Space

This post is inspired by one of Robert's jokes. There was something about space on TV, not sure what it was about. In any case, Robert was trying to think about different jokes. I told him he didn't have to try since he was tired, but he offered to make it a math joke, and asked about what kinds of spaces there are. This reminded me of something one of my professors at Wooster said. He said that if he were to get a job, he would name it Hilbert so that the doghouse could be called Hilbert's Space.

Combinatorics this time

So I was listening to a CD that was given to me as a gift (Thanks Mandy). Anyways, I saw that I put it on random order. Eventually, track 1 came up, so I got a little excited. Clearly, this song was designed to be first on the CD, so it must be highly respected. It's gotta be good. However, I didn't like it (no offense, it just doesn't compare to the other songs).

This got me thinking. I put this CD on random order, so instantly by placing no preference on the order of the songs, I'm giving preference to the later songs. These are the songs that the CD creator didn't think were as good (otherwise they'd appear earlier on the CD). Thus, random order is actual an insult to the creator of the CD and their taste of music. The only way to give credit to the author of the CD is for the different songs to randomly come in at the time they were intended to (ie. track 7 coming up 7th).

Using this idea, once we figure out how many songs are necessary to give enough credit to the author, we can use inclusion exclusion to determine the probability of getting that many in the correct location, and therefore determine how likely it is that we will offend the creator, and if the p-value is above or below 0.05, and thus if any offense is due to the listener's decision to put the songs on random order, or pure coincidence, depending on the number of songs required to match/mistmatch.

Note: I am too lazy to do any of these calculations, so I don't know if I'm offending Mandy or not.

Practical Application of Cyclic Groups

Jeff and I are walking back from somewhere. I'll go ahead and say we were playing games at Boris's house. Anyways, we're walking to my house. He has his bike, and there were times when there isn't enough room on the sidewalk for both of us and his bike, so I am being a gentleman and letting him lead the way. Since he's leading the way and doesn't know my normal route home, I'm telling him directions. We need to turn right, and due to some confusion, I felt it necessary to repeat my directions to him by saying right multiple times. Jeff, being the jokester he is, makes some comment about the number of times I said right. This led to a mathematical realization.

Clearly, turning right 4 times merely keeps you facing the same direction. There was some discrepancy about if right meant turn right and walk a block. Either way it'd result in the same thing. However, when dealing with just turning and not walking a block, we would have a cyclic group of 4 elements, $C_4$ (for those LaTeX users out there). And it works. The multiplicative group generated by $r$, with the relation $r^4=1$, thus we have $$.

So next time you wonder how groups play a role in the real world, just spin around a few times (I recommend a spinny chair with somebody spinning you).