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 π2 ≈ 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 × 107 seconds in a year to better than a half percent
° the speed of light in a vacuum is close to 3 × 108 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 second2. 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: 13 + 123 and 93 + 103. 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 eπ - π is just a bit less than 20. The number on the plate had to be an approximation of eπ. The connection with the ad is perhaps the most beautiful relation in mathematics : 3
eiπ + 1 = 0
While eπ isn't the real thing, it is suggestive enough to bring a smile. Of course there is the possibility the owner just considered eπ cool without thinking of eiπ, but this was the JPL parking lot.
If only vanity plates allowed superscripts...
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 -eiπ...
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 ... ez 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:
° 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
° 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