A few years ago I was walking through the parking lot of JPL in Pasadena when a license plate caught my eye. It took me a few seconds to figure it out, but it had to be a vanity plate and one I wouldn't mind having. I pointed and exclaimed something like 'how brilliant' to a rather perplexed friend. Just a few days ago someone set a rather amazing Internet ad. It had an easter egg that grabbed my eye for the same reason, although this was much more explicit. There is a connection between the license plate and the ad. I'll get to that, but first I need to talk about some playful tools that physics and astrophysics types use.

Any sort of creative work needs a playful component. Its how you develop your intuition. I'm sure many fields have their own power tools for play and you may want to think about yours as mostly they are so natural that they become part of how we think. You need to try and discard a large number of ideas without fear of failure. Some of these can be rather silly. Often you build toy models in your head or at the blackboard. Typically they aren't practical .. if someone were to ask how many cattle are required to provide shoes for New York City you might say to yourself 'consider a spherical cow'.. Physics problems tend to exclude much of the real world so you can focus on just the bit of interest. Mental or blackboard math rules. You aren't worried about exact answer. I love the power of computer simulations but anything you have to type into would just slow you down and break the flow during the play stage. You want to create little visual models that seem like an extension of your mind so you can dance with them. People develop a certain affinity for numbers and you carry around a few rough rules of thumb. Things like π^{2} ≈ 10 to about 1.3% .. when you're only worried about 10% errors the arithmetic is easy. Some bits of Nature are stored in this fashion:

° there are π × 10^{7} seconds in a year to better than a half percent

° the speed of light in a vacuum is close to 3 × 10^{8} meters/second. An interesting coincidence is the original definition of a meter was 1/10,000,000 of the distance from the pole to a point on the equator. Distances can be measured in the time light takes to travel the distance. We're used to light days and light years, but light goes about a foot in a billionth of a second or a nanosecond. A very tall friend sometimes gives her height as six and two thirds nanoseconds .. about the time light takes to traverse her length.^{1 }

° a light foot is sort of practical when thinking about connecting parts of a computer. If modules are separated by 10 feet, a delay of at least 10 nanoseconds is introduced. This gets interesting when you're using a lot of fiber optics or physical wire as the speed of light is slower.^{2} The speed of light in the fiber is about 2/3s that of in a vacuum or air. A signal will traverse 1000 km in about 5 thousandths of a second, while a radio wave going through the air only requires 3.3 thousandths of a second. A very long time for a computer. The difference was large enough to justify the construction of a specialized microwave network between Chicago and New York City to give some traders an advantage based on the difference between the index of refraction of air and optical fiber.

° the acceleration caused by Earth's gravity is denoted by g and is roughly 10 meters per second^{2}. It varies depending where you are on the surface, but unless you want to weigh a pound less or more, the rough figure works for quick calculations.

° the diameter of the Sun divided by the height of a person is roughly the same as the height of a person divided by the diameter of a hydrogen atom.

° it takes about 500 seconds for light to travel from the surface of the Sun to the Earth

° photosynthesis in crops and trees is usually under 0.5% efficient

° a cow requires about 10 times as much energy as the plants it eats to produce the same amount of energy

° ...

There are hundreds of little relations like this and a few curious numbers too. One of the most famous is the fine structure constant which describes the strength of electromagnetic interactions between charged particles. It is close to 1/137 and appears in enough calculations that every physicist knows it. If you want to flag one down in a stream of people (airports for example), just write 137 or 1/137 on a piece of paper and hold it in view. I've experimentally found this to be remarkably effective.

Some of the relationships aren't directly connected with Nature, but are just fun and sometimes are even useful. There are too many to list, but here are two..

° e is the base of the natural logarithms and is of fundamental importance to economics and science. It happens to be irrational and transcendental, but the first few digits are easy to remember: 2.7 1828 1828 45 90 45 (1828 repeats twice, and the angles in a right angled isosceles triangle)

° of course there is Hardy–Ramanujan number. The story goes that G. H. Hardy was visiting his friend and colleague Srinivasa Ramanujan in the hospital. He had ridden in cab number 1729 and remarked that it was such a boring number. Ramanujan counted, pointing out that it was the smallest number you can express by cubes in two ways: 1^{3} + 12^{3} and 9^{3} + 10^{3}. This has slipped into popular culture and has appeared in both the Simpsons and Futurama in the form of easter eggs. But then again some of the principals of those shows are mathematicians.

And number on a taxi brings us back to the number on a vanity plate. My eye was probably caught by the initial 23, which happens to be my favorite prime, but it looked suspicious - something close to 20 plus π ... I remembered e^{π} - π is just a bit less than 20. The number on the plate had to be an approximation of e^{π}. The connection with the ad is perhaps the most beautiful relation in mathematics : ^{3}

e^{iπ} + 1 = 0

While e^{π} isn't the real thing, it is suggestive enough to bring a smile. Of course there is the possibility the owner just considered e^{π} cool without thinking of e^{iπ}, but this **was** the JPL parking lot.

If only vanity plates allowed superscripts...

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^{ 1} again an approximation. If you want to do this more exactly the speed is about 0.984 ft/ns ..

^{2} the speed of light in a material is c/n, where n is the index of refraction. n is close to 1.0 for air, but about 1.5 for optical fiber

In physics experiments signals are delayed with respect others to make logic gates work properly. You develop a sense that 8 inches or 20 centimeters is about a nanosecond in coax.

^{3} It links the two most important irrational numerical constants e and π, number i which is the base of the imaginary numbers which gives us complex numbers, the additive identity 0 and the multiplicative identity 1. It also answers the question "how many mathematicians does it take to change a light bulb?" Then answer, of course, is -e^{iπ}...

If it seems strange that the exponential of a complex number could be equal to 1, consider what the number e is. It is closely related to compound interest ... e^{z} is the limit of (1 + z/N)^{N} as N goes to ∞. If you set z = iπ and do the math you get -1 + 0i or just -1. Here's an animated gif:

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Recipe Corner

**Caramelized Carrots**

**Ingredients**

° 3 pounds of carrots - go for the fancy colored ones if you want it to look good

° salt and freshly ground pepper

° 2 clementines or one large orange

° 1 tbl red wine vinegar

° 3 tbl butter or vegan margarine

° half a bunch of fresh thyme

**Technique**

° peel the carrots. you can quarter of halve them, but if they're pretty whole carrots with a bit of the tops on are beautiful

° put carrots in a large pan and just cover with water. Add a bit of salt and ground pepper and the clementine juice, the vinegar and butter

° bring to a boil and cook until nearly all the liquid has evaporated

° add the thyme sprigs, reduce heat to low and cook for about 5 minutes to caramelize

## precision twinklers

Fifty years ago this month Jocelyn Bell was going through the chart recorder output from a new radio telescope she and a few others had built at Cambridge . She had worked herself up to about a hundred feet of it a day looking for anything interesting in the squiggly line of ink. Very boring work. Lots of noise and bits of the expected.

But there was this "bit of scruff" when the telescope was pointed in a particular direction. The scruff was repeating every 1.3 seconds. Very regularly.

^{1 }You always doubt your equipment. Regular noise is usually something man-made or an artifact from the apparatus. And there were those new fangled satellites beeping away,. Carefully you try to eliminate the possibilities. She and her advisor eliminated most possibilities, but they weren't

thatcertain. Then she found another bit of scruff. That regular pulsing .. now at a different rate, but still very accurate. Better than the clocks they had in the lab. She had found something - something outthere.There was a flurry of activity when they announced. Astronomers around the world dropped what they were doing. Theoretical physicists and astrophysicists conjectured. A few of the conjectures advanced to hypotheses. And in this swirl of activity she found and third ... and then a forth. And in theoryland of the hypotheses made sense. It was called a pulsar.

She had found the crack in the door to some of the deepest Nature yet encountered.

In 2017 something like two thousand have been found. Shortly after the discovery it was suggested a star could collapse so dramatically that it's core was nothing but neutrons. Imagine a mass greater than the Sun compressed down to something about a dozen kilometers in diameter. Like a skater pulling in her arms it spins much faster than the star it started out as. The first went around every 1.3 seconds, but some spin more than a thousand times a second. Their magnetic fields are trillions of times stronger than the Earth's and, combined with the rotation a beacon-like signal forms. It sweeps through the sky like the light from a lighthouse as it rotates.

She and her advisor opened up an entirely new and unexpected branch of astrophysics that has led to a much deeper understanding of both astronomy and physics. Pulsars and their close relatives are hot areas of research. And they can even be useful. A GPS system is just a group of accurate clocks with transmitters orbiting the Earth. Pulsars are as accurate as atomic clocks. You could use them to build a galactic positioning system that would work anywhere in the Milky Way and not than just on Earth. And you can make use of their regular beat to build another kind of gravity wave detector. One that is complementary to the current interferometry technique.

Some of the techniques developed since then have trickled down into important technology we use and many hundreds of astronomers and astrophysicists have spent some time in the private sector making their contributions to the economy. Pure science is a very inexpensive mechanism for creating future value. The "problem" is you can't predict where it might lead.

Hewish went on to receive a Nobel Prize in Physics for the discovery. It is widely felt in the astronomy community that Jocelyn Bell should have shared in the discovery as her contributions were enormous. But: 1967 and female. She spoke of it ten years after the discovery:

"demarcation disputes between supervisor and student are always difficult, probably impossible to resolve. Secondly, it is the supervisor who has the final responsibility for the success or failure of the project. We hear of cases where a supervisor blames his student for a failure, but we know that it is largely the fault of the supervisor. It seems only fair to me that he should benefit from the successes, too. Thirdly, I believe it would demean Nobel Prizes if they were awarded to research students, except in very exceptional cases, and I do not believe this is one of them. Finally, I am not myself upset about it – after all, I am in good company, am I not!"

She's being far too generous. This

wasone of those exceptional cases. At least she's received other significant honors she's had throughout her career.____

And something sad.

I rarely use the term genius to describe anyone alive. Maryam Mirzakhani was an exception. A fearless mathematician, she was the only women to receive the Field Medal: math's highest recognition. She died on Saturday at age forty. She was just warming up. Here's a well-written piece about her that appeared in

Quantaa few years ago.Excuse my political comment, but Mirzakhani was female, brilliant and from an Islamic county. I doubt someone with these "liabilities" would be welcome here now ...

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^{1}She recognized that pulsars are astronomical sources where others had failed because she noticed that the pulses in her data (Figure 6.1) didn’t look like other forms of interference and they reappeared exactly once per sidereal day, indicating an origin outside the Solar System. She and Hewish, “decided initially not to computerize the output because until we were familiar with the behavior of our telescope and receivers we thought it better to inspect the data visually, and because a human can recognize signals of different character whereas it is difficult to program a computer to do so.” Other people were using software to filter out noise and were throwing out the interesting signal in the process..Posted at 06:05 PM in general comments, history of science, math | Permalink | Comments (0)

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