Top Ten Albums – January 2011

 

A few months ago I started keeping a mental list of what my favorite albums of all time are. It's a list based on my opinion at the time and also weighted by how often I am listening to that album. It's changed a fair amount over the last few months, and I've thought about how it would be fun to track these trends, especially looking back in the future.

So, both to share what I'm listening to and for my personal historical record, I am starting a new 'feature' on this blog, my top ten albums. Starting with a new year seems like a clean way to go, and installments will be monthly.

Here we go:

1. A Thousand Suns – Linkin Park
2. This Is War – 30 Seconds to Mars
3. Sing the Sorrow – AFI
4. Fallen – Evanescence
5. Alchemy Index, Vols. III & IV: Air & Earth – Thrice
6. Siren Song of the Counter Culture – Rise Against
7. Infinity on High – Fall Out Boy
8. 21st Century Breakdown – Green Day
9. American Idiot – Green Day
10. How to Start a Fire – Further Seems Forever

There were some others in the running for the list, but I had to make my decisions. After #5 it became a little more difficult, since I spend a vast amount of time listening to the top albums on the list.

The Math of Dream Time in "Inception"

I'm a big fan of Christopher Nolan's newest film "Inception."


I am also a big fan of math. I know that makes me a nerd. (When I was home for Christmas, my family was discussing complicated gift-giving timetables, and getting confused. So I drew a Venn Diagram where the universe was 'my family,' and gift-giving times were shown as color-coded functions describing the relationships between various parts of the family. I thought it cleared everything up nicely, although my family did not entirely agree.)


So, I decided, with the help of a partner, to try to figure out the Real Time to Dream Time conversion from the movie. I figured at first it would be a nice linear function, but alas, no.


Throwing out the off-the-cuff estimations that are made at various points in the movie, I considered my reliable data points to be:

Real Time	1st Dream	2nd Dream	3rd Dream
5	60	?	?
600	10080	259200	5256000

This comes from two scenes. In one, they are teaching Ariadne about the dream world, and they say "Five minutes in the real world gives you one hour in the dream world."


The second comes from when they are planning for the flight. They say that the flight is 10 hours, which gives them 1 week in the first dream, 6 months in the second dream, and 10 years in the third dream.


I have converted everything to minutes as our base unit. Given this information, we plotted the second (more complete) set into a curve to predict the final piece of that set, the time in Limbo. We got y = 26.276e^3.0481x  with an R² value of 0.9993.
This gives us a Limbo time value for that set at 109250719, or approximately 39,000 years.


The two data sets don't exactly work together, though. Any equation you find to approximate the relationship does not correctly predict our other given case. This is partly because we are working in 3 dimensions, so we need to define some variables:

• x is the dream level, given by 0 for real time, 1 for first dream, etc.
• z is the amount of real time, and
• y is the amount of dream time.

Then we can say that f(x,z) is the amount of dream time, and we have 

f(x, 600) = 26.276e^3.0481x.

We tried very many ways to fit our two data sets together, but we were not finding any good correlations. We are especially interested in the fact that they seem to be able to do the math quickly (or at least estimate quickly) in the movie, and we had not been able to find an elegant equation.

Eventually, we started to think about it another way. What if we graph the amount of real time passed for each minute in dream time, based on the dream level? This gives us a curve approaching zero on the y-axis.

We used our given information to graph this:

Dream Level	Real Time	Dream Time
0	1	1
1	1	600/10080
2	1	600/259200
3	1	600/5256000

Which gave us this relationship: a = 1.0835e^-3.048x

where a is the real time passed for each minute of dream time at dream level x.

I realized that a = z/y, since a is essentially the z per 1y.

So, with some algebra we come up with 

f(x,z) = z/(1.0835e^-3.048x)

Now, this is far from perfect, and as an example, in my table below I have a "Real Time" column, which is the real time as predicted by our function. You can see that the accuracy of the real time prediction goes down as real time increases, but that the prediction of dream time values gets more accurate as we approach our base case of 600 minutes real time. Also, "Real Time" is rounded to two decimal places, and the dream time minutes are rounded to the whole number.

You can see, though, that we are still off from our given values:





I am very interested in whether or not someone can come up with a better approximation than ours, and I'd love to see it.


Probably will be more on this to come.