Thursday, February 24, 2011
Space Shuttle Discovery launch
I'm currently in Florida with friends watching the launch of Discovery. This will be the second to last shuttle launch (there's a small chance that this will be the third to last launch if they get enough funding for a final launch). I and Tyrol5 wrote an article for Wikinews about the launch which includes a picture taken by me of the launch. There are additional photos on my Facebook page.
Monday, February 14, 2011
On rotations of spheres
Consider a sphere living in some number of dimensions. The sphere we are used to is the 2-sphere, denoted by S2. Mathematicians call it this because even though it lives in 3-dimensions, it is itself a 2-dimensional object (in the sense that small sections of it look like the 2-dimensional plane). One can similarly talk about the n-sphere, which lives in n+1 dimensions. Thus for example, the 1-sphere, S1, can be thought of all all points on a plane that are of distance one from the origin.
When one has a geometric object one of the most obvious things to do is to ask what rigid movements of the object will take it to itself. Thus, for example, for a sphere, a rotation about some axis through the sphere's center rigidly moves the sphere to itself.
Rotations seem simple, but they can be surprisingly tricky. For the 1-sphere if one does a rotation and then another rotation one is left with a rotation. This is an important and helpful property. It says essentially that rotations form what mathematicians call a subgroup of the group of all rigid motions. It turns out that this is still true for the 2-sphere, S2 even when one uses different axises for the two rotation Now, you might find surprising the fact that for S3, the sphere of three dimensions living in four dimensions, this breaks down. The composition of rotations is not necessarily a rotation. In some sense the not obvious fact is not why this breaks down for three dimensions, but why it still holds true in two dimensions.
The above is well known to mathematicians or to many people who have played around a bit with geometric objects. But, I recently learned a related fact that I found startling. Call a rotation "periodic" if we eventually get every point back to where we started. So for example, if one repeat a 90 degree rotation (π/2 for those using radians) four times we will have every point back where we started. Now, it turns out that for rotations of S2 even though the composition of rotations is a rotation, the composition of periodic rotations is not necessarily periodic. Once one knows that this is true it is easy construct examples. Consider two rotations around perpendicular axises each of 30 degrees (π/6 in radians). It isn't difficult to show that their composition although still a rotation is not periodic. This is a good example of how even basic geometry can surprise us even when we think we understand it.
When one has a geometric object one of the most obvious things to do is to ask what rigid movements of the object will take it to itself. Thus, for example, for a sphere, a rotation about some axis through the sphere's center rigidly moves the sphere to itself.
Rotations seem simple, but they can be surprisingly tricky. For the 1-sphere if one does a rotation and then another rotation one is left with a rotation. This is an important and helpful property. It says essentially that rotations form what mathematicians call a subgroup of the group of all rigid motions. It turns out that this is still true for the 2-sphere, S2 even when one uses different axises for the two rotation Now, you might find surprising the fact that for S3, the sphere of three dimensions living in four dimensions, this breaks down. The composition of rotations is not necessarily a rotation. In some sense the not obvious fact is not why this breaks down for three dimensions, but why it still holds true in two dimensions.
The above is well known to mathematicians or to many people who have played around a bit with geometric objects. But, I recently learned a related fact that I found startling. Call a rotation "periodic" if we eventually get every point back to where we started. So for example, if one repeat a 90 degree rotation (π/2 for those using radians) four times we will have every point back where we started. Now, it turns out that for rotations of S2 even though the composition of rotations is a rotation, the composition of periodic rotations is not necessarily periodic. Once one knows that this is true it is easy construct examples. Consider two rotations around perpendicular axises each of 30 degrees (π/6 in radians). It isn't difficult to show that their composition although still a rotation is not periodic. This is a good example of how even basic geometry can surprise us even when we think we understand it.
Wednesday, February 2, 2011
Obama's State of the Union Address: A Response to Nathaniel
President Obama delivered his State of the Union address last week. My younger brother has a piece in the Yale Daily News which extensively criticizes the speech. Nathaniel complains that the speech started with an inspirational message but "soon fell into a quagmire of policy." Nathaniel criticizes the speech for focusing too much on policy minutia and argues that, while referencing Kennedy, Obama fails to understand that inspiration requires large goals, not policy details. Overall, Nathaniel is correct that this was a disappointing speech. However, it was not disappointing because it went into detail: it was disappointing because the details were unimpressive.
While Nathaniel is correct that the speech was not very inspirational, he misses a broader point: We need policy expertise, and a President who is willing to present a State of the Union that discusses real policy issues is a good thing. After 8 years of George W Bush's flag-wrapping jingoism, after 8 years of Bill Clinton's contentless addresses, the American people should be happy that a President is willing to discuss serious policy issues. And if the American people don't like that, that's a problem with the American people, not a problem with their President.
Nathaniel has two other criticisms of the speech: that Obama was unwilling to take a stand on anything remotely controversial, and that Obama called for largescale spending that the recent elections show is not desired by the American people. These criticisms seem contradictory. One cannot in one paragraph complain that Obama was a coward and then in the next paragraph complain that Obama is too willing to engage in spending that many people don't want.
Some parts of Nathaniel's column do have merit. Nathaniel correctly identifies that Obama's comments about school reform were close to toothless. And Nathaniel correctly points out that Obama did not emphasize the degree to which the proposed initiative will place heavy burdens on the American people. And it seems disingenuous for Obama to claim to be willing to fund the " Apollo Projects of our time" while calling for a freeze in domestic spending.
While I disagree with much of Nathaniel's criticism, I am far from happy with Obama's speech. It certainly had its good points, but it had many failings. Obama failed badly in his discussion of energy policy. He gets points for mentioning nuclear power, something he has in the past downplayed. However, when discussing biofuels, he made no mention of the fact that the most prominent attempt at biofuel, corn based ethanol, is inefficient, environmentally damaging, and raises food prices which hurts the poor here as well as people in the developing world.
Similarly, Obama's emphasis on electric cars was similarly unpersuasive. The President did not discuss that electric cars’ power has to come from the general grid. While electric cars do overall pollute less and use less energy, this stems from economies of scale more than anything else. And the impact is not high. While Obama did discuss the energy of the future, he proposed no funding for research into genuinely new energy sources, such as fusion power. (In the particular case of fusion power, the US is putting resources into ITER, the international tokamak reactor, but the US is putting no money into other forms of fusion such as stellarators.)
Obama's discussion of school systems used metrics which are less than ideal. In particular, the fraction of a population which is going to college is an awful metric for success of students. Students who go to college are often unprepared and often come out not much better prepared than they went in. Moreover, many jobs, even high-tech jobs, don't require college degrees. We have serious problems with people going to college on loans they are unable to repay.
Obama was correct to talk about policy proposals in his address. The problem is not the discussion but the content of those policies.
While Nathaniel is correct that the speech was not very inspirational, he misses a broader point: We need policy expertise, and a President who is willing to present a State of the Union that discusses real policy issues is a good thing. After 8 years of George W Bush's flag-wrapping jingoism, after 8 years of Bill Clinton's contentless addresses, the American people should be happy that a President is willing to discuss serious policy issues. And if the American people don't like that, that's a problem with the American people, not a problem with their President.
Nathaniel has two other criticisms of the speech: that Obama was unwilling to take a stand on anything remotely controversial, and that Obama called for largescale spending that the recent elections show is not desired by the American people. These criticisms seem contradictory. One cannot in one paragraph complain that Obama was a coward and then in the next paragraph complain that Obama is too willing to engage in spending that many people don't want.
Some parts of Nathaniel's column do have merit. Nathaniel correctly identifies that Obama's comments about school reform were close to toothless. And Nathaniel correctly points out that Obama did not emphasize the degree to which the proposed initiative will place heavy burdens on the American people. And it seems disingenuous for Obama to claim to be willing to fund the " Apollo Projects of our time" while calling for a freeze in domestic spending.
While I disagree with much of Nathaniel's criticism, I am far from happy with Obama's speech. It certainly had its good points, but it had many failings. Obama failed badly in his discussion of energy policy. He gets points for mentioning nuclear power, something he has in the past downplayed. However, when discussing biofuels, he made no mention of the fact that the most prominent attempt at biofuel, corn based ethanol, is inefficient, environmentally damaging, and raises food prices which hurts the poor here as well as people in the developing world.
Similarly, Obama's emphasis on electric cars was similarly unpersuasive. The President did not discuss that electric cars’ power has to come from the general grid. While electric cars do overall pollute less and use less energy, this stems from economies of scale more than anything else. And the impact is not high. While Obama did discuss the energy of the future, he proposed no funding for research into genuinely new energy sources, such as fusion power. (In the particular case of fusion power, the US is putting resources into ITER, the international tokamak reactor, but the US is putting no money into other forms of fusion such as stellarators.)
Obama's discussion of school systems used metrics which are less than ideal. In particular, the fraction of a population which is going to college is an awful metric for success of students. Students who go to college are often unprepared and often come out not much better prepared than they went in. Moreover, many jobs, even high-tech jobs, don't require college degrees. We have serious problems with people going to college on loans they are unable to repay.
Obama was correct to talk about policy proposals in his address. The problem is not the discussion but the content of those policies.
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