The Autumnal Equinox

We live on the surface of a giant spinning ball. From our perspective, everything else appears to move around us as we spin. The sun, the moon, the planets, the stars. As we spin to face the sun, it appears to rise the sky. As we spin to face away is appears to set. It is no coincidence that we spin around exactly once each day. It is our spinning that determines the length of the day.

But we don't just spin in place. As we spin, we move in a circle around the sun. But the way we spin is tilted compared to the way we move around the sun. And the direction we're tilted doesn't change as we move around the sun, which means that sometimes we're tilted toward the sun, and sometimes we're tilted away from it. When you're tilted away from the sun, the days are shorter and the nights are longer, which makes it colder. The winter solstice is the day when you're tilted exactly away from the sun. When you're tilted toward the sun, the days are longer and the nights are shorter, which makes it hotter. The summer solstice is the day when you're tilted exactly toward the sun. It is no coincidence we move around the sun exactly once each year. It is our moving around the sun that determines the length of the year.

Not to scale. Credit: NOAA

Keep in mind that when the northern hemisphere is tilted toward the sun, the southern hemisphere is tilted away, and vice versa. The summer solstice in the northern hemisphere is the winter solstice in southern hemisphere, and the winter solstice in the northern hemisphere is the summer solstice in the southern hemisphere.

But today, we aren't tilted toward the sun, or away from it. Today, we are tilted perpendicular to the sun. Today is the autumnal equinox, the day when the day is equal to the night.

The equinox is a time of change. It marks the midpoint in the transition from the summer solstice to the winter solstice. Beyond that, it is also an inflection point. After the summer solstice, the days get shorter. At first, only a little bit. One day will be only a few seconds shorter than the day before it. But over time, the change increases, until one day will be minutes shorter than the day before it. The solstice is the time of the fastest change. After the solstice, the days will continue to get shorter, but the speed of the change will slow down again.

Free Will

If a tree falls in the forest, and no one is around to hear it, does it make a sound? Alice says yes, because it would produce a pressure wave through the air. Bob says no, because it wouldn't be perceived by anyone.

Alice and Bob aren't disagreeing about what happens when a tree falls down, they're disagreeing about the meaning of the word "sound".

Do we have free will?

Well, what do you mean by "free will"?

One notion of free will is that it is the ability to make decisions that are neither deterministically caused, nor random. This concept is called libertarian free will (unrelated to the political party), and it doesn't make much sense to me. To start off with, what's the middle ground there? How can something be neither determined, nor random?

Further, a non-determined decision wouldn't resemble a decision at all. Consider this scenario: you're in a building when a fire alarm goes off. So, you make a decision to evacuate the building. The only reason to evacuate the building was because the alarm went off. If the alarm hadn't played a role in determining your decision, then you wouldn't have made it.

So, that means we don't have free will, right? Well, it certainly means we don't have libertarian free will. But when you say things like "Free will doesn't exist.", people tend to react by saying things like "That means we can't make decisions!".

Which isn't true. Things that are completely physically determined can still make decisions. Consider a simple thermostat. If the temperature is below 18°C turn the heater on, otherwise, turn the heater off. Admittedly, that's in the fuzzy area between "decision" and "reaction", but I don't think it's fundamentally different than an undeniable decision, just simpler. (Which leads to another question of definition. What exactly is a decision?)

The fact that people tend to react that way implies that the libertarian notion of free will isn't the concept most people mean by the term "free will". So, we should find another definition that more closely matches people's free will. Here's my suggestion - Free will is the ability to make very complicated decisions that are determined by a very large number of factors and is very difficult to predict in advance.

Under this definition, we obviously do have free will. This doesn't resolve the issue. I'm still not certain exactly what a decision is. But I think it's a good step forward that helps avoid common misconceptions.

Consciousness

What is consciousness? Where does it come from?

I don't know. But I do firmly believe that it is not magic. Whatever it is, it is purely physical.

And I do know a story about how it may have arisen. Don't take it as a scientific explanation. It's not. It's just a story, which might have a hint of validity about it, but then again, might not.

It begins with life. Living organisms capable of reproducing and reacting to stimuli. Organisms that react to stimuli appropriately would be more likely to survive and reproduce than organisms that didn't.

But what response is appropriate can be hard to determine. Is that a predator, or a mate? Should I go towards it, or away from it, or play dead? And so the brain evolved. An organic computer to make decisions.

For a brain to make good decisions, it needs to model the world around it, and predict consequences of different possible actions. If I jump over this hole, can I make it to the other side? If I chase that prey, will I be able to catch it? The more accurately the brain models the world, the better the decisions it can make.

Then a species of social primates evolved. For social creatures, it's helpful to model the others in your group with more detail than other animals, because you interact with them more often. The better your model of them, the more likely you can get them to share their food with you.

But since you're all the same species, they're modelling you too. So the most accurate model of them is to recursively model them modelling you. In other words your brain has to have a model of itself. And that's essentially what consciousness is. It's the brain thinking about itself, trying to predict its own actions before it makes them.

This story doesn't answer everything. Particularly not the hard problem of consciousness. But I think it does a decent job of addressing the easy ones. Assuming, of course, that's it's even remotely true, which it might not be.

Science and Wonder

It is a common notion that science takes the wonder out of life. A prime example is John Keats's poem, Lamia.

Do not all charms fly
At the mere touch of cold philosophy?
There was an awful rainbow once in heaven:
We know her woof, her texture; she is given
In the dull catalogue of common things.
Philosophy will clip an Angel's wings,
Conquer all mysteries by rule and line,
Empty the haunted air, and gnomèd mine—
Unweave a rainbow, as it erewhile made
The tender-person'd Lamia melt into a shade.

I think this notion is wrong. Science, when properly understood, doesn't destroy wonder, it enhances it.

First, I'd like to clearly separate two relevant meanings of the word wonder. The first is synonymous with awe, the feeling you get when you think, "That's really really cool!". The second is synonymous with curiosity, the feeling you get when you think, "I wonder how that works...". They frequently come together, but they don't have to. It's entirely possible to feel awe at something that you understand completely, or to feel curious about something isn't particularly awe-inspiring.

Science enhances the feeling awe, because it reveals nature, and nature is, well, awesome. The real world is far cooler and more interesting than any fictional world I've ever read about (which is not to say that fictional worlds can't also be cool and interesting). I've written about this before, and given several examples of real awe-inspiring things. Most of those things would never be known about without science. And you can't have a feeling of awe towards something you don't know exists.

Science enhances curiosity in much the same way. Every question answered by science uncovers still more to be asked. Questions you wouldn't even be able to ask before, since you wouldn't have known the concepts they apply to.

I think the reason Keats, and others who make this claim do so because of two mistakes. First, they don't realize that the feeling of awe can be separated from the feeling of curiosity. Second, they don't realize that answering questions you're curious about can uncover deeper questions. If those two things weren't the case, then science would destroy wonder. Fortunately they're not.

Law and Morality

People sometimes claim, usually in the context of things like gay marriage or abortion, that you shouldn't legislate morality. Usually, this is in response to the claim that gay marriage (or abortion, or prostitution, or whatever the topic at hand is) is immoral, and therefore should be illegal. But that's a bad response, because morality should be legislated.

Opponents of abortion are actually pretty good about pointing out the foolishness of this stance, by comparing abortion to murder. No one would say "If you don't approve of murder, then don't murder people." or "I don't like murder, but if my neighbor does, who am I to judge?" The response to that is usually that it's ok to have laws against murder because murder hurts people.

But so what? Why is ok to make a law banning something that hurts people? Because hurting people is immoral.

I think the main reason this argument is compelling is that it's usually used against people who are arguing against something that's not immoral. People say, "I think gay sex is gross, therefore it's immoral, therefore it should be illegal!". The argument is clearly flawed, but the flaw isn't with the second "therefore", it's with the first.

The problem is not with saying that something immoral should be made illegal. The problem is with misidentifying what is moral in the first place.

So, what is morality? How do you correctly tell if something is moral or not? That's a topic for another post, or perhaps a book.

Membranes

As you may be aware, I like making geometrical constructions. Ever since I made an AlDraw app for Android, I've been using it to post constructions on Facebook. The other day I posted this construction, titled "Membranes".

This wasn't the first time I constructed this image, but when I reuse a construction, I still reconstruct it, rather than simply copying the original. But when I did that this time I discovered that I made the original one incorrectly. It wasn't exactly a mistake. I made it the way I had intended to, but I found out when I made a small change, it made the whole thing more elegant.

To explain how, I'll need to explain how I constructed it in the first place. I started with these two circles. The radius of the outer circle is R. The radius of the inner circle is R/2. They are both centered at (0, 0).

Next I drew circular arcs between the inner and outer circles. All three have radius R and are centered on the edge of the outer circle. The red at 0°, the blue at 30° and the green at 60°. I've shown the centers of the circles in the same color as the circles.

Then I added those same circles again, reflected around the x and y axes.

Then I wanted to connect the different arcs, and I want it to look smooth, so I want to make circular arcs that are tangent to the arcs they're connecting. How do I do that?

Well, if two circles are tangent, then a line drawn through the centers of both will pass through the point of tangency. So if you have a circle, and you know what point you want to be tangent, then you can draw a line through the center of that circle and through that point. Then any circle that is centered on that line and that goes through that point will be tangent to the first circle. And if you have two circles like that, and you want to find one circle that is tangent to both, then you can draw two lines. Where they intersect will be the center of the third circle.

So I'll start with the red circles. I drew lines through the centers of the red circles and the points where the red circles intersect the inner circle. Where they intersect is the center of the circular arc that connect the two arcs. Note that each red line comes close to where the other red circle intersects the outer circle, but not quite. Also, each red line comes close to where the blue circles intersect the inner circle, but again, not quite.

The I did the same thing with the blue circles. I drew a line through the center of each blue circle and the point where that circle intersects the inner circle. Note the intersection of the blue lines is close to the outer circle, but not quite on it. Also, each blue line comes close to where the green circles intersect the inner circle, but again, not quite.

The next and final step is where I noticed the mistake. Because the next step I simply drew straight line segments to connect the green arcs. The straight lines are nearly tangent to the green arcs. But not quite.

Now at this point, I could have simply found the circular arcs that would be tangent to the green arcs the same way I did for the blue and red. That would be one way of fixing it. But I wanted those green arcs to be connected by straight lines and for those line to actually be tangent to the arcs. And the only way to do that is the change the radius of the inner circle, which changes where the red and blue circles intersect it, which changes where the connecting arcs need to be. In other words, changing nearly the whole image.

So this time, I started not with the outer circle and inner circle, but rather the outer circle and one of the green circles.

Now I need to find the point on the green circle where a horizontal line will be tangent to it. A line drawn through the center of a circle will always be perpendicular to the circle and hence perpendicular to the tangent line where it intersect the circle. So what I need to do is draw a vertical line through the center of the green circle, and where it intersects the green circle is where a horizontal line will be tangent to it.

Then I can draw the inner circle with the same center as the outer circle and that goes through that point. A little bit of trigonometry shows that the radius of the new inner circle is R, which is approximately .5176R.

Then I drew the other circles the same as before, between the outer circle and the new inner circle.

Again, I drew lines between the centers of the red circles and the points where they intersect the inner circle. But notice, this time they don't pass close to where the other red circle intersect the outer circle, they pass right through it. Also, although the difference is too small to notice at this scale, the red lines also pass directly through where the point where the blue circles intersect the inner circle.

And again, I drew the lines between the centers of the blue circles and the points where they intersect the inner circle. And again, notice that they don't intersect near the outer circle, they intersect directly on top of it. Also, they pass directly through the points where the green circles intersect the inner circle.

And finally, I drew the lines connecting the green arcs, and this time, they're actually tangent.

It's interesting how that one little almost imperceptible change made the whole thing more elegant.

Superman and the Physics of Collapsing Buildings

I saw the new Superman movie this weekend, and I liked it. But it wasn't perfect, and the thing that bothered me the most was the bad physics. I'm not talking about Superman being able to fly or the Kryptonian terraforming machine being able to increase Earth's mass. That kind of thing is expected in a superhero movie. I'm talking about more everyday physics. The most egregious example is skyscrapers falling over.

It happens multiple times in the movie. Superman throws a bad guy (or a bad guy throws Superman) through a skyscraper, part of the building is damaged, it tips and falls over like a tree. You might be wondering what's wrong with that. After all, trees fall down like that. If you build a tower out of Legos and knock it down, it falls down like that. But large buildings don't fall down like trees or Legos. They don't fall over sideways, they simply fall straight down.

So, why do large building fall down? Because gravity pulls them down. It does not pull sideways, so it doesn't tip sideways. But then why do Legos and trees fall sideways? Because there are other forces at work, namely the internal forces holding them together and in the same shape. Gravity is pulling down, but the internal forces prevent the top from simply collapsing into the bottom, so it falls sideways.

Here's a force diagram of a brick in a Lego tower tipping over. Gravity is pulling down. Normal force is pushing at the same angle the building is tipping. The total force is in blue. The vertical components mostly cancel, leaving the total force going mostly sideways.

But why don't large buildings do the same? Don't they have internal forces too? Well, yes, but they don't scale up. As the building gets bigger, it gets heavier, and gravity pulls more strongly. The internal forces of a large building will be stronger than those of a Lego building because it's made with steel rather than plastic, but it will be weaker relative to the force of gravity. The normal force will still be there, causing it to tip just a little bit, but gravity will dominate, so it will fall almost straight down. The top will simply collapse into the bottom, rather than being pushed to the side.

So why does this matter? It's just a movie, right? That's true, but understanding physics and how forces scale can be important. For example, there was a very well known case where some tall buildings fell down unexpectedly. As physics predicts, the buildings fell mostly straight down. (But not entirely. A lot of nearby buildings were hit by debris.) But a lot of people didn't understand the physics, and thought that the fact that the buildings fell down instead of over meant the buildings weren't brought down by airplanes, but rather by controlled demolition, and thus a conspiracy theory was born.