Biscuit denser than a neutron star

Over lunch, a couple of friends and I happened to be talking about biscuits, among other things. As we all know, a biscuit tends to absorb water from its surroundings, as it is generally drier than them. Now if we place a biscuit next to the ocean, it would slowly absorb all the water out of the worlds oceans, while maintaining the same volume (yes, we knew that this is completely impossible).

Now lets throw some numbers in. According to wolfram alpha, the volume of the world's oceans is 1.3 x 10^21 L, for an approximate mass of 1.3 x 10^21 kg. Biscuits vary in size, but we took ours to be a cylinder of radius 3 cm, and height 0.5 cm, for an approximate volume of pi x 3^2 x .5 = 14 cm^3 = 1.4 x 10^-5 m^3. Thus the density of the biscuit (as the mass of the biscuit is negligible compared to the water) is about 10^26 kg m^-3. For a comparison, this is much denser than the density of a neutron star, which is typically 8 x 10^16 to 2 x 10^18 kg m^-3 (this agrees with the value of 2 x 10^17 that I calculated in an assignment not so long ago), which, according to wikipedia is "approximately equivalent to the mass of the entire human population compressed to the size of a sugar cube."

So, there you have it. What's denser than a neutron star? A biscuit (containing all the world's oceans).

Sound arguments require basic logic

Sound arguments require basic logic. Sounds obvious, right? But it seems that some people just don't understand this. Or maybe they understand basic logic incorrectly. Anywho, when someone structures an argument in such a way that it doesn't actually prove the point they were trying to argue, I find it mildly amusing. It happens often too. I'm going to go on to describe one example of this that I encountered recently.

So, a little while ago, I went to a talk entitled, "Is religion inherently violent?". I don't care where you stand on this issue, it is just to illustrate my point. The speaker went on to demonstrate that violence exists outside of religion with a variety of examples, particularly from the 20th century. She also put forward one way that violence would develop in any group of people, religious or otherwise.

You see what she did? The issue itself wasn't addressed at all. By showing that violence exists outside of religion, you're not saying anything about whether or not religion is inherently violent. What she wanted to prove was that religion does not imply violence, but what was shown was that violence can result outside religion.

On the other hand, if you couldn't show that there was violence outside of religion, and there existed some violence, that would be a proof that all violence comes from religion. Not a proof that religion is inherently violent as there could still be some religions that don't cause violence. I can see how people can get confused, but it's not too hard really, if you just think about it.

Lastly, I'm not saying the talk was bad, she did bring forward an interesting point. I'm just saying that she didn't prove the argument the talk was supposed to be based on.

A proof that there is no onto homomorphism from S_5 to S_4

Warning: contains maths, algebra even.

This is probably one of the most satisfying proofs that I've ever come up with (or at least remember coming up with). This is partly due to the fact it is non trivial, and partly due to the amount of time it took to come up with it.

Anyway, here's how it goes:

Suppose there is an onto homomorphism \(\varphi\) from \(S_5\) to \(S_4\).

By the first isomorphism theorem, \(S_4 \cong \, ^{S_5}\!/_{\mathrm{ker} \varphi}\), so \(\left| \mathrm{ker} \varphi \right| = \frac{\left| S_5 \right|}{\left| S_4 \right|} = 5\).

As this is prime, \(\mathrm{ker} \varphi\) is cyclic.

Let \(x\) be an element that generates \(\mathrm{ker} \varphi\). \(x\) is an even permutation as \(x^5 = 1_{\mathrm{ker} \varphi}\), so \(\mathrm{sgn}(x)^5 = 1\) and \(\mathrm{sgn}(x) = 1\).

Therefore all elements of \(\mathrm{ker} \varphi\) are even, so \(\mathrm{ker} \varphi \subseteq A_5\), the alternating group.

But \(\mathrm{ker} \varphi \lhd S_5\), so \(\mathrm{ker} \varphi \lhd A_5\), which is a contradiction as \(A_5\) is simple.

So there does not exist such a homomorphism.

Sure, I assumed that \(A_5\) was simple, but we all know that, right?