most irrational number

Today I was first exposed to the idea that, in a sense, we can say a number is "more irrational" than another... and even that there is a "most irrational number". The idea is to ask, "how well can we approximate the irrational with a simple rational?" We can approximate as close as we want, if we're willing to have huge numerators and denominators. But I can see that some irrationals will be approximated very well with a "small" fraction, and others won't. The ones that aren't are "more irrational".

This gets quantified by analyzing continuous fractions, and it turns out that ϕ (the golden ratio, or (1 + √5)/2) is the "most irrational". For more, see David Rutter's answer here.

kinematic vector patterns for circular movement

When we consider a point moving in a circle at constant speed, we usually limit the discussion to position, velocity, and acceleration. But if we keep going (that is, consider the derivative of acceleration, and the derivative of that, etc) a simple pattern emerges.

A quick review: 

The position vector points from the circle's center to the moving point. 

The velocity vector (which is the derivative of the position vector) is at a right angle to the position vector, pointing in the direction of motion. 

The acceleration vector (which is the derivative of the velocity vector) is at a right angle to the velocity vector, pointing toward the circle's center.

Note that the position and velocity vectors have constant magnitude but changing direction. The acceleration vector is no different; as the point moves in the circle, the acceleration vector changes direction to keep pointing to the circle's center. So we can ask about the jerk vector (which is the derivative of the acceleration vector). The jerk vector is at a right angle to the acceleration vector, and points opposite the velocity vector.

Notice that each successive derivative vector is rotated 90° from the preceding vector. If the moving point is rotating, say, clockwise, each successive derivative vector is rotated 90° clockwise. We can keep taking derivatives as much we want, and each one will be rotated another 90° from the last one.

A little analysis reveals that this falls out from the derivatives of sine and cosine. Just like taking the derivative of sine 4 times brings you back to sine, rotating a vector 90° either clockwise or clockwise 4 times will bring the vector back to its original direction.

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For reference, the successive derivatives of position are:

velocity

acceleration

jerk (or: jolt (UK), surge, lurch)

jounce (or snap)

[no concensus]

radioactive dating

When we know all of these:

• a rock contains A
• A decays into B
• A's half-life
• and both of these:
• how much A was in the rock when it formed
• how much A is in it now 
• or one from the above list, and both of these:
• how much B was in the rock when it formed
• how much B is in it now

we can accurately date the rock. But how do we know how much A or B was originally in the rock?

I should have gotten a handle on this a long time ago.

• some minerals strongly reject B, so we can be reasonably sure that all B found in such a mineral is from decay.
• in some materials, A and B would separate when forming (eg, from settling), so that all B found at A would be from decay. 

 Another explanation is still sketchy to me:

•  In the case of A being $C^{14}$, A is continuously generated by cosmic rays. So when we know all the carbon found in a rock, we know that a certain fraction of it was A ($C^{14}$).

My questions: Isn't that implying that the ratio of $C^{14}$ to all carbon has been constant over time? If so, how do we know that? And if the answer to that last question only tells us how we know the ratio of $C^{14}$ to all carbon has been constant in the atmosphere, how do we know that the atmosphere was the only source of $C^{14}$ trapped in rocks during their formation?

Derivative and Integral: the play

[Background: Given a graph, I can visualize its derivative much more easily than I can visualize any of its antiderivatives. Time for that to change, even if I have to resort to something as ridiculous as giving those operations personalities.]
  
Suppose a graph represents how good the day is going versus time. Positive values mean a good day; negative values mean a bad day. A derivative is like a person who only cares about whether things are getting better or worse, and whose mood is based solely on that direction. It doesn't care about the past. It doesn't care about how good things actually are right now. The integral is like a person whose mood gets gradually better or worse depending solely on how good things are in the moment. It's affected very much by the past. It doesn't care if things are improving or worsening.

Let's graph these characters' moods over a day that has its up and downs, as represented by the graph of cosine. Because Integral's mood depends on his past mood, I pick an arbitrary starting mood for him at 0.

t=0

Integral: Wow, what a day, huh? This is tops!
Derivative: If you think so, why aren't you happier? Your happiness is at 0.
Integral: Oh, I'm getting happy very quickly! Look how quickly my mood is climbing! Besides, you're at 0 also.
Derivative: But I'm at 0 because the day has stagnated. Who knows what the future holds...
Integral: Pessimist!

t=0.6

Integral: Why so negative?
Derivative: It's all downhill from here. At this rate, we'll start having a bad day.
Integral: Just enjoy the present! It's still a great day!
Derivative: Well, I notice you're not getting happier quite as fast.
Integral: Because it's not quite as nice a day. But it's still pretty great!

t=1.2

Derivative: Wow, the day's really taken a turn for the worst, hasn't it? I'm so bummed.
Integral: You're so worried about the future that you can't enjoy the present. I'm getting happier all the time!
Derivative: Yes, but not nearly as quickly.
Integral: Who cares? The day's still good, and I'm still getting happier.

t=π/2 (about 1.6)

Integral: Well, it doesn't get better than this! I'm so happy!
Derivative: You do realize you're coasting, right? The day's no longer good, and you're not getting happier.
Integral: I'm high from all the good stuff that just happened. I can't understand why you're so depressed.
Derivative: Isn't it obvious? This is the fastest rate at which the day has declined!
Integral: Why do I even talk to you?

t=2.5

Integral: I'm still happy, but I have to admit the day's sucked lately. It's bringing me down.
Derivative: Well, I don't feel quite as bad. The day's not declining as quickly.

t=π (about 3.14)

Derivative: Did you notice? The day stopped getting worse. I'm no longer sad. If things start looking up, I'll actually be happy.
Integral: Yeah, well the last part of this day sucked so much, it wiped out all my happiness from the first part. I'm no longer happy. This could start to irritate me.

t=4

Derivative: Hey, the day's finally headed in the right direction. This is great!
Integral: Shut up. It sucks. I'm feeling worse and worse.
Derivative: But look at the trend! It's improving.
Integral: You're even more irritating when you're happy.

t=3π/2 (about 4.7)

Derivative: Wow, things are really looking up! I couldn't be happier!
Integral: We just came through so much suck. Twice as much suck as good stuff, in fact. This is misery.

t=5.5

Derivative: You have to admit, things have been good lately.
Integral: Yes, so well that it's pulling me out of this bad mood. But you're not as happy. 
Derivative: I'm still happy, just not quite as much, because things aren't looking up quite as much.
  

t=2π: repeat

GPS error

Previously posted in a Facebook discussion. I thought I'd archive it here.

Article: Due to statistical bias, GPS distance estimates tend to always be about 10-20% higher than is correct.  (http://www.i-programmer.info/news/145-mapping-a-gis/9164-gps-always-over-estimates-distances.html)

My comments:  

This is surprising and counter-intuitive. I didn't follow the details of the explanation, so I did my own test: take points A and B, and place each in the center of a 3x3 grid. This grid represents possible error (each cell representing a place where the point might be measured), so there are 81 possible ways to measure the distance between A and B. The result: 27/81 (or 1/3) chance of under-measuring the distance; 9/81 (or 1/9) chance of measuring an equivalent distance; 45/81 (or 5/9) chance of over-measuring distance. How about that!
 --------------------
As I worked through it, I could see the above falls out of the fact that at small angles, sin(θ) changes more quickly than cos(θ).
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I thought of a way to think about this for a continuous range of error. Say the true distance between A and B is D. B will be erroneously taken to be at a random point on a circle of radius R centered on B. Call this circle B-false.

Draw the circle of radius D centered at A. This divides B-false into 2 arcs. The smaller arc is where B can be falsely placed and result in a smaller measured distance between A and B. The larger arc, where B is more likely to be taken to be, results in a larger measured distance between A and B.

calculus really likes radians

I remember being introduced to radians by my teacher: "If we met an alien civilization, they wouldn't be using degrees. Why would they divide a circle into 360 parts? There's no mathematical reason to do that. But everyone would know to use an arc length equal to the radius. The radius is universal." That made sense to me. There's nothing arbitrary about the radius!

Or is there? Why not use a unit that's the same length as the diameter? I'll call it the "diametan". So whereas 1 radian is the angle needed to make an arc whose length is as long as the radius, 1 diametan is the angle needed to make an arc whose length is as long as the diameter. (An angle of 1 diametan = 2 radians.)

So... radians, diametans... one sounded as good as the other to me. Until I graphed them, and thought about their derivatives.

Compared to the graph of sin(radians) (which peaks at π/2), the graph of sin(diametans) will be squashed (it peaks at π/4). This will mess up its derivative (and its anti-derivative, and its derivative's derivative, etc), instead of keeping the nice alternations between ±sin and ±cos.

Perhaps this should have been obvious to me; the same problem is to be seen with the graph of sin(degrees) (but instead of being squashed, it's stretched).

What makes radians special is the same thing that makes e special: they balance things just right such that derivatives (or anti-derivatives) find their way back to the starting function.

I got on this line of thought by working through the derivation of the Maclaurin series for sin x, and wondering what would change if the trig functions expected diametans instead of radians.

$e^{iθ}$

Once one sees the series expansions of $\cos θ$, $\sin θ$, and $e^x$, it's easy to see why
$e^{iθ} = \cos θ + i\sin θ$,
and from there, why
$e^{iπ} + 1 = 0$
which I always thought appeared mystical.

However, I'm not satisfied; I'd like to understand it at a visual level. Also, I'd like to understand how to derive the related Taylor/Maclaurin series.

I learned Gaussian integers ($m + ni, | m, n ∈ ℤ$) and proved an equilateral triangle cannot have all vertices at Gaussian integers.

Maxwell's religious views

Maxwell's mindset seems to be that faraway stars are composed of molecules exactly the same as we find on Earth (which he would know through spectroscopy). Only created objects stay the same, whereas natural objects continually change. Therefore, the universe was created by a Creator. Therefore God.

http://www.victorianweb.org/science/maxwell/molecules.html

Einstein's religious views

1923: "My comprehension of God comes from the deeply felt conviction of a superior intelligence that reveals itself in the knowable world. In common terms, one can describe it as pantheistic."

1929: "I believe in Spinoza's God who reveals himself in the harmony all that exists, but not a God who concerns himself with the fate and actions of human beings."

1931: "It is very difficult to elucidate this cosmic religious feeling to anyone who is entirely without it. The religious geniuses of all ages have been distinguished by this kind of religious feeling, which knows no dogma and no God conceived in man's image; so that there can be no church whose central teachings are based on it. In my view, it is the most important function of art and science to awaken this feeling and keep it alive in those who are receptive to it."

1934: "You will hardly find one among the profounder sort of scientific minds without a religious feeling of his own. But it is different from the religiosity of the naive man. For the latter, God is a being from whose care one hopes to benefit and whose punishment one fears; a sublimation of a feeling similar to that of a child for its father."

Date unknown by me: "In every true searcher of nature, there is a kind of religious reverence, for he finds it impossible to imagine that he is the first to have thought out the exceedingly delicate threads that connect his perceptions."

Date unknown by me: "A contemporary has said, not unjustly, that in this materialistic age of ours the serious scientific workers are the only profoundly religious people."

Einstein's activism

Einstein was active in civil rights, including equality for gays, but particularly for African Americans. Of course the FBI noted such suspicious behavior as Einstein signing a fundraising appeal for the NAACP.

A conservative columnist, Jimmy Tarantino, wrote, "Red-fronting Einstein should be deported. The Senate Internal Security Committee would most likely discover that Einstein's Commie connections have been so wide and vast in scope that it may be safer to deport him to his native Europe. Who needs this Einstein? Not the American people, and that's for sure."

the return of vacuum tubes?

Qubits deteriorate easily. One of the mainstream areas of research into how to preserve them is with the use of vacuum tubes.

QM foundation discussion banned; no-cloning theorem

According to HarvardX: EMC2x The Einstein Revolution:

The principal physics journal in the United States, Physical Review, for a time more or less banned discussion of the foundations of quantum mechanics. [hyperbole?]

Also, I've now heard of the no-cloning theorem which states it is impossible to create an identical copy of an arbitrary unknown quantum state. (1982) Supposedly it's comparable in importance to the uncertainty principle.

Bohr vs Einstein

(Taking Kaiser's article at face value)

Einstein and Bohr had very different styles that were reflected in their approach to QM (quantum mechanics). (In fact, there were a series of Bohr-Einstein debates about the nature of QM.)

Bohr valued plain language, community, and teamwork.
Einstein valued visualization supplemented with math, and solitary thought.

Bohr was content with descriptions and wasn't bothered by QM stopping with probability.
Einstein valued axioms and wanted underlying principles that led to QM's uncertainty.

Source: David Kaiser's "Bringing the Human Actors Back On Stage: The Personal Context of the Einstein-Bohr Debate". 

Big Bang Theory linked to divine creation

The Big Bang Theory was originally seen as supportive or quasi-supportive of religion by both religious factions and by skeptical scientists.

So we have Pope Pius embracing Big Bang Theory saying:

"it would seem that present-day science has succeeded in bearing witness to the august instant of the primordial Fiat Lux, when, along with matter, there burst forth from nothing a sea of light and radiation. What, then, is the importance of modern science in the argument for the existence of God based on change in the universe? With exact and detailed research into the large-scale and small-scale worlds it has considerably broadened and deepened the empirical foundation on which the argument rests. Thus, with that concreteness which is characteristic of physical proofs, it has confirmed the contingency of the universe and also the well-founded deduction as to the epoch when the world came forth from the hands of the Creator. Hence, creation took place. We say, therefore, there is a Creator. Therefore, God exists.”

And we have Fred Hoyle (leading astronomer) arguing for the competing Steady State Theory, writing

"for it is against the spirit of scientific inquiry to regard observable effects as arising from causes unknown to science, and this is in principle what creation-in-the-past implies."

Source: HarvardX: EMC2x The Einstein Revolution

Physicist assassin

Einstein and his friend Friedrich Adler both applied for the chair of the University of Zurich’s physics department. Adler told the university that if they had the chance to hire Einstein, they should hire Einstein instead of himself.

Adler soon turned his attention to politics, as he hated Austria’s support of WWI. This lead Adler to assassinate the Minister-President of Austria. Adler’s father tried to save his son by arguing Friedrich was insane, but Friedrich instead argued his political position, and I presume he appeared quite sane while doing so. Einstein wrote to help but stopped short of saying Adler was insane.

From prison Adler delighted that he could think about physics again, and wrote to Einstein describing what he thought was a flaw in relativity. Adler’s father seized on this as further evidence his son was insane (because he dared to argue with Einstein)! The father wrote to leading physicists for confirmation of this. The physicists were in a bind because they could see Adler made a normal mistake, but not an insane one.

Adler was sentenced to death. Turbulent times lead to an offer of amnesty for all political prisoners in exchange for an apology, but Adler refused to apologize. Eventually Adler’s sentence was commuted.

Hubble discovered that other galaxies existed

I knew Edwin Hubble for discovering galaxy redshifts (1929), showing the universe was expanding. What I did not know was that just 6 years prior, it wasn't known that other galaxies even existed! It was Hubble who proved some there were galaxies beyond our own. This had long been suspected, yet it had not been the prevailing view.

Source: https://en.wikipedia.org/wiki/Edwin_Hubble