## Archive for the ‘**toggles**’ Category

## Geekgasm

So I’ve been fiddling round with Ubuntu a bit. I’ve added a whole bunch of useless crap to my top panel thing. CPU monitor, network traffic monitor, it shows me the weather, there’s a sticky note widget thing. It’s great. Pimp my desktop. I only did it because I had two of the same thing for some reason and then I deleted the entire top panel and had to recreate it. So I decided to knock it up a notch. Bam!

Now that my desktop is pimped, I feel so much nerdier. It’s been a good day…

Have I mentioned that Gasm should be a tog?

## Multiplicative subsets of the Natural Numbers.

“Vert” should be another tog, I’ll come up with a definition later I suppose.

OK, here’s some terminology; a multiplicative subset of the natural numbers is one where if i and j are both in it then so is i.j. An additive subset is one where if i and j are in it then so is i+j. Obviously all additive and multiplicative subsets of the naturals have infinite cardinality. They are all elements of the power set of the natural numbers.

Apparently, there are uncountably many multiplicative subsets of the natural numbers. I’m not entirely convinced. There are at least countably many, since {the numbers greater than n} form a countably infinite family of multiplicative subsets. There are only countably many additive subsets of the naturals, however. Additive implies multiplicative since multiplication is just iterated addition. A multiplicative subset is a member of the power set of N. But there are uncountably many M-sets, so there must be a bijection between **P(N)** and M-sets. But there are infinitely many subsets of **N** that are not multiplicative…In fact for any M-set there is an infinite family of subsets which are not multiplicative. (i.e. for some set M, any set of the form M\{x} where x is in M and x=i.j some i,j in M) Are there really only countably many non-multiplicative subsets of **N**? I know infinity is thoroughly weird, but I’m having trouble with this.

The easiest explanation is that my Symbolic Logic notes are faulty. I’ll email Mr. Dude about that later…

In other news that question I was pondering a while ago about the set of all circles with a given arc as a chord has been solved! Well, we think that the answer is that the only parts of the plane not in the set are points on a line which is an extension of the chord.

Speaking of maths problems I’ve pondered before I realised that there are several other ways to define the sierpinski gasket, some of which were mentioned in a talk Ian Stewart gave at a maths society event last term… I’ll write some more about that later, right now it’s Simpsons O’clock!

## “Toggle” gets sexed up.

So, we were talking about duffel coats on the way back from campus, and then we got on to talking about how toggle is a great word and deserves to mean something better than this. Then, after some random tangent we ended up talking about what the words “inflect, deflect, reflect and genuflect” have in common. We decided some kind of generalisation would lead to a new word, something like this;

- Flect: to distort or alter the motion or direction of something, normally something non-physical

This idea of generalising similar sounding words and creating new words in much the same way as mathematical generalisation works was such a good idea that it deserved a new word; to toggle.

- Toggle: to extract the similarity in meaning from words of similar sounds and to coin a new word with that meaning with that sound

With this word coined, we realised that togs (the word formed by a toggling) are everywhere. “ology” “ism” and so on. Great.

On another note, I’ve decided that I should stop calling my knot theory revision “revision”, because most of this stuff is new to me. So I should call it knot theory vision. There’s no sense in which it is *re*-vision… But if I went round saying “I’m going to do some maths vision” people would think I’m a weirdo… Not that I’m not a weirdo…