Tangent of a Googol

There's an interesting math problem (with a fun backstory) in Feynman's autobiography "Surely You're Joking, Mr. Feynman!" The question is simply to calculate

tan(10¹⁰⁰).

I'm considering the problem in radians, but it's also challenging in degrees.
The number 10¹⁰⁰  can also be written as

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

This is a very big number that is also known as "a googol," which is pronounced the same as "Google."
Usual methods of calculating the tangent ("tan") function don't work well on very large numbers. If you Google "tan(googol)" or "tan(10^100)" you actually get different results (both wrong). You do get the correct result if you use Wolfram Alpha.
One way to get the right answer is to find a multiple of π that is very near a googol (call it nπ for some integer n). Then, because tangent is a π-periodic function,

tan(10¹⁰⁰)=tan(10¹⁰⁰-nπ),

and, since 10¹⁰⁰-nπ is a fairly small number, its tangent can be calculated accurately by normal methods.

Finding the number n and, say, 9 digits of 10¹⁰⁰-nπ requires knowing 109 digits of π, but this is easy to look up.

After some Googling, I couldn't find anyone that actually stated a decimal answer to this question, which is my main purpose here. It's about 0.401231962.

After the Ball in "My Fair Lady"

Before you go judging Professor Higgins and Colonel Pickering for ignoring Eliza after the ball: did you pay enough attention to her to notice that she said nothing (except a quick "Thank you" to Pickering) between the Ascot opening race and her moment alone after the ball? I have to wonder if the filmmakers did this deliberately.

Trump's Hot Mic Moment

This weekend, I've read about the 2005 recording of Trump's graphic, lecherous, swaggering conversation about adultery and sexual assault. People are reacting strongly, as though this recording has given us new information about Trump. We already knew he was a foul-mouthed, sexist braggart. It's not surprising that he says these things in private. It's only remarkable that we haven't caught him at it before.
Also, it took me a while to realize it, but Trump is insulting men as he defends himself. He has been trying to dismiss his behavior as "locker-room banter," as though men talk like this all the time. It's possible that I've heard talk like this in a locker-room, but I didn't pay much attention to middle-school delinquents. And to say that he has more respect for women than anyone? How degrading to the men he pretends to be above. Sorry Mr. Trump, but there are men who have more respect for women than you do—including all the decent ones.

Crushed

This blog is the closest thing I have to an influence on a culture and government that sometimes disgusts me. I write for my benefit. My mother might read this because she's subscribed, but I doubt anyone else will. If you've read this far, you know that I don't even write well, which compounds my problem. Like most people, many things about my life are dictated to me and I must accept them or be crushed.
I am at the mercy of forces too powerful for me to affect, defy, or ignore. Have you ever really pushed against a large boulder? It's heavy, hard and totally indifferent to the living consciousness seeking to influence it. If some other force were to set it in motion and you got in it's way, it would crush you. It would never know or care that a spark of life—which had thought itself important and unique—just became so much goo.
The immediate cause of these thoughts is that, due to a change of employer and not understanding that "open enrollment" means "the only time health insurance can be purchased," I don't currently have health insurance. This probably means I am stupid, but more importantly, I am young and healthy and likely to remain so. A couple of years ago, not buying insurance would have simply meant that I saved a few hundred dollars. Due to the Affordable Care Act, I am still uninsured but will be paying a fine. I am a student with a small income, and I won't get to keep that money.
I am experiencing a tiny sample of a life-crushing scenario faced by the truly poor. The "individual mandate" of the Affordable Care Act takes away the option to save money by not buying health insurance. Those who did not maintain insurance because they could not afford it are now worse off: they must now pay a fine with the money they risked their health to save. Some of the decisions made by my "representative" government offend me to my core.
I have a personal philosophy that I believe should rule my life. Instead, I will be ruled by other people's philosophies. Most of the people who will impose rules on my life will never hear me. They are a boulder.
I have a soul. How I feel about what happens to me should be regarded as an "end"—others should view it as a severe negative consequence that they have arranged my life circumstances to conflict with my values.  I believe in cooperation by common consent and noninterference by unaffected parties. I have the moral right to make a choice regarding the cash value of my health, and it is no one's concern but mine if I choose to risk death for the sake of another resource. I do it every day by crossing the street, which is also none of their business.
That's what I would say if I had a voice.

How Awesome is Picard?

Captain Jean-Luc Picard of "Star Trek: the Next Generation" is pretty amazing. I've collected a few facts illustrating why I think he may be among the greatest men who will ever live.

• Captain Picard has his own holiday.
• Captain Picard met the devil and wasn't impressed. He sued her. And won.
• Captain Picard was mistaken for God. Twice.
• Captain Picard lived a lifetime in 25 minutes.
• Captain Picard can make stating the time sound impressive.
• Captain Picard regularly lectures a god on morality.
• Captain Picard threw a saddle at a pirate.
• Captain Picard went back in time in order to get stabbed through the heart. Again. And it still tickled.
• Captain Picard hired Whoopi Goldberg to tend his bar.
• Captain Picard mind-melded with a vulcan to help him with self-control.
• And he did it while overcoming a prim French name.

Chess in "Donald in Mathmagic Land"

In the Disney short "Donald in Mathmagic Land," Donald watches a game of chess. This is supposed to show how math is used in an everyday setting.
The game is shown without players, pieces moving on their own through stop motion. The game is played rapidly, lasting only about 24 seconds. In order to finish the game in such a short time, some moves begin to take place before the previous ones have finished.
The game begins as the Blackmar Gambit, but quickly devolves into a shallow novice game designed to remove pieces as quickly as possible rather than exhibit strong play. The simple discovered check in 16...Nc5+ seems to have come as a complete surprise to white. It is a flurry of movement to entertain and impress the uninitiated, but it is useless for serious study. I record the details here for the curious.

1. d4 d5
2. e4 dxe4
3. d5 c6
4. dxc6 Qxd1+
5. Kxd1 bxc6
6. Nc3 Nf6
7. f3 exf3
8. gxf3 e5
9. Bh3 Bxh3
10. Nxh3 Bb4
11. Bd2 Bxc3
12. Bxc3 e4
13. Bxf6 gxf6
14. fxe4 Nd7
15. Nf4 O-O-O
16. c3 Nc5+
17. Kc2 Nxe4
18. Rad1 Rxd1
19. Rxd1 Nf2
20. Rd6 Kc7
21. Rxf6 Rf8
22. Nd5+ cxd5
23. Rxf2 Re8
24. Rxf7+ Kc6
25. Rxa7 Re2+
26. Kb3 Rxh2
27. a4 Kb6
28. Rd7 Rd2
29. Rxh7 d4
30. cxd4 Rxd4
31. Rh6+ Ka5
32. Rc6 Rd3+
33. Ka2 Kxa4
34. Rc4+ Kb5
35. b3 Rd5
36. Ka3 Rd8
37. Rb4+ Kc5
38. Ra4 Kc6
39. b4 Kb5
40. Ra5+ Kc6
41. Ka4 Kb7
42. b5 Ra8
43. Rxa8 Kxa8
44. Ka5 Kb7
45. b6 Kb8
46. Ka6 Kc8
47. Ka7 Kd8
48. b7 Ke8
49. b8=Q+ Ke7
50. Qe5+ Kf8
51. Qe6 Kg7
52. Kb7 Kf8
53. Kc7 Kg7
54. Kd7 Kf8
55. Qe7+ Kg8
56. Qf6 Kh7
57. Ke7 Kg8
58. Qe6+ Kh8
59. Kf7 Kh7
60. Qg6+ Kh8
61. Qg7#
The white queen is shown capturing the red king.

Why Forward Must Be Up

When signs need to indicate a forward direction, they typically use an upward pointing arrow. "Forward" is actually perpendicular to the face of the sign, so an up- or down-arrow is equally wrong. However, roads leading away from you rise in your field of vision as they get farther away, so an upward vector is the more sensible choice.
I've wondered for years about the apparently arbitrary "up equals forward" convention, and I just today realized this satisfying explanation.

The Parking Gods Reward Hubris

On Tuesday, I was exhausted. I had worked hard all day and only came back to my apartment for dinner. I still had a major project to finish and needed to go back to my office. Normally, I'm only too happy to walk to school since it only takes 15 minutes, provides the only exercise I get these days, and saves gas. This day, however, I felt I owed myself a ride.
Parking regulations in the lot next to the math building are only enforced until 6, so I could safely park in any spot. Even after hours, people seem to avoid parking in the spot reserved for the dean. Feeling a bit cheeky, I took it myself.
I worked long and hard on my project. As I left, security guards were prowling the building and there was a danger of tripping an alarm if I exited through the wrong door. I was pretty tired, so to keep from sulking over the sleep I had lost, I picked some cheerful music from my iPod for the walk home.
Walk home.
The significance of these last words did not occur to me until Saturday afternoon when I found that my truck was not parked in its usual spot by my apartment. A black horror overcame me as I realized that—in my exhaustion—I had forgotten that I needed to extract my vehicle from "Reserved for the Dean: $50 fine, towing enforced."
I trudged to the campus police office listening to the march from "1941"—normally a sure-fire way to cheer me up. Passing the math building, I noticed something odd: my truck was still parked in the dean's spot.
"Great," I thought, "so, instead of a single $50 ticket and a towing/storage fee I get three $50 tickets?"
I checked the wipers. No tickets.
"Then it's the boot! Oh, no! Anything but the boot!"
I looked down. No boot.
"So... do I just drive away?"
I checked the rest of my truck. Nothing out of the ordinary. I looked around for cops (sometimes they toy you by letting you think you've gotten away with it). There was no indication that my truck and I had violated anything.
Safely on the road home, I said a prayer of thanks. Specifically, I promised three of my first four children to any god that could prove his involvement, but suffice to say I was relieved.
Now, I happen to know several people who have been ticketed (or worse) for parking in that lot without a permit. They parked in the ordinary stalls (aware of the risks) and came back in a couple of hours only to discover that justice had been delivered in their absence. I parked in the prime location (flippantly enough that I forgot) and wandered back half a week later without consequences. Since every other factor is stacked in favor of my receiving a harsher punishment, I can only conclude that whatever forces guard wayward parkers are fond of my insolence, irony in general, or both.

Presidential Consumer Review

I wondered today, what if we reviewed our elected officials like we review books, personal electronics and pet shampoos? Being the creative sort, I made up an answer:

Definition of Combinatorics

Those who study combinatorics are working to discover how many different ways they can suffer. The binomial coefficient that dominates their work is


.
This expression does not mean that I am over the pain. It is pronounced "I choose Pain."

Gain and Loss

Everyone seems to believe that one man's gain must be another man's loss. The only exception they allow is government action in which case the poor gain and somehow, nobody loses. Well, maybe some rich pigs get taxed, but we hate them anyway.
Reality is precisely the opposite: unless coercive forces like the government get involved, people only interact by choice. Every exchange has expected gains for all parties. You can tell that they expected to gain because if someone expected to lose, they wouldn't have participated.

Subconscious

My brain comes up with some pretty weird stuff while I sleep, but this one is pretty interesting even a few hours after waking up. I dreamed that I was vandalizing a Port-a-John by writing on the wall:

Parapsychologists and parasitologists: one makes your ghosts flee, the other makes your fleas ghosts.

You've Been Mugged

If you have ever been to a large city, you have probably been mugged, and you might not have even realized it. I'm not saying you failed to notice a guy in a ski-mask jumping out of an alley and pointing a knife in your face; it was probably more subtle than that. It could be as simple as someone politely asking for money. To see why I call this a mugging, let's identify the essential features of a traditional mugger and see how a panhandler compares.
In a mugging, there are usually two parties: the mugger and the victim. The victim has something the mugger wants (say money). The mugger cannot or chooses not to try to exchange anything for the money. Instead, the mugger forces the victim into a dangerous position (such as the business end of a gun) and offers to release the victim—in exchange for the money. The mugger has produced nothing of value. He created a problem and expects to be paid to take it away.
Notice how the paragraph above becomes a typical begging situation when the gun is replaced with social pressure. Beggars create awkward an social situation and expect to be paid for taking it away.
A defender of begging would probably point out that awkwardness is not deadly whereas muggers' weapons are. This is certainly true, but they operate on the same principal, and while I prefer to have a nasty feeling in my gut rather than a knife, you can successfully threaten me with either.
Furthermore, while beggars are less physically dangerous, I suggest that there's a special sort of repugnance in using someone's own pity as a weapon to against them. Imagine if muggers did the same: they would ask to borrow your pocket knife and then held it to your throat saying, "Give me your wallet!"
I am not suggesting that we arrest people for begging. Asking for money (like most forms of speech) is a right that should be protected. Denying people the right to solicite "donations" could easily lead to outlawing certain types of fundraising for non-profit organizations (I think we can agree to avoid that). However, if individuals recognize begging as a sort of coercion, then we will feel less of the social pressure beggars attempt to apply. We will be disarming our muggers.

Friendly Airport Service

There are wonderful people in the world; we just meet them so seldom.

I once flew into the Salt Lake airport with some LDS missionaries returning from their two years of service. An airport employee (janitor I think) approached one of them and asked, "You're just getting back from a mission?"
"Yes."
"Your family hasn't seen you in two years?"
"Yes."
"Are they waiting for you in the lobby?"
"They said they would be."
"Would you like to surprise them?"
The employee then proceeded to show the missionary a hallway through a door marked "Employees Only."
Later, in the lobby, I quickly identified the young man's relations: a huge group with signs reading "Welcome Home Elder ______!" (they matched his name-tag). They were anxiously watching the passengers arriving from his flight.
I stuck around to see what happened. When the family had waited long enough to seem a bit confused by his absence, I saw the missionary emerge from another "Employees Only" door—behind his family. He was able to get right in among them before shouting, "How's everybody doing?"

Let's hear it for offering quirky services to strangers.

Shouting in the Darkness

I post to this blog for the same reasons that a castaway writes in the sand or that men on mountaintops talk to the stars. I have no hope that the words will  reach anyone else, but I derive some pleasure from translating my ideas into a physical form. It is a protest against the tragedy that so many worthwhile thoughts die without notice even from the thinker.

Return of the Jug

Has anyone else noticed that in "The Return of the King" (2003) the pitcher Pippin puts in Gandalf's arms when he takes the palantír looks just like the one he has in Minas Tirith a few minutes later? The first time is about 00:30:09 into the film, the second is 00:52:46. I'm not suggesting that the filmmakers made a mistake, only I was surprised to see that Pippin or Gandalf would bother to pack it. Or that such identical jugs were in mass production in a pre-industrial society.

Greed

Regardless of the system, the people at the top are capitalists.

The only question is whether they allow the other people to be.

Milton Friedman is unstoppable. He taught at Chicago University (one of the world's finest school for economics) where not even the professors could find room to disagree with him once he got started explaining things. He went on to win the Nobel Prize in economics, largely for his explanations of American monetary policy and explaining why the government should basically stay out of businesses' way.
Here's part of a chat he had with Phil Donahue:

The first video for this interview can be found below. The entire show was uploaded by the same user.

Math Majors

What does it mean to be a math major? Well...

• It means you make jokes that rely on the Banach-Tarski Paradox (and your "friends" get it).
• It means you're more comfortable finding numerical solutions to nonlinear differential equations than calculating a 15% tip.
• It means you enjoy pain.
• It means you keep track of strange things and post them on your Blog.
• It means you can't show pie charts to your peers because they'll criticize you for using a cute, unscientific presentations of data.
• It'll be even worse if they notice that you used Excel instead of Matlab.

Why a Franklin Planner is Better than a Girlfriend

This list is difficult to find, so I thought I'd help get it out there:

10. A Franklin will never make you late.
9. You don't have to pay attention to your Franklin if you don't want to.
8. Your Franklin will still be there when you get off your tour of duty.
7. If you lose your Franklin, it won't keep your tapes and sweaters.
6. You only have to pay for your Franklin once.
5. You never have to worry about someone else using your Franklin.
4. Your Franklin will wait in car while you play basketball.
3. Your Franklin doesn't care if you have another planner.
2. Your Franklin already comes with rings.
1. If your Franklin gets too big, you can just remove some pages.

1984 in the 23rd Century

As a fan of "Star Trek: The Next Generation," I have watched most episodes several times. "Chain of Command" (both parts) was one of my favorites since both Picard and Riker get to show their metal.
However, I was disappointed to realize that half of "Chain of Command"seems to be borrowed from George Orwell's "Nineteen Eighty-Four."
In the episode, Picard is captured by Cardassians and imprisoned for espionage. He was working with Worf and Crusher, who escaped without Picard's knowledge. Picard is then treated with, shall we say, less dignity than befits his rank (claiming POW rights would have required admitting he was acting under Starfleet's orders).
The specifics of Picard's predicament, in many important ways, resemble Winston Smith's re-indoctrination. Observe:

• They are punished for insisting that the four objects before them don't make five
• They are tortured first by humiliation and starvation, then by mysterious devices that inject pain directly
• They are fed lies about the wars outside
• They are concerned for the safety of the women they were working with
• They are given the opportunity to save themselves by betraying their respective women
• Their captors express a mental kinship to them, wishing to be able to explore their minds further

While Riker's exploits in "Chain of Command" would make the episode worth watching anyway, the apparent copying is disappointing.