Tau Day

Happy Tau (τ) day! It’s June 28th, so here at Preciseish we celebrate 6/28 3:19 (6.283185) This is what we should use instead of pi. Everyone loves pi, but it really should not exist. Tau does everything pi does, but better.

Pi is what you get when you divide the circumference of a circle by it's diameter, but tau is the circumference divided by the radius. After you've defined pi, you'll almost never use the diameter again because it is the radius that makes a circle. A circle is defined as the set of points on a plane a constant distance (radius) from a point. If you draw a circle, you maybe do it by using a compass, or a bit of string, but always by enforcing a radius.

So what if we replaced pi with tau? Working in radians would become much more intuitive. We know that as a unit of angle measurement, 360° = 2 π. But why is one time around two times pi? It's confusing. 360° = 1 τ. Much better. Now you can immediately see how big the angle is. If you have ¼ τ, that's just one quarter of the way around, or 90°. Even better, sine and cosine become intuitive. Instead of memorizing some points like sin(π/2) = 1 without understanding what it means, now you can just learn that the sine is just the height on a unit circle at the angle. Now knowing sin(τ/4) is just knowing how high you are on a circle when you're a quarter the way around: you're on top, so 1.

One thing that seems, at first, like an advantage for pi is that the expression for the area of a circle comes out neater: π r² seems simpler than ½ τ r². But if you work with math or physics enough you'll realize that the ½ (constant variable)² form is very natural and makes sense. It comes from calculus. Consider falling. The fundamental constant for falling is gravitational acceleration, g, and our variable is time. To start with, all we know is g. Integrate that and get g t, that’s the speed you’re falling. integrate again and you get ½ g t² which is how far you've fallen. This is what is going on with the circle too. We start with the fundamental constant, tau. integrate and get τ r, that’s the circumference of the circle, integrate again and you get ½ τ r², the area of the circle. It's natural.

How about the beautiful and strange Euler's Identity that features pi so prominently? Euler's Identity is the following: e^{i π} + 1 = 0. It seems amazingly improbable that Euler's number, the imaginary square root of negative one, and pi should so succinctly combine to a non-imaginary, non-irrational zero. But would using tau do any damage? It turns out to give e^i τ = 1. Not only is this more elegant, in my opinion, but it results more directly from the underlying relationship involved: e^{i x} = cos(x) + i sin(x). If x is taken as pi, the result is an awkward -1. It takes some rearranging to get the more pleasing form above. If x is taken as tau, the result is a perfect 1 right off.

If you still aren't convinced, you must watch Vi Hart's quick and entertaining video on the subject, and enjoy this thorough but relative light and quick paced Tau Manifesto.

Acceleration

Acceleration is the rate of change in velocity. Remember that velocity is properly understood as a combination of speed and direction, so acceleration can refer to the rate of change in direction as well as change in speed, or some combination of both. For example, something orbiting at a constant speed is still accelerating. In the figure below, note that the length of the velocity vector (representing speed) is not changing, because the acceleration (due to gravity) is always orthogonal (perpendicular to) to the velocity.

That acceleration due to gravity is usually expressed by the constant, g = 9.8 meters per second per second. That number is approximate, and only valid near the surface of the planet, but it means that every second you're falling you'll be falling 9.8 m/s (22 mph) faster. In three seconds you'll be going highway speeds. The fastest production cars can just barely match that, doing 0-60 mph in just under three seconds. That's also about the acceleration that a cheetah can pull in a straight line. All of those things can accelerate at about 1 g. Impressive in a different way is the remarkable mantis shrimp. The mantis shrimp packs a punch that has it's foreleg accelerating at about 10,600 g! That's on the level of a bullet fired out of a gun. This motion is so tremendous that it can even cause cavitation for additional damage.

Food Origins

Before international shipping, most things that we ate were very local. For example, peaches were only found in China and no one else ate them. And this is true of most of the foods that we cultivate.

One of the most dramatic changes to this condition came with the exploration of the Americas by European explorers and conquerors. The plant life of the Americas was largely isolated from the rest of the world which led to the evolution of foodstuffs that could be found no where else, including much of what we consider standard fare now. It may be surprising to learn that Italians were not cooking with tomatoes until maybe 1592, for example. They are from Mexico and were completely unknown to Europeans before about 1550. Other examples of foods from the Americas:

• Potatoes
• Bell peppers
• Peanuts
• Chocolate
• Tobacco
• Vanilla
• Corn
• Squash (all kinds, including zucchini)
• Avocado
• Chili peppers
• Strawberries
• Pineapples
• Sweet potatoes 

And some other food origins:

• Oranges - southeast Asia
• Bananas - southeast Asia
• Rice - China
• Sugar - India
• Cucumbers - India
• Plums - east Europe
• Asperagus - central Eurasia (widespread)
• Cherries - Europe
• Cabbage - Europe and Britian
• Apples - Turkey
• Apricots - Armenia
• Spinach - Persia
• Wheat - the Levant
• Carrots - Iran (although they were bred orange in the Netherlands)
• Lettuce - Egypt
• Grapes - the near east
• Broccoli - bred from northern Mediterranean plants
• Peas - Mediterranean
• Olives - north Africa
• Coffee - Ethiopia
• Watermelon - southern Africa

Be glad that you live in such an interesting culinary time.

Charcoal

Charcoal can seem like a strange thing. Why do we want to burn something that has already been burnt? 

Well, charcoal is the result of burning, but a very specific and careful kind of burning. It might more accurately be considered refining. The original, raw, organic fuel found in nature (say, wood) is refined into a more effective fuel, charcoal.

The chemical reaction you really want when burning something organic for fuel is carbon and oxygen becoming carbon dioxide. Something like wood has a lot of carbon, but also a lot of other stuff that can get in the way of a good fire, like water. The slow way that wood is burnt into charcoal, mostly without oxygen, gets rid of most of that, leaving almost pure carbon.

Since charcoal is almost pure carbon, it: 

1. Has a lot more potential energy for it’s weight 
2. Burns hotter and 
3. Burns more cleanly with less smoke and less toxic smoke.

The higher heat obtainable is what made it most desirable through history because very high temperatures were required for working metal.

Hiccups

Many types of animals get hiccups besides humans. But all are mammals. A hiccup is a spasm of the diaphragm, and it seems only the mammalian diaphragm is susceptible. However, some amphibians have a similar reflexive gulp which leads some to suspect that hiccups are a vestigial amphibian trait. This is further supported by some embryonic similarities between amphibians and mammals. Another theory observes that only mammals get hiccups, and that babies and the young are especially afflicted, and suggests that it is related to mammalian reflexes that coordinate breathing and milk sucking. 

The longest known case of hiccups belonged to Charles Osborne, who had them from 1922 to 1990 - 68 years. That would be about 430 million hiccups. 

Utilitarianism

One of the major modern philosophical systems of ethics is Utilitarianism. The basic idea is that 1. everyone is fundamentally equal and has an equal claim to happiness. To be selfishly interested in helping only yourself or your family or friends is not ethically defensible. 2. The best thing to do is what maximizes happiness. Usually taken as the most happiness for the most people, although there have been other interpretations. 

This system is attractive because it’s relatively simple and relatively difficult to counter. Consider the volunteer asking you to donate a trifling amount of money to provide vaccines to children in struggling countries. How do you justify not doing it? Is it more important to buy yourself a can of soda with that dollar than to potentially save the life of a stranger? Utilitarianism is the recognition that obviously it is not and you should donate the dollar.

So okay, now you’re a utilitarian. You donate the dollar. What next? Well, for every subsequent dollar you think of spending you have to consider whether it would better be spent fighting the serious problems of the world and saving innocent lives. Starbucks? Nope. See a movie? Nope. Go out to dinner? Nope. It seems like maybe you should give of yourself until there is no one who needs help more than you. Or maybe you can do more good by being successful so you have more to give, but surely every luxury is still immoral.

Okay, so maybe you aren't a utilitarian. It’s hard to argue with the logic of the system, but at the same time it seems self-apparent that it is flawed; that’s just not how people work.

Probably, the value of utilitarianism, as with most philosophical ideas, is in the contemplation of them. And when faced with a difficult ethical question, it can be valuable to have a number of carefully thought through perspectives to consider. For example, maybe utilitarianism isn't very useful for an individual human, but maybe it can be of use to something less personal like a government or other organization.

Fun With Archaic Measurements

A league is a unit of distance that used to be more useful. It's the distance one could walk in an hour, reckoned to be about three miles.

Similarly, a league in a nautical context is the distance one can see from the deck of a boat and about equal to three nautical miles.

A nautical mile, then, is about 1 arc minute of latitude, or 6076 ft. This is about 15% longer than a regular mile.

The name "mile" comes from the Latin word millia which means thousand, as in a thousand paces. Isn't that more satisfying than the awkward 5280 ft? A mile is also an even 8 furlongs.

The name furlong is just "furrow long" rendered in Old English. It's the fairly standard length of a plowed furrow. It turns out that a square one furlong on each side is ten acres.

Acre is Old English for "open field", and is about the area that can be plowed with a pair of oxen in one day. Picture a football field excluding endzones and sidelines. It's an area one furlong by one chain.

A chain is a length taken from an actual chain a clergyman named Gunter used for measuring land. It had 100 links and was the length of four rods.

A rod is about 16.5 ft. People who did a lot of measuring for engineering or land surveying would have an actual metal rod of that length to use for their standard.

This was more rigourous than the ancient cubit. The word comes from the Latin word cubitum for "elbow" and the length was the distance along one's forearm from the elbow to the tip of the middle finger, about two spans. 

A span is the distance from the tip of one's thumb to the tip of one's pinkie finger with the fingers spread wide, approximately 9 inches. 

The inch has, at one point, been defined as the length of three grains of barley laid end to end, but the word "inch" is from the Latin word uncia for one twelfth part because it was (and now is again) one twelfth of a foot. 

The foot is, guess what, about the length of a person's foot.