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Fun with Mathematics: The Finite and Infinite Universe of YouTube

If you've ever sat down to watch one YouTube video, you've no doubt spent a lot of time watching several.  Ever since it launched, YouTube has grown into a monolith of the digital media and social networking age.  ComScore estimates that between August 2009 and August 2010, the average web surfer spent approximately four to four-and-a-half hours watching YouTube videos each month, or eight to nine minutes every single day (Website Monitoring estimates the average at 15 minutes a day).  But have you ever wondered just how big the YouTube space really is?

I thought it might be a fun little exercise to contemplate the theoretical boundaries and mathematical possibilities of YouTube.   A brief review of YouTube's impact on the internet since its inception in 2005 is a fascinating glimpse into a gigantic and as yet undiscovered universe.

How many YouTube videos are there today?

It's nearly impossible to say.  The exact number of videos on YouTube fluctuates all the time, given the number of new videos that are added every day.  Dr. Michael Wesch of Kansas State University reported that a March 2008 wildcard search on the website (performed by typing "*" into the YouTube search bar) yielded nearly 80 million results.  Programmer Artem Russakovskii estimated that there were 141-144 million videos on YouTube as of August 2008.  A wildcard search as of the publication of this diary yields approximately 275 million results, but that estimate will surely grow pretty rapidly and it's very difficult to determine how accurate it is even now.

How long would it take one person to watch them all?

Wesch wrote that the average YouTube video clip is 2 minutes 46.17 seconds long, or 2.77 minutes.  Let's assume that there are indeed 275 million videos currently on YouTube.  So if you sat at your computer, constantly looking at these clips without a single break:

275 million x 2.77 minutes = 761,500,000 minutes

761,500,000 minutes = 12.7 million hours

12.7 million hours = 529,000 days

529,000 days = 1450 years

More than one millenium would pass before you'd be able to watch all YouTube videos from beginning to end.  So you should either take that goal off your bucket list, or find a permanent medical cure for death.  Maybe that knight who was guarding the Holy Grail in Indiana Jones and the Last Crusade could do it.  What else could he do to pass the time?

How many new YouTube videos are uploaded on a daily basis?

Again, this number is pretty hard to track down.  Using YouTube data, Website Monitoring reported in May 2010 that at least 24 hours' worth of content is uploaded every single minute.  But, YouTube's own press room states that 35 hours of new content is added each minute.  We'll call these the low and high estimates for now.  Again, using Wesch's estimated average length of YouTube clips:

Low estimate:
24 hours x 60 minutes = 1440 minutes of video uploaded per minute

1440 minutes / 2.77 minutes per video = 520 videos uploaded per minute

520 videos per minute x 1440 minutes per day = 748,800 videos uploaded per day

High estimate:
35 hours x 60 minutes = 2100 minutes of video uploaded per minute

2100 minutes / 2.77 minutes per video = 758 videos uploaded per minute

758 videos per minute x 1440 minutes per day = 1,091,520 videos uploaded per day

These figures estimate that between 750,000-1.1 million videos are added to YouTube in a single 24-hour period.  For comparison, Wesch estimated it as 200,000 new videos uploaded per day, but that was back in March 2008, so the rate of increase has jumped with the booming popularity of YouTube.  Either way, the number of new uploads at a given time is big, and it keeps growing.

What's even more interesting is that the number of new uploaded videos is dwarfed by the number of times these videos were played.  YouTube claims that their website received 700 billion playbacks in 2010.  Just imagine the resources and bandwidth required to support that much new content and user participation on a daily basis, and to do it efficiently enough so that the servers didn't crash every couple of seconds.  It's really an amazing operation.

YouTube doesn't have infinite bandwidth, so it must have limits.  What are they?

Obviously, even a website as widespread and expansive as YouTube has to have a finite space with which to operate.  But it's very difficult to know exactly how large this space is or how big it can potentially be.  Since the number of new uploads and the number of playbacks have increased each year, YouTube's bandwidth has also had to increase just to sustain its servers.  The website Willy Dobbe estimated the total bandwidth usage in July 2006 as 25 petabytes per month, which is equal to 25,000 terabytes or 25 million gigabytes.  Whatever the exact number is today, it is obviously large enough to get YouTube to work properly.

But leaving aside the issue of bandwidth, there's another way we can conceptualize the theoretical, finite limits of YouTube -- and the scale becomes exponentially larger when we do.  Each upload contains a random string of 11 characters at the end of the URL address that serves as a unique ID for the video.  For example, the clip from Indiana Jones that I linked to earlier has the following 11-character string:


If you change the string by even one character, you'll get a different video.  For the purposes of this exercise, we will assume that YouTube will never increase or decrease the length of the string, and that the population of all 11-character strings represents the maximum limit of YouTube uploads.  You'll soon see that the current number of YouTube uploads, no matter how large it is, will never approach the number of possible video IDs, not until long after the human race has died out.

The string can contain lowercase letters (26 characters), uppercase letters (another 26 characters), all digits from 0 to 9, -, and _.  Each character can appear at any position in the string.  That's 64 different characters that can be distributed at 11 different positions, which yields a huge number of possible permutations of the string:

6411 = 7.38 x 1019

Look at that!  7.38 x 1019 different possible YouTube strings!  So in theory, if YouTube had an infinitely large bandwidth, and if it were to keep coding its uploads with random 11-character strings indefinitely using only those 64 specified characters, the website could upload as many as 7.38 x 1019 videos.

If that huge number is the total size of the YouTube universe, how long would it take to fill it up?

Much longer than you might think.  Let's go back to one of our earlier figures -- the current wildcard search.  If 275 million is the real number of YouTube uploads existing today, then the YouTube universe is barely populated at all!

275 million / 7.38 x 1019 = 3.73 x 10-12

That means the current number YouTube videos on the internet comprises only 0.000000000373% of its potential space, which is essentially nothing at all.  Put another way: If you visited YouTube and blindly typed in a random 11-character string at the end of the URL address using any of the allowed 64 characters at any position, you'd have no greater than a 0.000000000373% chance of actually hitting a YouTube video -- so don't bother trying.

Now, look at the current rate of increase in new YouTube content.  If YouTube really added as much as 1.1 million videos every single day with different video IDs indefinitely, it would take a very, very, very long time to reach the limit of 7.38 x 1019 strings.

7.38 x 1019 strings / 1.1 million videos per day = 6.71 x 1013 days to reach the limit

6.71 x 1013 days = 184 billion years

Yes, at the current rate of adding new videos, it would take 184 billion years just to use up all possible strings!  The universe isn't even that old -- it's thought to be "only" 13.7 billion years of age.  YouTube users will obviously have to pick up the pace to super-cosmic speed to fill the YouTube universe.

In fact, let's test that theory out.  Let's say YouTube was somehow able to increase their upload rate by an order of 1 billion.  In other words, new videos were added at a rate of:

1.1 million x 1 billion = 1.1 x 1015 videos per day

How long would it take to fill the YouTube universe now?

7.38 x 1019 strings / 1.1 x 1015 videos per day = 67,091 days to reach the limit

67,091 days = 184 years

Ah, that's much better!  Just 184 years with that gigantic upload rate running constantly and the YouTube universe will be full.  Not that any of us will be alive to see that day come.  It's hard enough for any server just to double its bandwidth, much less expand it a billion-fold.

Oh, and you remember how it's impossible for any one person to watch all 275 million videos on YouTube today because it would take 1,450 years to do it?  That's nothing compared to how long it would take to watch 7.38 x 1019 videos from beginning to end without stopping:

7.38 x 1019 videos x 2.77 minutes = 2.04 x 1020 minutes

2.04 x 1020 minutes = 3.41 x 1018 hours

3.41 x 1018 hours = 1.42 x 1017 days

1.42 x 1017 days = 3.89 x 1014 years = 0.389 Petayears

It's a bit early to plan this far ahead, but does anyone want to go to the End of Watching YouTube Clips Party in the year 389000000000000 A.D.?  I think it will fall on a Thursday.

This is a nice little thought experiment and all, but what does it mean?

I don't really know.  This question of YouTube's finite space and potential was born from simple curiosity more than anything.  One thing that I find to be very fascinating about YouTube is that despite the arbitrary condition I've set for its theoretical limit, it really is an engine for infinite possibility.  It doesn't matter if the code string at the end of a URL address is 11, 12, 13, or 100 characters long.  Whatever finite limits a website like YouTube has, it's still a place where you could theoretically film anything at any time in any way you want and post it for the entire world to see it instantly -- well, at least until copyright lawyers come knocking on your door!

The real genius of YouTube is that it takes advantage of film, an artistic, visual medium that allows people to express themselves in an infinite number of ways.  Consider this: If you shot a one-minute video of your house from 12:00 pm to 12:01 pm, and then shot a separate one-minute video of the same house from 4:00 to 4:01 pm, then you would have two different films, albeit with very subtle differences.  Now add in the various other variables that comprise a finished film product -- running time, camera angles, lighting, sound, mise-en-scene, characters, costumes, dialogue, special effects, editing, and any combination of the above in any order -- and you could literally imagine an endless number of things that could be depicted in a film, regardless of the limitations of technology, environment, resources, and time.

This infinite nature of film artistry was true even during the beginning of the 20th century when moving pictures were first being shot.  A complex, tightly edited four-minute music video from 2011 and a twelve-second clip of New York City stock footage from the 1920s may be vastly different in quality -- but in quantity, they are both equivalent as one YouTube video, and when you consider just how many different ways the same films could have been shot and edited together, then the number of possible films from those two subjects alone grows infinitely large.  In addition, the fact that filmmakers in the 1920s did not have the capability of creating special effects like you might see today in The Matrix or Avatar did not mean that the human race would never achieve that kind of technological milestone.  It just meant that we hadn't gotten there yet.

At its most basic level, YouTube is just a website, and like all websites, it is the product of human ingenuity with human limitations.  However, I think that when one considers just how much information, content, and creativity that this one website has provided across the globe in the relatively short period of time it has been around, one can appreciate both its finite limits and its infinite imagination, its big influence on the internet and yet its smallness in the grand scheme we call life.  In human terms, every video clip is a small glimpse of the world around us, a brief window into our emotions, relationships, and experiences.  In mathematical terms, a single YouTube video is just one webpage in the vast ocean of more than 13 billion webpages.

It's a story much like our own cosmic evolution.  The scale might be much smaller, but imagine that a single YouTube video is like a single grain of sand in all the beaches on earth, where both the video and that one grain of sand have their own unique properties, details, and history.  Imagine that a single YouTube video is like a single star in all of outer space, where both the video and that star have their own brightness, lifespan, and place in the universe.  Much like a young Carl Sagan staring into the night sky imagining the vast boundaries of the cosmos, I wanted to approach this exercise with the same wonderment.

I thought I might end this diary with -- what else? -- a short YouTube video that nicely sums up my appreciation for mathematics, technology, information, and the boundless possibilities of our universe.  The late Sagan himself makes an appearance in it, as do other prominent scientists like Bill Nye, Neil deGrasse Tyson, and Richard Feynman.  All credit to the YouTube user melodysheep for creating this Symphony of Science.

Updated by SuperBowlXX at Fri May 20, 2011 at 04:39 PM EDT

A few other things I realized about the math of watching all 275 million YouTube videos from beginning to end:

1,450 years assumes that there's no lag time between each one.  You know how it takes at least a couple of seconds to load a new webpage after you've clicked on a link?  Just two seconds of lag time to load each individual webpage with a new YouTube clip would add more than 9 million minutes (17 years) to your objective.  And if you had a dial-up connection that loads YouTube clips really slowly?  Ho boy.

Now, you could save yourself some time if you wanted to skip duplicate videos.  Because of pirates downloading videos from third party websites and uploading them to YouTube, lots of YouTube clips are copies of one another.  If we assume that 10% of all 275 million videos were duplicates, you could cut out 10% of the total time if you skipped the duplicates, which would save you at least 145 years.  But you'd still be deader than dead before you were done.

Updated by SuperBowlXX at Fri May 20, 2011 at 08:38 PM EDT

One other thing about those 275 million videos which is so obvious that I completely forgot to consider it:

Even if you somehow achieved immortality and completed your original goal of watching all YouTube videos from beginning to end.....by the time you finished watching the 275 million videos currently in existence today, you'd still be nowhere close to finishing your goal.  That's because while you were busy spending 1,450 years watching those videos, new ones were added the whole time.  Unless the rate at which you could watch YouTube videos was greater than the rate at which YouTube uploaded new ones, you'd never reach the end.  It's like an endless hallway that gets longer and longer the closer you get to what you think is the finish line.

So to recap: In order to watch all YouTube videos, you'd have to be able to a) live forever, and b) stop time.  Call me when you achieve either superpower.