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.

Rare Earth Hypothesis vs. Reason

As you probably have gleaned from the title of this post, I disagree with the Rare Earth hypothesis-- that is, the theory that the spectacular coincidences and conditions that led to complex (not even necessarily intelligent) life on this planet is so statistically remote that it is extremely unlikely that any other planet in the universe has evolved complex life. The term (and hypothesis) comes from a book by Peter Ward and Donald Brownlee.

Now, I have not read this book, and therefore I am not fully versed in their arguments. I plan to read the book and make a more detailed argument, but I wanted to lay down my base hypothesis first. Also, Wikipedia does an ok job at summarizing their argument.

So, before we get to their equation, let's look over some data:

• It is estimated that there are 200-400 billion stars in the Milky Way Galaxy.
• It is estimated that there are about 100 billion (10¹¹) galaxies in the universe.
• Galaxies tend to contain between ten million (10⁷) and one trillion (10¹²) stars.

Now let's crunch those numbers. With simple math, we can conclude that the average galaxy has 10^9.5 stars. So, to calculate the total number of stars in the universe we need 10¹¹ * 10^9.5, which gives us, with rounding, 10²⁰ stars. (That's 100,000,000,000,000,000,000 stars.) That's a lot. If each star was the size of a tennis ball, stacked up they could reach the moon from the earth (avg. distance) almost 17.5 million times.

Now let's make some guesses. In the famous Drake Equation, it is estimated that half of all stars will have planets. So that gives us a more manageable number of 10¹⁰ stars with planets. For the sake of argument, let's say that our solar system is fairly typical, and maybe the average number of planets in a solar system is around 6. I think I am being generous to the Rare Earth-ists here, as a solar system could easily have tens of planets, not to mention life-supporting moons. That gives us 6 * 10¹⁰ planets in the universe. Disregarding the Drake estimate of 2 planets per star supporting life, let's say that, I don't know, 1/8 of planets support complex life.

So, we have finally arrived: 60,000,000,000 planets could support life. Sixty billion seems to be far from the estimate of 1. If my estimates have an error of 99.9%, there would still be 7,492,500,000 planets supporting complex life. Of course, some of that may have died off already, and most (statistically) we would never be able to find. (If you read the numbers above, and you're good with visualization, you should already have realized the size of the haystack in which we are searching for a proverbial, uh, 60 billion needles.)

As I have said, I will in fact be making a more detailed argument later. However, I assert that the vast immensity of the universe makes the Rare Earth theory extremely unlikely, statistically. And I haven't even broached the subject of carbon-based vs. non-carbon-based life or the general assumptions of what a 'life-supporting planet' requires.

QED.