The Colors

In 1665, the plague closed Cambridge for two years. Isaac Newton, who had just gotten his bachelors there, was stuck at home. To fill the time, he made a series of the biggest breakthroughs in physics, ever.

One of these regarded the nature of light and color. People knew that white light shined though a prism resulted in a rainbow of colored light, but they thought that the white light was perfectly pure and the colors were the result of the prism muddying the light. Well, Newton got a hold of two prisms. He used one to split the light, observed the different colored beams of light, and then tried to use the second prism to split one of the colored beams of light. If the prevailing theory was right, more colors should have been added, but no; it remained one color (red). Newton's new, correct, theory was that all colors are already present within the white light and the prism just separates them. In fact, he was able to use mirrors and lenses to recombine the separated colors back into white light. Incidentally, if you've ever heard of ROY G BIV you can thank Newton. He decided the colors of the rainbow were: Red, Orange, Yellow, Green, Blue, Indigo, and Violet. If indigo seems a bit awkward there, you're right. He just added it because he wanted there to be seven colors.

In fact, quite a few languages have done without "blue" entirely. If you ever read the ancient Greek Iliad and Odyssey you can count the usages of each color.  William Gladstone (the British politician) did just that and counted 170 uses of black, 100 of white, 13 uses of red, less than ten each of yellow and green, and not a single use of blue. Homer famously described the sea as "wine dark." If you read the Torah and New Testament in their original ancient Greek and Hebrew you'll, again, find no blue. In 1969 Berlin and Kay proposed a theory based on the study of dozens of languages suggesting that they always follow the same progression of adding color terms. A primitive language will only have black and white. Any dark or cool color will be lumped into black and any light or warm color will be white. If the language has three basic color terms, the next is always red. Then you get green and yellow, then blue. If you think about it, in a way blue is very rare in nature. It's by far the least common color in plants, animals, rocks, or dirt. It's the most difficult pigment to produce. But what about the sky, the sea? Well, we call those blue, but what about air, or water? They're clear, colorless. And these vast features are always background. One can imagine that if one had no concept of blue, no word for it, one could fail to really see that background. See this episode of Radiolab for a good discussion. 

The Lever of Mahomet

Imagine a game: it uses a cart that can move forward and backward along a straight track. The cart can move with any finite speed and acceleration. A straight, uniform rod is attached to the bed of the cart with a hinge. We'll assume the hinge is frictionless, there is no air resistance, etc.

Player 1 assigns the cart a motion that gets it from Point A to Point B. It may start and stop several times, it may reverse direction, but it has to eventually get to Point B. Player 2 is given the motion Player 1 came up with and has to try to find an initial position of the rod such that the rod will never quite fall all the way down. He gets as many tries and as much time as he wants to try and accomplish it. 

The question: Is there any motion that Player 1 can choose that Player 2 can not eventually beat.

It turns out the answer is no. At least as a thought experiment, there is no dance Player 1 can come up with that can not be beat by Player 2 choosing just the right starting position. Think of the extremes. Given the motion, Player 2 knows that if he starts the rod far enough over to the back it will end up all the way down in back. He also knows that if he starts the rod far enough forward it end up all the way down in front. Well, everything we're talking about is continuous, so as he gradually changes the rod angle from back to front there must be some small range of angles where the result transitions from ending up down in back to ending up down in front. Any one of those transition points is a solution where it doesn't end up down at all. 

This problem is from an article by Richard Courant and Herbert Robbins called "The Lever of Mahomet" and can be found in The World of Mathematics.

Taste

We perceive taste though chemical reactions that take place on the tongue. The tongue has different receptors that react with different types on molecules and ions in the food. This gives us information about the chemical content of the food. 

We now generally recognize five categories of taste:

• Sweet taste is a reaction with dissolved glucose (or sucrose or some other similar molecule). This tells us the food has a lot of easily accessible energy content. If you're an animal this is a very good thing, so sweet tastes good.
• Salty taste is the detection of Na+ ions. Salt (NaCl) dissolves into Na+ and Cl- ions in water, so if our saliva has a lot of Na+ ions we must be eating something salty. Again, in nature salt can be scarce, but is critical for survival, so it tastes good. 

• Sour taste is detection of H+ ion concentration in the saliva. This means that sourness is the same as acidity. 
• Bitter is triggered by a variety of molecules, many are alkaloids which tend to be basic in pH. This usually comes from plants that are producing it as a defensive poison, so the body's first reaction tends to be to label the taste as bad or even gag. Bitter foods like coffee and beer tend to be acquired tastes.   
• Umami is detected by reacting with various proteins, and thus indicates protein-rich foods like meat. This is a savory taste. 

Earth Gravity

If you think about what we have previously learned about gravity, every bit of mass pulls on every other bit. So the dirt right under your feet is exerting gravity, and dirt on the other side of the globe is as well. But the r² term in that equation says that the stuff close to you pulls much harder. It seems that if you want to know how all the earth together pulls on you, you must sum the effects of each individual bit.

The problem is simpler if we can think of the earth (or whatever) as an onion of nested spherical shells, where each shell is about uniform density. This is a fair assumption because for any very large object, gravity itself will collapse the thing down into a sphere shape and roughly sort the material of the sphere with denser material closer to the center. The Shell Theorem, proved by the ubiquitous Issac Newton, says that for any one of these shells, for the purposes of calculating the gravitational effect of the whole, it is equivalent to assume all of the mass is at a point in the center. Now, as you add the shells together, each tells you to model it’s mass at the same center point, so it’s valid to do that for the whole solid sphere.

The interesting flip side of that theory is that at any point Inside a hollow shell, you will feel no gravity from the shell. At the center this is not surprising, it’s pulling you in all directions equally and it cancels out. And with some thought it makes sense when you’re near the edge (but still inside) also. If you consider the part of the shell that is pulling you generally out from the center, it’s small but close to you. The remainder pulling you in toward the center is much larger, but farther away, so the effects balance out.

So, if you were to tunnel down into the earth, the further you went the less gravity you would feel until weightlessness at the center. Similarly, the gravitational effect of the atmosphere on you is roughly cancelled out to zero, because you are inside that shell.

Some have held the theory that the earth actually is a hollow shell, and that others inhabit the interior face with their own atmosphere, and even their own small sun at the center. We see now one reason this is would not work well. The inhabitants of the interior would feel no gravity holding them to the ground, only a slight (about 2 oz) centrifugal force.  The gravitational effect of the atmosphere inside the hollow earth and the small sun would likely overcome that effect and cause everything to  drift up into the center.

G. K. Chesterton Says

Ideas are dangerous, but the man to whom they are least dangerous is the man of ideas. He is acquainted with ideas, and moves among them like a lion-tamer. Ideas are dangerous, but the man to whom they are most dangerous is the man of no ideas. The man of no ideas will find the first idea fly to his head like wine to the head of a teetotaller.

- Heretics

What fairy tales give the child is his first clear idea of the possible defeat of bogey. The baby has known the dragon intimately ever since he had an imagination. What the fairy tale provides for him is a St. George to kill the dragon.

- Tremendous Trifles

The above was aptly paraphrased by Neal Gaiman in Coraline as "Fairy Tales are more than true; not because they tell us that dragons exist, but because they tell us that dragons can be beaten."

Truth must of necessity be stranger than fiction ... for fiction is the creation of the human mind, and therefore is congenial to it. 

- The Club of Queer Trades

The only way to be sure of catching a train is to miss the one before it.

- According to Pierre Daninos

The traveler sees what he sees, the tourist sees what he has come to see.

- The Temple of Silence & Other Stories

I say that a man must be certain of his morality for the simple reason that he has to suffer for it.

- Illustrated London News, Aug. 4, 1906

To have a right to do a thing is not at all the same as to be right in doing it. 

- A Short History of England

The whole modern world has divided itself into Conservatives and Progressives. The business of Progressives is to go on making mistakes. The business of the Conservatives is to prevent the mistakes from being corrected. 

- Illustrated London News, Apr. 19, 1924

I believe what really happens in history is this: the old man is always wrong; and the young people are always wrong about what is wrong with him. The practical form it takes is this: that, while the old man may stand by some stupid custom, the young man always attacks it with some theory that turns out to be equally stupid.

- Illustrated London News, Jun. 3, 1922

Without a gentle contempt for education, no gentleman's education is complete. 

- The Common Man

Precisely because our political speeches are meant to be reported, they are not worth reporting. Precisely because they are carefully designed to be read, nobody reads them. 

- "On the Cryptic and the Elliptic," All Things Considered

Nine times out of ten, the coarse word is the word that condemns an evil and the refined word the word that excuses it.

- The Everyman Chesterton

Coin Toss

Consider the coin toss. Flip a coin; the result is heads (H) or tails (T).

Each time we flip it there are two possible outcomes with equal likelihood, so the probability of getting heads is 50% or 0.5 and the same is true for tails. But what if we want to know the probability of flipping twice and getting heads the first time and tails the second time? If we look at column B below, we can see all the possibilities. If we flip twice, there are four: heads-heads, heads-tails, tails-heads, and tails-tails. One of those equaly possible four outcomes is the one we're looking for, so the probibility is 1/4 = 0.25 or 25%. Note that for each flip the total number of possible outcomes (Column D) doubles. This is because for each of the previous outcomes we've added two variations, two branches on the probability tree we see in Column B. It turns out that if we want to know the probability of multiple events occurring, we can multiply the individual probabilities. So for if we want the chance of getting HT it's the chance of getting H first (1/2) times the chance of getting tails on the second flip (1/2), (1/2) * (1/2) = (1/4).

Okay, but what if we don't care about the order? What if we just want to know the chance of getting one heads and one tails in two flips? Either HT or TH counts, so it's 2/4 = 0.5. In Column C we can see the groupings of outcomes if we don't care about order. They present an interesting pattern. Look at the coefficients, the multiple of each item. They follow a pattern we call Pascal's Triangle.

Coefficients also follow Pascal's Triangle when we do binomial expansion. To find (A + B)³ we look at the fourth row of the triangle (we consider the first row to be more like the zeroth row) for the coefficients and get A³ + 3*(A²*B) + 3*(A*B²) + B³. But this is basically row three of our Column C above, but with A and B instead of H and T, if we say HHH is like H³. If we extend that metaphor, then flipping three times is like (H + T)³. It's interesting that the math seems to work on events like it does on numbers or variables. Really, that's the power of statistics and probability, doing math on events.

Hair

A hair is essentially a very tall thin stack of flattish cells. The strength and toughness of hair comes from a protein coating on the surface. This protein has long tough filaments that tie the column of cells together along it's length. This is the same protein (keratin) that makes skin and fingernails tough.

The individual strands of hair can be round in cross section or more of a flattened oval, with flatter sections resulting in curlier hair. Think of string vs ribbon. 

Each hair goes through a phase of growth, followed by a period of near static length, before being shed and growing anew. Your maximum hair length is determined by the duration and rate of the growth period. Typical values would be a duration of five years at six inches a year, resulting in hair two and a half feet long. But depending on the individual it could be twice as long, mostly depending on a longer growth duration. World record hair lengths can be over 18 feet long, but it's unclear how much such lengths rely on braiding or matting. 

If you have a tabby cat, look closely at it's hairs. You may observe that in addition to body stripes, the individual hairs may, themselves, be striped. These are called agouti hairs and result in a finely speckled pattern.