I’ve always enjoyed imagining higher dimensional geometry. Not that it’s easy to visualize, but it’s easy to discern what some of the properties would be. Today I thought of an example of such a property.

Consider the different ways two (infinite) lines can be positioned relative to each other in different dimensions:

0. Lines can’t exist in 0-dimensional space
1. Any two lines in 1-dimensional space are necessarily the same line
2. In 2-dimensional space, two distinct lines can relate to each other in two different ways:
   • They can intersect
   • They can be parallel

3. In 3-dimensional space, there is a third option:
   • They can intersect
   • They can be parallel (which means they are always the same distance apart)
   • They can be skew (if this one isn’t immediately obvious, imagine a bridge, where one line (the road) goes over the other line (the river), and they neither intersect, nor run parallel)

   The first two options imply that the lines are co-planar: there is some flat (2D) plane containing both of them. In the third option the two lines are necessarily not co-planar.

4. If we add a 4^th dimension, we should expect to be able to distinguish between two different kinds of skew lines:
   • Skew lines that can be contained within the same 3-dimensional sub-space, in the same way that parallel lines can be contained within a 2-dimensional subspace. We can refer to them as Skew₃
   • Skew lines that cannot be contained within the same 3-dimensional sub-space, in the same way that Skew₃ lines cannot be contained within a 2-dimensional subspace. We can refer to them as Skew₄

   Intersecting and parallel lines will both still be possible, of course.

5. With the 5^th dimension too, we should be able to have Skew₅ lines, which cannot be contained in the same 4-dimensional subspace.

And up and up, as high as you like. Our fellow beings in some 307-dimensional world probably spend most of their high school geometry class memorizing the 307 different ways that two lines can relate to each other (my favorite is Skew₁₀₈), and developing their ability to quickly recognize them when projected in a 307-dimensional hologram.

Apparently compilers can get confused:

Excuse me for a minute while I go on a bit of a rant. I’m tired of all these liberal-wearing hippie-popping gun-toting bigot-eating abortion bankers trying to tell us what color skin we have. If I want the government to stay out of my foreign policy, I won’t get a six-pack of mortgages while I’m at the convenience store! Our president and congress need to spend more money defending our own citizens from external threats, and less money on the military. Take your family-values and go back to the country you illegally came from! The government thinks it can completely ignore the rights of women and force them to endure unwanted marriages that were the result of violent-crime-weddings, all in the name of preserving sanctity or some other hogwash. Well I say, wash those hogs! Get the lipstick off those pigs!

If all those damn front-wing Harvard and Princeton momma’s boys with their fancy schmancy family values and religion think they can run my life, I’ve got just three words for them: The Constitution! This country was founded on the principles of right and wrong, not of black and white. The longer we’ve got a muslim-cowboy fatherless Texan ethnic man in the oval office, the more money this country spends on Medicaid! Do you know why everybody’s so fat? K-mart! If marijuana companies weren’t allowed to advertise to children, drunkenness wouldn’t be so epidemic among all these preschool-dropout droopy-pants-wearing redneck KKK-members blasting their rap music so loud you can’t hear yourself gay-marry!

The Supreme Court (which is behaving more like a fast-food menu item than a judicial body) thinks that it’s ok to judge people by the color of their t-shirts, but I say the Supreme Court can go back to Canada where it came from. If you want to take jobs from honest, hard-working Wall Street con-artists, then you have no place in my America. I wish all these stupid arm-chair bloggers would quit pretending Ronald Reagan was still turning this country into a socialist paradise. When are people going to learn to think for themselves? Do you think the media has your best interests in mind? There are seventy million people in this country who are forced to get their education from Burger King, and if the government can’t fix it, then it’s time for a new government. Get out of my country, NOW!

A list of the research interests of all the Computer Science professors at MIT:

The Lady and I watched Doubt last night, and I thought it was excellent. I don’t remember seeing a movie with Philip Seymour Hoffman that I didn’t think was excellent. If he starred in Spy Kids 4, I’d probably think it was great. That’s an exaggeration.

I was doing my online Physics homework, and when I got a problem wrong, the little gray box that sometimes pops up with a hint asked me to supply my own:

I’ve added a “lists” feature to the blog (linked at the top). It will allow me to have running lists of anything that I feel ought to be enumerated. Starting with topics I’m interested in learning about. There will be a list of good names for a rock band just as soon as I think of the first one.

One of my favorite mathematical theorems that I've been exposed to so far is the divergence of the sum of the prime reciprocals (as referenced here and proven here). It says that if you take all the prime numbers (which are infinite) and turn them each upside down so that they're in the denominator of a fraction, and add up all those fractions, the resulting sum will be infinite. If this doesn't sound remarkable, let's try a few.

The first ten primes:

	1/2     0.50000000000000000000
	1/3     0.33333333333333333333
	1/5     0.20000000000000000000
	1/7     0.14285714285714285714
	1/11    0.09090909090909090909
	1/13    0.07692307692307692307
	1/17    0.05882352941176470588
	1/19    0.05263157894736842105
	1/23    0.04347826086956521739
	1/29    0.03448275862068965517
	Total=  1.53343877187203202214

Not very big, but of course ten primes is not that many either. Let's try some more

The first 500 primes, 2 --> 3571

	...     .......
	1/3541  0.00028240609997175939
	1/3547  0.00028192839018889202
	1/3557  0.00028113578858588698
	1/3559  0.00028097780275358246
	1/3571  0.00028003360403248389
	Total=  2.36532946588456968519

By adding 490 more primes, we increased our total sum from 1.533 to 2.365. Finally, if we add the reciprocals of the first million primes, the result is only 3.06821904805 (that's as far as my computer is interested in calculating). I doubt it would ever count high enough to get to 4. And yet, the theorem says that if the process continues indefinitely, the sum will get as high as you like.

I like it.

Somebody stopped me downtown today to ask me for directions to the beach. I asked if they were referring to the beach at the ocean, and they said yes.

But the ocean is 115 miles from here.

If we can agree that

Let us consider this subject completely laid to rest.

Opening sentence about mildly aggravating thing that happened! Assertion that I’d prefer it hadn’t happened that way! Interrogative regarding these things always happening to me! Further developing various contours of negative feelings! Conveniently omitting positive aspects of situation! Stomp! Jump! Stomp! AAAAAHH!! Gripe! Complain! Moan! Threats not directed at anybody in particular! Lamentation about the state of things! EXCLAMATION POINT!!!

This post is a satircal look at political corruption.

In the interest of furthering Google’s ability to solve minor development issues, I should show what I recently learned by trial and error about Clojure’s syntax for instantiating Java’s inner classes. Java distinguishes between static nested classes and non-static nested classes, and the only information I could find on Clojure syntax only applied to the static nested classes, which incidentally are instantiated like this:

(new OuterClass$InnerClass)

Not too hard. Non-static classes on the other hand must be constructed on a member of the outer class. And so

(OuterClass$InnerClass. my-outer-class)

I assume appropriate arguments could be appended to either of those statements if the constructor required it.

I just had my first Physics 2 lecture. According the schedule on the registration server, the class consists of a 2:15 hour period in the morning, a 2 hour lab in the afternoon with another 1 hour period immediately following, four days a week. I went to the first lecture this morning, and the professor spoke for 2 hours and fifteen minutes purely about physics, with almost no information about the class, and no syllabus. I had my hunches about the structure of the class from taking Physics 1 last fall, but I was hoping for some indication that those suspicions were correct. Instead all I got was a class decision that exams should be on Thursdays rather than Mondays, and a casual reference to the existence of online homework problems. He didn’t say anything about labs, or about the 1 hour period at the end of the day (which I concluded wasn’t a lecture because at the end of the class he said “see you tomorrow”, but at the very least was the time period during which the exams would be held, which I luckily figured out from another casual reference). When I took Physics 1 there was an extra hour session on the schedule for help with homework, but I was able to neglect attending it altogether with no major issues. All the exams were during the regular lecture time period.

Naturally I went up to the professor after class to ask for a syllabus, or at least some more information about how the class was going to be run. He told me that he doesn’t have a syllabus and doesn’t normally make them for summer classes, and when I asked for specific information in place of the syllabus, he refused to answer any of my questions, and told me to simply ask a classmate who had been in his Physics 1 class last month. I generally don’t socialize with the other students as much as I can help it, due to the fact that I’m several years older than most of them, and have an irrationally age-centered view of social status, so I didn’t like his suggestion all that much. However, he seemed pretty firm on that solution, so I decided to just forget it and make my best guess about everything. Evidently though, it upset me enough to motivate me to write two decently-sized paragraphs about it.

Trying to fix a software bug that you can’t reliably reproduce is like chasing a ghost.

That’s a good analogy because it’s something that everybody has experienced.