Inception Math

This 2010 contemporary sci-fi actioner follows a subconscious security team around the globe and into the intimate and infinite world of dreams.
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caramie wrote:5 minutes in real time = 1 hour of dreaming = 60 minutes = factor of 12!
So EVERYTHING is wrong! hahaha math FAIL :lol: :lol: :lol: :lol:

10 hours in real time = 5 days in level 1 and so on....
Cobb estimates it's a factor of 12 without a sedative during Aradne's training. But they used sedatives in the inception.

Yusuf explains that the sedative speeds up brain function by a factor of 20. Meaning what we perceive as time normally will be 20 times faster in the dream.

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Momin wrote:How we get the proportion of 20 times?

According to my Math:
if 5 mins = 1 hour
=> 5 mins = 60 mins in L1

Getting the unit number:
1 MInute in Reality = 60/12 = 12 minutes in L1


therefore:
1 hour in Reality = 60 * 12 (where 60 is 1 hour in reality)
= 720 minutes in L1
= 12 hours in L1

for L2:

= 720 * 12 ( its recursion process: where 720 is 12 hours in L1 inception)
= 937440 minutes
= 21 days

does it make sense?
Just watched this film...so you may have come to the same conclusion by now. Taking into account all the variables - here's what I think is the correct answer to the maths (yes maths I'm British), behind Inception.

Early on in the film, it's stated (when Ellen Page and Leonardo DiCaprio's characters come out of the interview-like dream) that five minutes in the real world represents approximately 1 hour or 60 minutes, in Level One dream-space time.

To get the unit number, as you put it i.e. of what 1 minute of reality represents in dream-space time level one, you divide 60 minutes (where 60 is the number of minutes in dream-space time) by 5 minutes (where 5 is the number of minutes that have elapsed in reality).

Meaning that 1 minute in reality (at this point in the film) represents 12 minutes. A factor of 12 for each level.

Later on in the film, however, it's stated that while using the enhanced sedative, at each level real-time would appear to slow by a factor of 20.

Meaning that 1 minute, in the real world represents 20 minutes of Level One dream-space time.
Every time a person goes deeper by another level, the factor is increased again by 20.

In other words on every level 20 minutes elapse, for every 1 minute that passes on the previous level.

Therefore at Level Two, 1 minute of reality = 20 or (the number of level one minutes elapsed for 1 minute of reality) x 20 (factor of increase) = 400 minutes or 6.666* hours


At level Three, 1 minute of reality = 400 or (the number of level two minutes elapsed for 1 minute of reality) x 20 = 8,000 minutes or 133.333* hours or 5.5* days


At Level Four, 1 minute of reality = 8,000 or (the number of level three minutes elapsed for 1 minute of reality) x 20 = 160,0000 minutes or 2,666.666* hours or 111 days

and so on....

However, the film did also state that, if a person dies while using the heavier sedative, they would become trapped in 'Limbo', (a non-constructed dream-space) where the rules could change dramatically ... possibly further increasing the difference between the amount of time elapsing in reality and dream-space time, by factor of decades or even infinitely.
Last edited by Gecko21 on May 3rd, 2021, 10:26 am, edited 1 time in total.

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Wow. My brain hurts.

"Math has never been my strong subject..." :lol:

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Alright. This seems like a fun way to waste a few minutes. Here we go.

TIME FUNCTIONS FROM THE SHOOTING SCRIPT
ARTHUR: "Five minutes in the real world gives you an hour in the dream" page 68

Define N(x) as the dream time per time x in proceeding level.
N(5 minutes) = 60 minutes
Therefore, N(x) = 12x

COBB: How long do they dream?
YUSUF: Three, four hours. Every day.
COBB: How long in dream time?
YUSUF: With this compound...about forty hours. Each and every day. Page 89


Define C(x) as the dream time per time x in proceeding level while under the influence of Yusuf's general compound.
C(3-4 hours) ~ 40 hours
If C(3 hours) = 40 hours exactly, C(x) = 13.333x
If C(3.5 hours) = 40 hours exactly, C(x) = 11.429x
If C(4 hours) = 40 hours exactly, C(x) = 10x

Therefore C(x) ~ 11.5x

EAMES: Nothing they'd put him under for. Besides, we need a good ten hours.
SAITO: Sydney to Los Angeles. Twelve hours and forty-five minutes- pages 100-101


YUSUF: Brain function will be about twenty times normal. And when you go into a dream within that dream the effect is compounded.
ARIADNE: How much time?
YUSUF: Three dreams...that's 10 hours, times 20, times 20, times 20...
EAMES: Math was never my strong suit.
COBB: It's basically a week one layer down, six months two layers down-
ARIADNE: And ten years in the third level. Who wants to spend ten years in a dream? page 103


Define I(x) as the dream time per time x in proceeding level while under the influence of Yusuf's special Inception compound.
I(x) ~ 20x
First Level: I(10 hours) = 20 * 10 = 200 hours = 8 days 8 hours ~ 1 week
Second Level: I(200 hours) = 20 * 200 = 4000 hours = 23 weeks 5 days 16 hours ~ 6 months
Third Level: I(4000 hours) = 20 * 4000 = 80000 hours = 9 years 6 weeks 6 days 8 hours ~ 10 years
Limbo: I(800000 hours) = 20 * 80000 = 182 years 33 weeks 5 days 16 hours (NOTE: I(x) may not be valid on this level)

Therefore if a van falls for 3 seconds in reality, it will be 1 minute in level 1, 20 minutes in level 2, and 6 hours 40 minutes on level 3 (which is a fitting timetable for what is seen in the film).

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NMaudlin wrote:Alright. This seems like a fun way to waste a few minutes. Here we go.

TIME FUNCTIONS FROM THE SHOOTING SCRIPT
ARTHUR: "Five minutes in the real world gives you an hour in the dream" page 68

Define N(x) as the dream time per time x in proceeding level.
N(5 minutes) = 60 minutes
Therefore, N(x) = 12x

COBB: How long do they dream?
YUSUF: Three, four hours. Every day.
COBB: How long in dream time?
YUSUF: With this compound...about forty hours. Each and every day. Page 89


Define C(x) as the dream time per time x in proceeding level while under the influence of Yusuf's general compound.
C(3-4 hours) ~ 40 hours
If C(3 hours) = 40 hours exactly, C(x) = 13.333x
If C(3.5 hours) = 40 hours exactly, C(x) = 11.429x
If C(4 hours) = 40 hours exactly, C(x) = 10x

Therefore C(x) ~ 11.5x

EAMES: Nothing they'd put him under for. Besides, we need a good ten hours.
SAITO: Sydney to Los Angeles. Twelve hours and forty-five minutes- pages 100-101


YUSUF: Brain function will be about twenty times normal. And when you go into a dream within that dream the effect is compounded.
ARIADNE: How much time?
YUSUF: Three dreams...that's 10 hours, times 20, times 20, times 20...
EAMES: Math was never my strong suit.
COBB: It's basically a week one layer down, six months two layers down-
ARIADNE: And ten years in the third level. Who wants to spend ten years in a dream? page 103


Define I(x) as the dream time per time x in proceeding level while under the influence of Yusuf's special Inception compound.
I(x) ~ 20x
First Level: I(10 hours) = 20 * 10 = 200 hours = 8 days 8 hours ~ 1 week
Second Level: I(200 hours) = 20 * 200 = 4000 hours = 23 weeks 5 days 16 hours ~ 6 months
Third Level: I(4000 hours) = 20 * 4000 = 80000 hours = 9 years 6 weeks 6 days 8 hours ~ 10 years
Limbo: I(800000 hours) = 20 * 80000 = 182 years 33 weeks 5 days 16 hours (NOTE: I(x) may not be valid on this level)

Therefore if a van falls for 3 seconds in reality, it will be 1 minute in level 1, 20 minutes in level 2, and 6 hours 40 minutes on level 3 (which is a fitting timetable for what is seen in the film).
I don't think I could read that if I tried... but I'll take your word for it.

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NMaudlin wrote:Alright. This seems like a fun way to waste a few minutes. Here we go.

TIME FUNCTIONS FROM THE SHOOTING SCRIPT
ARTHUR: "Five minutes in the real world gives you an hour in the dream" page 68

Define N(x) as the dream time per time x in proceeding level.
N(5 minutes) = 60 minutes
Therefore, N(x) = 12x

COBB: How long do they dream?
YUSUF: Three, four hours. Every day.
COBB: How long in dream time?
YUSUF: With this compound...about forty hours. Each and every day. Page 89


Define C(x) as the dream time per time x in proceeding level while under the influence of Yusuf's general compound.
C(3-4 hours) ~ 40 hours
If C(3 hours) = 40 hours exactly, C(x) = 13.333x
If C(3.5 hours) = 40 hours exactly, C(x) = 11.429x
If C(4 hours) = 40 hours exactly, C(x) = 10x

Therefore C(x) ~ 11.5x

EAMES: Nothing they'd put him under for. Besides, we need a good ten hours.
SAITO: Sydney to Los Angeles. Twelve hours and forty-five minutes- pages 100-101


YUSUF: Brain function will be about twenty times normal. And when you go into a dream within that dream the effect is compounded.
ARIADNE: How much time?
YUSUF: Three dreams...that's 10 hours, times 20, times 20, times 20...
EAMES: Math was never my strong suit.
COBB: It's basically a week one layer down, six months two layers down-
ARIADNE: And ten years in the third level. Who wants to spend ten years in a dream? page 103


Define I(x) as the dream time per time x in proceeding level while under the influence of Yusuf's special Inception compound.
I(x) ~ 20x
First Level: I(10 hours) = 20 * 10 = 200 hours = 8 days 8 hours ~ 1 week
Second Level: I(200 hours) = 20 * 200 = 4000 hours = 23 weeks 5 days 16 hours ~ 6 months
Third Level: I(4000 hours) = 20 * 4000 = 80000 hours = 9 years 6 weeks 6 days 8 hours ~ 10 years
Limbo: I(800000 hours) = 20 * 80000 = 182 years 33 weeks 5 days 16 hours (NOTE: I(x) may not be valid on this level)

Therefore if a van falls for 3 seconds in reality, it will be 1 minute in level 1, 20 minutes in level 2, and 6 hours 40 minutes on level 3 (which is a fitting timetable for what is seen in the film).
Not to burst your bubble, but the van falls on the first level, not in reality.

-Vader

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Vader182 wrote:
NMaudlin wrote:
Therefore if a van falls for 3 seconds in reality, it will be 1 minute in level 1, 20 minutes in level 2, and 6 hours 40 minutes on level 3 (which is a fitting timetable for what is seen in the film).
Not to burst your bubble, but the van falls on the first level, not in reality.

-Vader
No bubble-bursting here. ;) I understand the van fell on Level 1 (for 1 minute). I merely hypothesized that if same said van were to fall in "reality" that from the height seen 3 seconds would be a reasonable amount of time for it to fall (exact calculations below). I had to do this in order to get an understanding of the physical laws of the first dream level. Had I started on Level 1, it would have been more difficult to understand why the van would fall for 1 minute of Level 1 dream time because that's seems ridiculous here in reality.

If we use the constant g=9.8m/s^2 for an object in free fall (completely ignoring the obvious effects of the van's initial speed and air resistance) we can apply these physical properties to each dream plane:

if h = the height of the bridge
Reality: The van falls for (h/4.9)^(1/2) seconds
1st Level: The van falls for 20[(h/4.9)^(1/2)] seconds
2nd Level: The van falls for 400[(h/4.9)^(1/2)] seconds
3rd Level: The van falls for 8000[(h/4.9)^(1/2)] seconds

Enter whatever bridge height you please and it should come out about right. I did a few below:

Approximate Height of the Golden Gate Bridge
152-meter bridge (~500 feet)
Reality: 5.57 seconds in free fall
1st Level: 111.39s = 1 minute 51.39 seconds in free fall
2nd Level: 2227.84s = 37 minutes 7.84 seconds in free fall
3rd Level: 44556.77s = 12 hours 22 minutes 36.77 seconds in free fall

Approximate Height of the Tower Bridge when closed
8.6 meters (~28 feet)
Reality: 1.32 seconds in free fall
1st Level: 26.49 seconds in free fall
2nd Level: 529.92s = 8 minutes 49.92 seconds in free fall
3rd Level: 44556.77s = 2 hours 56 minutes 38.4 seconds in free fall

And of course the Crown Jewel...the ACTUAL dimensions of the bridge on which filming took place: http://www.portoflosangeles.org/transpo ... ca_103.asp
Henry Ford Bridge (when vertically lifted as it was in the film)
49.68 meters (163 feet)
Reality: 3.18 seconds in free fall
1st Level: 63.68s = 1 minute 3.68 seconds in free fall
2nd Level: 1273.66s = 21 minutes 13.66 seconds in free fall
3rd Level: 25473.16s = 7 hours 4 minutes 33.16 seconds in free fall

The numbers don't get much more accurate than that. :ugeek:

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Wait, doesn't it work the opposite way? Time goes 20 times faster in every dream level, but people perceive time in their own dream level at a normal speed. So, those that Yusuf (on level 1) perceives as 3 seconds are actually 3:20=0,15 seconds in reality.

The van is shown in slow-motion because those 3 "layer 1" seconds (=0,15 "real" seconds) become a much longer time for the people who are in the lower dream levels.

I understand it this way:

Let's say that, in Yusuf's perception, it takes 3 seconds for the van to fall.
3 seconds on layer 1 --> 60 seconds on level 2.
1 minute on level 2 --> 20 minutes in level 3.

On the plane, only 0,15 seconds have passed.

But, if I remember correctly, at a certain point they say that Arthur has "a couple of minutes left" on level 2, so the fall of the van probably lasts 5-6 seconds on layer 1.

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Lynn wrote:Wait, doesn't it work the opposite way? Time goes 20 times faster in every dream level, but people perceive time in their own dream level at a normal speed. So, those that Yusuf (on level 1) perceives as 3 seconds are actually 3:20=0,15 seconds in reality.

The van is shown in slow-motion because those 3 "layer 1" seconds (=0,15 "real" seconds) become a much longer time for the people who are in the lower dream levels.

I understand it this way:

Let's say that, in Yusuf's perception, it takes 3 seconds for the van to fall.
3 seconds on layer 1 --> 60 seconds on level 2.
1 minute on level 2 --> 20 minutes in level 3.

On the plane, only 0,15 seconds have passed.

But, if I remember correctly, at a certain point they say that Arthur has "a couple of minutes left" on level 2, so the fall of the van probably lasts 5-6 seconds on layer 1.
The times listed are the amount of perceived time on each dream level for the equivalent time in reality. So as you said, 0.15 seconds in reality is perceived as 3 seconds on Level 1, 1 minute on Level 2, and 20 minutes on Level 3.

I believe the lines you are referring to are immediately after Eames in Level 3 reveals he heard the music start 20 minutes earlier:

COBB: Yusuf's about ten seconds from the jump, which gives Arthur about three minutes, which gives us about-
ARIADNE: Sixty minutes. (page 180)


and after they miss the first kick (hitting the bridge barrier) and have to resort to the backup kick:

COBB: When the van hits the water. I figure Arthur's got a couple minutes and we've got about twenty. (page 184)

So based on Cobb's estimates the timeline is as follows:

From Yusuf starting the music to slamming into the barrier:
Reality: 0.5 seconds
1st Level: 10 seconds
2nd Level: 3 minutes 20 seconds
3rd Level: 1 hour 3 minutes 20 seconds

From the slamming into the barrier to hitting the water:
Reality: ~0.15 seconds
1st Level: ~3 seconds
2nd Level: ~1 minute
3rd Level: ~20 minutes

Note that Cobb's numbers are a little sloppy. The more accurate numbers are easily found using my formulas in my previous post:

Reality: 0.16 seconds
1st Level: 3.18 seconds
2nd Level: 63.68s = 1 minute 3.68 seconds
3rd Level: 1273.66s = 21 minutes 13.66 seconds

So in reality from music playing to hitting the water, only about 0.66 seconds have passed!
Last edited by NMaudlin on July 21st, 2011, 2:44 pm, edited 1 time in total.

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^ Great, I think that's a good analysis! :)
NMaudlin wrote:So in reality from music playing to hitting the water, only about 0.66 seconds have passed!
I love this film. :D

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