Yesterday Om made a comment noting a sneaker with equations inscribed in it. It makes you wonder what would be appropriate. Perhaps simple Newtonian physics like F = ma, v = at, d = at2/2. You might also try to build a model of running. I scribbled the equations of a very simple running model. They wouldn’t speak to a teen and they wouldn’t speak to an expert either. It turns out that an exact model of human running don’t exist. The same is true for walking. This despite quite a bit of money from athletic shoe companies. While a good deal is understood is understood, the situation isn’t as nice as swimming was in the 70s when computational hydrodynamic models revolutionized technique as evidenced by the record books.
The basics, however, are known fairly well. To run you need to strike the ground very hard with a minimal contact time. Using the basic physics of motion you can glean a lot of information using video analysis without a detailed model of running. A pair of sports science people did this for Usain Bolt’s 9.58 second run. The specifics of his run are amazing. A few notes:
° He generated a maximum power of about 2,620 watts (about 3.5 horsepower) 0.9 seconds into the run. For a bit over a second he developed over 2,000 watts.
° He is very tall and aerodynamically inefficient - his coefficient of drag is about 1.2. A little worse than a falling piano. He’s also about 10% worse than the average world class sprinter, so he has to make it up with raw muscle power. It is good to be a freak of nature.
° During the run Usain's muscles delivered about 81.6 kilojoules of usable energy … 19.5 nutritional calories. If you assume an average metabolic efficiency of about 24%, you get to about 40 calories:-) That may describe the run, but his core has heated so much that he will be spending a lot of energy as his body attempts to cool down over the next 15 to 20 minutes. Even so the amount of energy he would have burned including the recovery period is probably less than one Snicker’s bar.
° Most of the power generated by his muscles, about 92% of it, goes into dealing with air friction. Power needed to move through the air goes up with the cube of velocity and coefficient of drag or Cd for the running human body is awful. So rather than moving forward, most of his effort is going into parting the air and creating vortices. Since his muscles are anaerobic during the run it would be interesting to set up a oxygen tent on the Tibetan plateau or in the high Andes in Chile and cut it open a second before the start. Lower air resistance would allow a faster run, but you’d want to get him back into sea level oxygen asap after the run.
° Back to that 2,600 watt burst. Assuming the 24% metabolic efficiency it means he has to dump over eight kilowatts of waste heat. For the 2,000 one second rate he has about 6 kilowatt-seconds of heat energy cooking away inside him and the amount over the run when he’s dropped back to around a kilowatt of output is still huge. It turns out the limiting factor on ten second runs like this is core body temperature which is gets to dangerous levels before the anaerobic portion of the metabolism fades. If he could keep up the rate for a 200 meter run he might not be able to recover. We have some natural limits.
° A few athletes have higher power bursts — weight lifters for extremely short periods of a few tens of milliseconds. Probably the best are indoor cyclists who can deliver 1,500 watts for about 20 seconds. It takes a very special body type.
I am so far from the limits:-) I’m really happy when I can average 200 watts to a rowing machines flywheel for an hour. (a few notes in an earlier post)
Those who probe the limits raise some interesting questions. What could Usain have done to slightly improve his time? It turns out he's a slow starter and has the Cd issue, but are there runners that have his other gifts and a slightly better number? How does wind influence these numbers (that is easy to calculate)? And the general very fascinating question of how important is body and muscles type to different types of sport?
What a beautiful blizzard. Last week it was great fun to watch and even more fun to walk in, but in the back of your mind you know that there is warm and safe shelter. A few years ago Sandy shattered the illusion of guaranteed safety. We were without power for nearly two weeks. Two weeks that saw subfreezing temperatures and a drop of household temperature to figures commonly associated with the refrigerator. Although the furnace burns natural gas, it requires electricity to run a large fan and to power the controller circuitry. Saturday morning, as the storm intensified, that memory returned and I turned the thermostat up a few degrees to buy a bit more time 'just in case'...
Somewhere, probably within a hundred miles, a large turbine was spinning under the blast of superheated steam from boiling water. The turbine was coupled to a massive complex of copper and iron all intricately interwoven into a twisted form. It has two basic pieces - one that spins and one that doesn't. A large magnetic field is created by one part and the other interacts with it generating an electric current. That's it. There are only two major moving parts - the turbine wheel and half of the generator. Once created the power needs to be connected to users. Large copper bars are connected to the generator at a special location. They extend to a large transformer and wind around a specially shaped piece of iron. Another piece of copper winds around another part of the transformer to boost the voltage of the electricity so it can efficiently be sent a long distance over high voltage transmission lines. As it nears the house it enters another set of transformers that reduce the voltage to levels household devices can use.
In principal it is just that simple. I turn up the thermostat and the blower motor on the furnace comes on putting a slightly higher load on the turbine wheel. It's all connected and responds at a good fraction of the speed of light. If one of the wires breaks everything comes to a halt along that path that wire is on. A tree branch falling on a power line can do just that. If load can't be moved more frightening and expensive damage is done - transformers and other components overload and burn out. There aren't that many spare large transformers in the world and production takes time.
We usually think of electricity as the transmission of energy from a source to load. Because it is ephemeral it is convenient to think of it in terms of power .. the rate at which energy is used. Since our needs vary throughout the day, there needs to be a near instantaneous management of supply to meet the demand. It practice the system is very complex and poorly understood by non-specialists. Much of the initial engineering was done in the 20s and 30s using large and very expensive physical models of portions of the grid. These were effectively large analog computers - among the most impressive ever made. Expensive, but driven by a huge desire. We talk about how important our mobile devices and computers are, but try living without electricity in Winter conditions for a few weeks and make a choice between your phone and electricity.
To deal with instantaneous changes in demand there needs to be a massive overcapacity relative to average demand. This is most apparent in during the Summer in hot cities where special turbine powered generators are used to supply peak air conditioning loads that go well beyond the normal capacity of the grid. In fact average demand during the year is about 30% of total generating capacity. Some of the capacity is very inexpensive - a few cents per kilowatt-hour - while turbine powered generation during peak demand can go for over $2.00 per kilowatt-hour. There are seasonal and weekly components to the demand curves that are regionally predictable and used to adjust the generators that respond slowly. But on a minute to minute level there is a spiky "fur" to the curve. This is difficult and expensive to manage. The chart shown is power demand in the UK during Andy Murray’s 2013 Wimbledon victory against Novak Djokovic and its corresponding demand-reducing powers (red line, below). The dip is 1.6 gigawatts - about two large power stations. A larger increase in demand occurred in 2011 when millions switched on their televisions to watch a royal wedding. In the US there is a massive impact during the Superbowl - particularly during halftime.
During the blackout of Sandy two expensive sources of local electricity became important. Primary batteries and fossil fuel powered generators. The batteries went into flashlights and portable radios. An alkaline C cell stores about ten watt-hours of energy - that translates to about $100 per kilowatt-hour .. a big change from the 15 cents or so you probably pay your power company.1 Home scale generators are also be expensive. A rugged natural gas unit in sized to run a furnace fan, refrigerator, a few lights and a computer or two, can cost as much as a furnace and requires maintenance. Electricity from them is too expensive when the grid is working, so you can be talking about hundred of dollars per hour for electricity over the lifetime of the unit. That said, electricity is so important than these and cheap portable gasoline generators sold like hotcakes in the Northeast for about a year after the storm.
Storage can be part of a practical electricity solution, but you have to pay attention to costs and scale. Microgrids with combined heat and power generation can work well for industrial complexes, universities and even large apartments. There has been a surge of construction in the Northeast in the past few years - the cost of being without power is just too high. Larger scale storage can be practical, but only to a point. Mostly it is pumped hydro. Water is pumped against gravity to a physically higher reservoir when power is inexpensive and, when power is needed, water from the high reservoir is allowed to run downhill spinning a turbine attached to a generator along the way. The problem is there aren't many location with the right geography to do this inexpensively.
Dozens of other schemes exist. Apart from leveling the cost of power and reducing the need for new generating capacity, it is often seen as necessary for intermittent renewable sources of power. After all, the Sun doesn't shine all day and the wind isn't always blowing. Batteries have been rapidly dropping in price. There are a few niches where they currently make sense, but in general they need to come down by at least an order of magnitude. For this reason many consider renewables capped at a rather small scale.
With solar and wind you worry about capacity factor - system's predicted electrical output in a year of operation to the nameplate output. For solar this is generally in the 20 to 25% range with wind being somewhat higher. You also have to worry about the capacity of the portion of the grid these systems are coupled to. In many parts of the US and Europe wind turbines need to be feathered in high winds as there is insufficient demand in the portion of the grid they serve for the power they could be generating at that moment.
It turns out, if you just rethink the grid, the need for storage can be greatly reduced - at least in the US.
The US has a section in the Southwest that can generate enormous amount of power from photovoltaics. There are also several regions that with rich wind power potential. The current grid fragmented and difficult to manage. Shipping power more than five hundred miles is not very common and the capacity isn't great. The structure was largely designed from the 20s to the 50s and I'd characterize it as held together with piano wire and chewing gum at this point. It needs to be replaced.
Consider these high resolution (in time and space) animations of solar and wind potential in the 48 states2
There rush by, but it turns out that you can make use of the production in various parts of the country if you can just move the power over great distances. We now have the technology to do that with ultrahigh voltage direct current power transmission. I've done a bit of modeling in the area suggesting that storage isn't as important as most people think. Now more detailed studies suggest you could get rid of all coal power generation, assume no growth in nuclear, and come in somewhat under current energy costs in about 15 years with massive increases in large wind turbines and PV farms - although the wind turbines are the real key. You still need to have some large natural gas plants, but in one study CO2 emissions are down 78% over 1990 levels.3 Other types of pollution drop drastically too. Here's an animation showing power transmission:
Think of it as an interstate electric transmission system on par with the Interstate Highway System. It isn't clear if the mandate to build it can be mustered up, but it may be a solid path forward. A huge and important opportunity.
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1 If we didn't have rechargeable batteries, primary cells would be a major expense for cellphones - figure about $300 to $400 a year in batteries.
2 From the Cooperative Institute in Environmental Research partnership of NOAA and CU-Boulder. From Nature Climate Change - a paper
Future cost-competitive electricity systems and their impact on US CO emissions
Alexander E. MacDonald, Christopher T. M. Clack, Anneliese Alexander, Adam Dunbar, James Wilczak & Yuanfu Xie
Abstract
Carbon dioxide emissions from electricity generation are a major cause of anthropogenic climate change. The deployment of wind and solar power reduces these emissions, but is subject to the variability of the weather. In the present study, we calculate the cost-optimized configuration of variable electrical power generators using weather data with high spatial (13-km) and temporal (60-min) resolution over the contiguous US. Our results show that when using future anticipated costs for wind and solar, carbon dioxide emissions from the US electricity sector can be reduced by up to 80% relative to 1990 levels, without an increase in the levelized cost of electricity. The reductions are possible with current technologies and without electrical storage. Wind and solar power increase their share of electricity production as the system grows to encompass large-scale weather patterns. This reduction in carbon emissions is achieved by moving away from a regionally divided electricity sector to a national system enabled by high-voltage direct-current transmission.
3 Electric power generation is about 40% of total power use in the US. A shift to electric vehicles requires more electricity - this technique is very useful for that as the cars have local battery backup that can be leveraged.
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Someone asked for my chili recipe. I tend to throw beans in together, but based things on an old recipe someone sent a decade or more ago. The base recipe is very good, but I usually am doing slight variations. Use it as a starting point and move out..
here's my response:
Here’s the base recipe I use for the chili … someone sent it to me years ago and I only follow it roughly. In the Winter sticking to canned veggies usually wins .. particularly for tomatoes. I also used canned beans for this as it is thrown together on snowy days.. all cans are the 12-15 oz size here. On beans just throw in whatever you want. I usually go mostly black and pinto, but pick one, all or off the list… On the peppers usually do two red and one green,, the author was going for color. It is way better if you stew plum tomatoes down into a sauce unless you have a great store bought sauce - I haven’t found a good plain one, but in a pinch its ok. The masa/cornmeal is the trick. I also usually go for 2 c of broth and 1 of beer (Sukie has the rest:-) Of course the chili powder section is where you may want to go wilder, but this is a good basic filling chili
• 2 Tablespoons Olive Oil • 4 cloves Garlic, Minced • 1 whole Large Onion, Diced • 1 whole Red Bell Pepper diced • 1 whole Yellow Bell Pepper diced • 1 whole Green Bell Pepper diced • 2 whole Carrots, Peeled And Diced • 2 stalks Celery, Diced • 1 whole Jalapeno, seeded and Finely Diced • 3 cups Vegetable Broth • 1 can Plain Tomato Sauce • 3 oz Tomato Sauce Paste • 1 can diced Tomatoes + Chiles • salt • 1 teaspoon Ground Oregano • 1 Tablespoon Ground Cumin • 2 Tablespoons Chili Powder • 1 can Kidney Beans, Drained And Rinsed • 1 can Pinto Beans, Drained And Rinsed • 1 can Garbanzo Beans, Drained And Rinsed • 1 can Black Beans, Drained And Rinsed • 1/4 cup Masa or Regular Cornmeal • 1/2 cup Warm Water
In a large pot, heat the oil over medium heat. Add the garlic, onion, 3 colors of bell pepper, carrots, celery, and jalapeno, then cook for about 5 minutes, stirring occasionally, until starting to soften. Add the oregano, cumin, chili powder, and salt. Stir and cook for a few more minutes.
Pour in the broth, tomato sauce, and tomato + chiles. Stir, bring to a boil, then reduce the heat to low, cover, and simmer for 30 minutes. Add the beans, stir, then cover and simmer for 30 more minutes.
Mix the masa with the warm water and stir it into the pot. Simmer for 15 more minutes. Taste and adjust seasonings.
Serve with cilantro. Non vegans go for cheese or yogurt
At the beginning of each year I treat myself to a week or so of study in something new to me. The intention is to learn enough that I might not embarrass myself asking questions afterwards ... This year I looked at the interplay between food systems, restaurants and transportation in 19th century America. It turned out to be a richer topic than I had imagined leading to a few rabbit holes. One was the emergence of the thin body type as desirable. First in men and twenty or thirty years later in women and its connection to the concept of muscular Christianity and the emerging middle class in America and England. Just as interesting was the realization that colonial America offered an abundance of food to immigrants (slaves not included). There was an easy self-sufficiency that didn't exist in Europe at the time - at least not for the commoner. In the 17th and 18th century the English spoke about the luxury of dietary self-sufficiency as self governance and that was probably the meaning of the term in America at the time of independence. I didn't pursue those lines very far as there was other fascinating game at hand, but perhaps I'll return. I was primed to think about some of our habits that would seem odd to those before the Civil War...
With the New Year many have started diet and exercise programs and almost as many won't make it more than a month or two. As I learn more about metabolism I tend to separate diet from exercise. Exercise by itself is terrific and is now considered a powerful drug with a mostly positive impact and few side effects. It is frequently prescribed, but many of us tend to under-medicate. We also misjudge its impact on weight maintenance. It takes a large amount of exercise to burn enough calories to lose weight, but we tend to reward our hard work with goodies or otherwise subconsciously eat more. Unless you are an athlete in training you probably can't outrun your fork. Other than noting any calorie counting scheme that includes exercise is probably suspect, I'll concentrate on diet.1
There are historical treatments that point out when thin became popular in Western Europe and the US and how the diet industry began. I've seen feminist pieces on the negative issues associated with body images, but I haven't seen a cultural examination of diets and fad diets. Many of the plans seem to be based on the epic heroic story arc with someone starting out suffering to the point where they start on a quest. The quest usually involves wander and sometimes invokes deep cultural myths (but rarely science). After difficult battles it ends with success and happiness. This is sometimes communicated with religious fervor. The cultural myths may be mirrors to deeply seated beliefs. Sometimes there is a retreat to earlier times which are painted as carefree and healthy. Modernity and 'chemicals' are foes to be avoided - if we can only return to these simpler times all will be well. Never mind that people rarely made it to thirty before the agricultural revolution and everything organic is chemical based.
This being a modern age, it is often assumed prescriptions for the best healthy diet exist. A numerical prescription for a healthy diet requires a predictive model that is appropriate for individuals. Unfortunately we aren't even close. There are thousands of studies on some food, nutritional element or diet, but most of them are limited and don't reach the entry level of being considered solid science. Human nutrition studies are expensive and many are funded by companies with clear conflicts of interest. A few are raised to public attention by undeserved publicity. Fish oil was highly touted as healthy with millions buying and using the product. The problem was that the excitement was based on a single weak study made a few decades ago. The result seemed impressive so it was publicized. When examined carefully it became clear that while the benefit was real, it only existed for people who were mostly Inuit and may be detrimental to other groups. Superfoods come and go yearly and there seems to be a fad diet of the year. Some larger trends - paleo comes to mind - have no solid basis in science and seem to be based on a mixture of myth and even politics.
The science situation is difficult. The type of experiments you would like to do where you start infants on vastly different diets and follow them to old age while holding all environmental conditions the same as other trial groups are impractical and unethical. Data is taken on massive longitudinal studies, but it takes at least thirty years to begin to get information and much of the data is self-reported and of poor quality. Furthermore individual metabolisms differ greatly and are now known to be influenced by our personal micro-biome and perhaps even the epigenetics coming down from our grandparents. Metabolic measurements can be made, but the precision is not terribly accurate. Try counting calories some time. I guarantee you won't get within twenty percent of reality. Even then the caloric assignment for food components is a hundred years old. Even though it is known to be flawed, there isn't anything much better that would make a difference.
So is there any hope?
While the quantified person approach is a bit silly at this point as nutritional models are too primitive there is something very useful.2 Many studies are large enough and well enough understood that metastudies lead to qualitative and semi-quantitative results. These have been relatively stable for a few decades, but are refining and becoming more accurate with time. While I'm a doctor, I'm not the right kind of doctor to offer dietary advice. So with that caveat here's a summary of the consensus of those who are expert:
° don't over-eat and try not to exceed a desirable weight range in your youth and middle age.
° eat a lot of vegetables and fresh fruit
° whole grains and legumes are good
° avoid unsaturated and trans-fats
° aovid excessive salt
° avoid excessive butter
° if you eat meat, do it in moderation (Mediterranean diets average on the order of than two ounces of meat a day)
° stay away from excessive sugar.
° if you drink, do it in moderation
° where possible, prepare your own food so you have knowledge and control of its contents
A few dietary types have emerged as statistically leading to populations that enjoy good health outcomes into their 80s. Mediterranean and vegetarian diets appear to be good over large populations. Strict adherence to a particular diet is probably unimportant and unadvised given the quality of current models. If you can find one that incorporates most or all of the bullet points that is easy for you to maintain, go for it. There are a few behaviors that seem to team up with the nutritional elements to lead to good outcomes.
° try to lead a low stress lifestyle
° value friendships
° get enough sleep
° get enough exercise
Not that this is easy given many of our lifestyles. If you're overweight getting back into a healthy weight range can be difficult. Keeping it off for the long term makes losing weight in the first place look easy. But the fad diets are wastes of your time and some can even be dangerous. If you want change, just go with something simple. The expert view seems to be getting away from the idea of healthy nutrients and getting to having a combination of foods that gives you the proper mix of nutrients. There will be many paths to achieve the goal.
Exercise is a different matter. Just find something you like to do and just do it. There is evidence that as little as a half hour a day of fast walking bestows major benefits. Countries like Denmark and the Netherlands partly fund their bicycle infrastructure from healthcare funds. There is good evidence that getting enough active transport, walking or cycling, leads to a higher quality of life in middle and late age. If a large enough segment of the population participates, healthcare costs drop.
If you are trying to lose weight you can be pseudo numerical. You won't be making accurate assignments to what you eat, but if you are internally consistent and weigh yourself a few times a week (psychologically Wednesday turns out to be best if you do it once a week), you can make adjustments that will fit your metabolism.3 Regular use of a scale is a powerful tool.
But in the end I suspect movement towards the healthier eating styles that are being identified will be glacial. We love good stories and the drama of the heroic narrative is a siren call and part of popular culture.
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1 Energy expenditure from exercise is poorly measured. Smart watches, Fitbits and exercise equipment - unless calibrated for you in laboratory conditions - are rarely better than 20% accurate and repeatability is poor. They can be useful for those who like to keep track of progress, but the connection to weight management is complex and non-linear. It appears that large amounts of exercise make the body more efficient and do not lead to weight loss - at least not as much as one would expect doing a simple tabulation. It should also be noted that no one knows how much exercise is needed. There are some good base level estimates, but it is certainly not true that doing two times as much as some basic level is two times as good for you. This is seriously complex so unless you like to keep track of numbers and try to make them increase because you like to see numbers increase, it is probably better find an exercise routine that is like and do it because you want to.
2 There are several medical conditions that offer a small number of repeatable and accurately measured variables to be useful. Diabetes is an example. It will get better with better sampling and models, but current techniques are valuable and life-saving.
Also don't hold your breath for genetically-targeted nutrition recommendations - at least those with any science behind them. Even if they did exist there is the problem of convincing people to adopt them.
3 Brian Wansink of Cornell offers a number of useful psychology based mindless eating suggestions in his book by that title.
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Recipe Corner
It is Winter and that means soup. I rarely use recipes, so here's a walkthrough of a lentil soup
Start off with a bag of lentils. All of them are different and all are good. Pick your favorite or just be random about it. Note the cooking time on the package and sort and rinse them. Add some olive oil to the bottom of a soup pot. I use a few glugs (how scientific) of an olive oil. You don't need a fancy $30 a bottle oil, but try to go with a real olive oil rather than a cheap mixed product. Turn the heat on low and start chopping the veggies. I generally start with a yellow onion, a couple of carrots, a stalk or two of celery, and a one or two smashed cloves of garlic (usually two:) Turn up the heat to medium and throw all of it in along with a pinch or two of sea salt. Stir as they get soft begin to smell good.
I usually add a few more veggies at this point. The last soup had fennel and the one before it had chopped acorn squash. Use your imagination. Toss them in and keep stirring.
Now add the spices. Decide on a direction. My last soup had a curry flavor from dried curry, cumin and turmeric. I pour some out onto my palm as a measure and dump them in. Just checking it worked out to about 1/2 tsp of each. For fun I added a bit of fresh lemon zest and stirred it all in. You don't want to heat them too much - just enough to toast them a bit - you'll know as they get very aromatic. Maybe a minute and a half to two minutes.
Add the lentils and a vegetable stock. I imagine chicken stock would work, but I'm vegetarian and don't go there. I make my own stocks and they're unsalted as it is too easy to over-salt a soup otherwise. I generally use a cup of lentils and four times as much stock. Feel free to vary if you like thicker or thinner soups.
Bring the soup to a boil and then cut the heat to a simmer uncovered. Go for softened lentils - don't let them get mushy. Also check any added veggies to make sure they're soft - a good reason for cutting them into small cubes.
Now add some chopped greens like spinach or swiss chard. They wilt in a minute or so and be bright green. Remove the pot from the heat when that happens. Now add a bit of lemon juice, sea salt to taste and serve. This is a good one for finishing with a fruity olive oil.
There are so many variations. Try it with some coconut milk for example... a bit of grated ginger.
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...
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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 -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:
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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