involved a flight from Albany to Baltimore, Kansas City, San Antonio, Las Vegas, and San Jose.

This always makes me wonder about the computation behind these flight plans. There's got to be some logic to it, that allows them to get all the planes they need to the places they need them, presumably as cheaply as they can manage. The result is most likely derived empirically from crunching some numbers on existing flight networks, then optimizing the results, but it's kind of amusing to imagine that the folks at Southwest are sitting on a solution to the Traveling Salesman Problem, and using it to gain an advantage over their competitors...

(It's also a good idea for a throw-away element of a science fiction story about Singularity-type stuff, but I'm not the guy to write that...)

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Are you a mental model, or do you have a mental model?

If so, then Farnam Street (their modest motto is “What the smartest people on the internet read”) wants to hear from you. They say:

Help! Mental Model Index

We need your help to ensure we have the most comprehensive list of mental models.
We’ve brainstormed the following list of mental models. This list will be the master list for the site. So, we need your help to make sure its complete. We think we’ve done a pretty decent job with Psychological misjudgments, but we need your help ensuring that we cover the big ideas in other disciplines such as Investing, Economics, Mathematics, Biology, Physics, etc.

Maybe not “worship” exactly, but at least great respect and deference.  By “efficient” I mean that it increases economists’ standard “cost-benefit” concept of welfare.  That is: as usually estimated, the benefits of deferring greatly to distant ancestors far outweigh its costs.  And while this does suggest that we should defer more to ancestors, it also shows just how much distorted prices can break economists’ favorite tools.

The economic welfare of a proposed change is the benefits minus the costs of that change, translated into cash terms, though of course changes don’t have to actually be cash transactions.  When available, market prices are commonly accepted as estimates of the benefits and costs of things gained and lost.  Economic welfare is a powerful heuristic for finding win-win deals: in many kinds of situations, the strategy of consistently making the changes that increase economic welfare tends to be usefully close to an actual win-win deal that gives most everyone more of what they want.

The efficient ancestor worship problem arises from two key facts:

1. Economic welfare cares not about giving people experiences but about satisfying their preferences, i.e., giving them what they want.  And even long dead people still have (or “had” if you prefer) preferences that we could now better satisfy.  If we do something a dead person would have wanted, that counts as a benefit.
2. At standard market interest rates, the magic of compound interest quickly gives astronomical priority to the preferences of folks who lived long ago.  For example, in historical records near risk-free interest rates (e.g., land rents over prices) consistently exceeded 9%/yr from 3000BC to 1350AD, for a total factor of over 10162.

Together, these facts suggest we would increase economic welfare if we spent less than 10162 dollars today to do anything for which a 3000BC ancestor would have been willing to pay a dollar (equivalent in their currency).

Clearly we would quickly bankrupt ourselves if we tried to implement such “efficient” changes, and doing so would not be remotely close to a win-win deal with our ancestors.  What goes wrong here?

Our contract law system refuses to enforce many win-win deals between distant generations.  Many folks would be willing to create trusts that accumulated funds long after their death and then paid distant descendants (perhaps indirectly) to do things like remember their ancestor’s name, pray to his gods, etc.  Unless stolen, such funds would eventually come to dominate the world economy and dramatically lower interest rates.  With lower interest rates, economic efficiency would count the preferences of distant ancestors as far less valuable, and as a bonus businesses and governments would have far stronger incentives to attend to the interests of distant future folks, such as via global warming policies.

But we in fact refused to enforce a great many such long term deals.  For example:

The rule against perpetuities at common law … prevents a person from putting qualifications and criteria in his will that will continue to control or affect the distribution of assets long after he has died, a concept often referred to as control by the “dead hand.”

Our unthinkingly repugnance at being controlled by the dead, and our eagerness to grab their resources, prevents us from enforcing long-term win-win deals.  This refusal to enforce deals increases interest rates, which distorts all our trade-offs across time, bringing economic welfare estimates into stark conflict with intuitive moral judgments about time trades (as in global warming), which then encourages people to turn to non-economic frameworks for policy analysis.

When policy distorts prices, it distorts calculation of economic welfare, which encourages people to ignore economic welfare when choosing policy, which reduces their reluctance to intervene to further distort prices, which leads to a sad spiral of increasing confusion.  Please, let’s enforce long-term win-win deals!

Added: A fascinating alternate history might start from a year 1300 English legal precedent enabling flexible growing long term trusts.  By 1800 early trusts grew a billion-fold, and trusts dominate the economy.  What else changed?!

Some folks admire the super-polymath Professor Mohamed El Naschie. Some gaze from afar in wonder. But at least one person devotes considerable energy to blogging, in a rather unfriendly manner, about the professor.

Professor El Naschie, unbothered, describes himself with simple modesty:

He is the principle advisor of the Ministry of Science and Technology of the Kingdom of Saudi Arabia (KACST – Riyadh) since many years…. His research interests include: Stability, Bifurcation, Atomic-engineering, Nonlinear Dynamics, Chaos, Fractals, High Energy Particle Physics, Quantum Mechanics and E-infinity theory. He is editor-in-chief and associate editor of numerous learned journals.

Longtime readers of this blog may remember last year's orgy of pies on the run-up to Pi Day (March 14th, or 3-14). This March at Casa Free-Ride, there's been less pie making, in large part due to the fact that I'm no longer on sabbatical (either from my job or from coaching soccer).

But the bake-off is on again, so I figured that I needed to feed you all one really good pie (or pie recipe, anyway).

This pie melds three flavors that play very well together: rich chocolate, tart cherries, and almonds. As a bonus, it puts those flavors together in a pie that is rich but not heavy, one that doesn't lean on eggs, or cream cheese, or butter, or milk.

I make this pie with a food processor, but if you don't have one, you can manage with a blender, a heavy rolling pin, and a knife and cutting board.

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Oh dear. Sometimes it’s so hard to let go.

And most importantly, don’t forget to join us MARCH 13, at 1pm for the PLUTO IS A PLANET PROTEST MARCH AND RALLY. The march starts at the Greenwood Space Travel Supply store (8414 Greenwood Ave N) and will end at Neptune Coffee (8415 Greenwood Ave N).

But really, Greenwood Space Travel Supply is all kinds of awesome, even if they’re weirdly co-dependent with small rocks in the outer solar system. They’re the Seattle branch of the 826 network, which is a non-profit writing center for kids.

They also have cool t-shirts.

The Nautilus curiosity shop in Torino, Italy describes this as “A very nice French conformateur, a device for registering the shape of a head so that a precisely-fitting hat may be made.”

The most basic explanation of Pi is that it is the ratio of the circumference to the diameter for a circle. That seems simple enough, but it turns out that Pi is an irrational number - so you can't just write it down. Oh, I know that you are an uber-geek and you could recite the first 80 digits of Pi. But the question is - how many digits are enough?

In this post, I am going to assume that we don't know the true value of Pi (which is essentially true). I can then use propagation of error techniques to see how dependent different calculations are on the value of Pi.

Super Brief Intro to Uncertainty

I still can't believe I haven't put a post together on the basics of measurement and uncertainty. Add that to the todo list. The most important idea in measurements is that they are not exact values. Let me start with my favorite example. Suppose I have a table that I want to know the area of. To do this, I measure the length and the width. The value I come up with for the length is 133.2 cm. But what does this mean? Is this the exact length of the table? No. Two problems.

• The table doesn't have an exact length. What does the length mean for a table? Is it a perfect rectangle? No. Is it even straight on the edges - probably not.
• Even if it were a perfect table, would my measurement be perfect? No.

Maybe I measured this length a whole bunch of times and at different locations. This would give me an estimate of how the measurements are spread out. If I do the same for the width, I might get something like:

This means that the length of the table is almost certainly between 133.0 cm and 133.4 cm. If a similar thing can be said about the width, then this diagram could represent the area.

The point I would like to make - since the width and the length have uncertainty, the calculated area would have uncertainty. How do you determine this calculated uncertainty? I have three ways:

• Use the extreme values of length and width to calculate the extreme values of the area (in this case the smallest area uses the smallest length and width). This is the method I use for my algebra-based physics labs.
• Assume the error is small, linear, and normally distributed. In this case, you can use the partial derivatives of the functions to determine the relationship of the uncertainty for the measured stuff on the calculated stuff. Here is wikipedia's page on this, but I am not really going to go into the details.
• Assume that if you measure the stuff a whole bunch of times, the data would be normally distributed. Write a program that generates normal data and use that to calculate tons of times the calculated value. Look at the spread of all these calculations to determine the uncertainty. I am not going to do this right now.

Back to Pi

Archimedes used 96 sided polygons to estimate the value of Pi. He showed that Pi was greater than 3 and 10/71 and less than 3 and 1/7th. This gives a decimal value from 3.14084507 to 3.142857143 (with no rounding). I could write this as an average and an uncertainty of about:

That is not too bad of a value. But what about pi = 3? Is that bad? First - according to Snopes, no state has ever proposed a law that would officially change Pi to 3. It is still a fun story. Anyway, in this case I could perhaps say:

I chose the uncertainty in this fictional Pi to be +/- 0.2 so that the range would cover the true value of Pi. Really, though you could in general write Pi as:

Where Delta pi is the uncertainty in pi.

Some uses of Pi

So what effect does the uncertainty in Pi have on different uses of Pi? Let me start with something practical - the speedometer in your car. Basically, your speedometer needs Pi to make the conversion between angular velocity and linear velocity using:

I know, there is no pi in that equation. But, how do you know the angular velocity (omega)? If this is measured in revolutions per second (or minute) then you have to convert units. Let me write this as:

Now, I will assume that omega, r, and pi all have uncertainty. Then the uncertainty in the velocity would be (using the max-min method from above for simplicity):

And I would do a similar thing for the minimum value. I could average the difference between average and the max and the average and the min. (I will put these calculations in a spreadsheet for you).

What about the volume of a sphere? This same thing is used for calculating things such as - the volume of the sun or the volume of a spherical cow. Here is the volume of a sphere:

These two uses of Pi seem boring - but really this is the basis for many applications of pi. There are tons of others, but they are maybe more abstract (but just as important). Now, on the to the spreadsheet. I will put in some values for the stuff, but you can change them if you like.

Note - I don't know how to change the number of digits presented in google docs. Also, I seem to have hit a creative wall with uses of pi. How about you list your favorite use of Pi in the comments?

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In tribute of a sort to tonight’s Ig Nobel show at the University of Dundee, not far from the Tay Bridge, here are the concluding lines of William McGonagall’s mortal poem “The Tay Bridge Disaster“:

Oh! ill-fated Bridge of the Silv’ry Tay,
I must now conclude my lay
By telling the world fearlessly without the least dismay,
That your central girders would not have given way,
At least many sensible men do say,
Had they been supported on each side with buttresses,
At least many sensible men confesses,
For the stronger we our houses do build,
The less chance we have of being killed.

The best of the rest from the Physics arXiv this week:

Observation of an Antimatter Hypernucleus

A Signature of Cosmic Strings Wakes in the CMB Polarization

Self-Assembled Granular Walkers

Quantum Dating Market

A Computational Algorithm based on Empirical Analysis, that Composes Sanskrit Poetry

Another scientific breakthrough seen likely to improve lives almost immediately:  This one is reported in The Guardian, under the headline “Grow-your-own to replace false teeth“:

“Tests have shown the technique to work in mice, where new teeth took weeks to grow. There’s no reason why it shouldn’t work in humans, the principles are the same,” said Prof Sharpe.

That news article was published on May 3, 2004.

The real numbers, complex numbers, quaternions and octonions give Lie 2-superalgebras that describe the parallel transport of superstrings, and Lie 3-superalgebras that describe the parallel transport of 2-branes! john http://math.ucr.edu/home/baez/ baez@math.ucr.edu

Earlier this week, I wrote about an inclined treadmill, and talked about physical work. Physically, the amount of work that you do is equal to the amount of force that you exert in a certain direction multiplied by the distance you move in that direction.

If you walk up an incline, as opposed to moving on a level surface, you have to also fight the force of gravity to get up that hill, hence you have to do extra work.

I contended that, in order to walk up an inclined treadmill, you also have to do extra physical work, the same way you have to do extra physical work to walk up a real hill.

"Not so fast," said many of the commenters! Unlike a real hill, where you actually wind up at a higher elevation than you started at, you don't travel any real distance on a treadmill, and you certainly don't end at a different elevation than you started at! Therefore, the argument goes, you don't do any extra physical work.

So, I ask you, what would Einstein have to say about this? Einstein -- mastermind of the equivalence principle -- realized that as far as forces go, the only thing that you feel is acceleration. Your environment doesn't matter at all.

What's a good, analogous thought experiment for our treadmill/hill problem here? Imagine an escalator. Now, you're going to walk up this escalator, and you're going to walk up 50 steps on this escalator for three different cases, like Fred here.

If you walk up 50 steps, and each step is 0.2 meters high, then you will raise your elevation by 10 meters relative to not taking those 50 steps. In other words, it doesn't matter whether that escalator is moving up while you do it, in which case you might wind up raising your elevation by 20 meters total, or the escalator is moving down, in which case you might wind up not changing your elevation at all.

In either case, you have to do work against the force of gravity, regardless of what your velocity is. This even works in the case of a broken escalator, a.k.a. stairs.

And that's why, regardless of whether your elevation changes or not, you do work whenever you walk up an incline! Hope you've had a great week, and I wish you all a great weekend!

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The APS March meeting is next week as 10000 physicists invade Portland, Oregon. I hope Powell's bookstore has stocked their science sections well! GQI, the topical group on quantum information, sponsors a good number of sessions at the meeting including sessions with invited talks, focus sessions, and general sessions. Below the fold I'm assembling a list of quantum computing sessions, but before the fold I'd like to point out the invited sessions, which have longer speaking slots where one can actually learn more than the speakers name and research project title, that are sponsored or cosponsored by GQI (also below note the Focus sessions listed below have invited speakers)

• Monday, March 15 8:00am-11:00am Session A8: Quantum Opto-Mechanics
  Room: Portland Ballroom 255
  (Jointly sponsored with DAMOP)
  Invited speakers: Jack Harris, Klemens Hammerer, Philipp Treutlein, Nathaniel Brahms, Keith Schwab
• Monday, March 15 11:15am-2:15pm Session B6: Controlling Dissipation in Quantum Systems
  Room: Portland Ballroom 253
  (Jointly sponsored with DAMOP)
  Invited Speakers: Frank Verstraete, Hans Peter Buechler, Matthias Lettner, Luis A. Orozco, Sergio Boixo
• Monday, March 15 2:30pm-5:30pm Session D4: Quantum Computer Science
  Room: Oregon Ballroom 204
  Invited Speakers: Graeme Smith, Aram Harrow, Ben Reichardt, Sandy Irani, Stephanie Wehner
• Thursday, March 18 11:15am-1:40pm Session W6: Superconducting Qubits
  Room: Portland Ballroom 253
  Invited Speakers: Radoslaw Bialczak, Franco Nori, Leonardo DiCarlo, Sahel Ashhab

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