Backward in Time
Backward in Time
When I have a boring task to get through — a three-hour lecture, a giant file download, or a long term point goal in fitocracy — I use this formula to convert the percentage completed (p) into a date:
T=(Current Date) - (e^(20.3444*p^3+3) - e^3) years
When the task is 0% done, it gives today's date, and as I make progress, I move further and further back in time
(inverse given in lighter colors) Inverse: p = sqrt((ln(T+e^3)-3)/20.3444)
[Line Graph explaining the correlation between completion percentages and temporal deltas. 0% = now (Date of comic is 2012-02-14T00:00-0500, approx. 1329195600 UNIX) 10% = September 2011 20% = 2008 30% = 1997 40% = 1958 50% = 1776 60% = 405 AD 70% = 22,000 years ago 80% = 671,000 years ago 90% = 55 million years ago 100% = 13.8 billion years ago ]
It moves slowly through the first few years, then steadily accelerates. I tuned the formula so the time spent in each part of the past is loosely proportional to how well I know it. This means I hit familiar landmarks with each bit of progress, giving me a satisfying sense of movement.
[The following are panels detailing completion percentages, correlated time periods, and notable events from this time period.]
7.308% December 18, 2011 Around this time: Kim Jong-Il dies. US leaves Iraq.
31.12% February 1995 Around this time: Windows 95 debuts. OJ found not guilty.
47.91% 1844 Around this time: Rubber vulcanized, bicycle invented, wrench patented.
70.33% 24,000 years ago Around this time: Caves painted, ceramic art made. Neanderthals extinct.
90.42% 68 million years ago Around this time: First flowering plants.  Chicxulub impact kills off most dinosaurs.
100% 13.76 billion years ago Around this time: Universe begins. First stars ignite.
Download complete.
[Cueball watches a download progress on a laptop in amazement and happiness.  Megan stands nearby and looks at Cueball with a bemused posture.] Cueball: Swoosh! Watching all that time blur past is such a rush! Megan: So...you've tried to make an extreme sport out of... waiting . Cueball: Swoosh!

Cueball / Randall creates this formula which helps him wait for long stretches of time which goes increasingly faster into the past as more time goes by, which gives him the effect of looking like the time goes by quickly.  Which assists in the waiting process.
As far as the actual math is concerned, the formula is an exponential function (i.e. the variable appears in the exponent). The effect that the function grows faster and faster as p grows, is due to T(p) being exponential. More precisely, when you repeatedly add some constant to the exponent, you will repeatedly multiply some (other) constant with the value of the function. Compare how "slow" a value grows by adding even high values (1, 1001, 2001, 3001, 4001, 5001…) and how fast it grows by multiplying even low values (1, 10, 100, 1000, 10000, 100000…)
Now, the function has to be adjusted so that, as Randall put it, "the time spent in each part of the past is loosely proportional to how well I know it." The most important adjustment is putting p to the power of three. That lowers the amount added to the exponent for low values (0.1³=0.001, 0.2³=0.008, i.e. only 7/1000 have been added for 10% workflow) and increases the amount for high values (0.8³=0.512, 0.9³=0.729, i.e. more than 1/5 has been added for 10% workflow). That means the recent past will pass even slower and the historic past even faster than it already does by choosing an exponential function.
The remaining adjustments are technical. The coefficient in front of p³ adjusts the constant by which the result will be multiplied while adding some constant to p, while it also roughly ensures that p=1 yields the lifetime of the universe. The 3 added to the product in the exponent further adjusts the actual values of the power without touching the slope (the multiplicative constant). In the parentheses, e³ is subtracted to put the time to 0 when p=0. Otherwise the function would start approx. 20 yrs and 1 month ago. For bigger p, this offset does not matter much. Imagine subtracting 20 yrs from the lifetime of the universe!
Finally, the result is subtracted from the current date for aesthetical reasons. The formula could tell you "20 yrs ago", or it could read "February 1992". Randall decided the latter would be better.
There is actually a mathematical error in this comic; the inverse function in grey writing off at the bottom right of the main formula involves a square root, when the actual inverse of Randall's main function would involve a cube root.  In addition, this function does not contain the current date, meaning that T, in the inverse, refers to how long ago a point in time was, rather than the point in time itself.  When the T in the inverse is 20, it means that the date referenced by T is 20 years ago.
The punchline "Swoosh!" is about how fast the last few percents of Cueball's download happen in "such a rush". For most humans waiting for a download to complete tends to become really boring and progress would instead seem to get slower and slower.
(Also, the workout website, Fitocracy has been mentioned previously in xkcd.)
Note that as of the time that this page was last cached, the comic was uploaded at 14.278499285807% progress.
When I have a boring task to get through — a three-hour lecture, a giant file download, or a long term point goal in fitocracy — I use this formula to convert the percentage completed (p) into a date:
T=(Current Date) - (e^(20.3444*p^3+3) - e^3) years
When the task is 0% done, it gives today's date, and as I make progress, I move further and further back in time
(inverse given in lighter colors) Inverse: p = sqrt((ln(T+e^3)-3)/20.3444)
[Line Graph explaining the correlation between completion percentages and temporal deltas. 0% = now (Date of comic is 2012-02-14T00:00-0500, approx. 1329195600 UNIX) 10% = September 2011 20% = 2008 30% = 1997 40% = 1958 50% = 1776 60% = 405 AD 70% = 22,000 years ago 80% = 671,000 years ago 90% = 55 million years ago 100% = 13.8 billion years ago ]
It moves slowly through the first few years, then steadily accelerates. I tuned the formula so the time spent in each part of the past is loosely proportional to how well I know it. This means I hit familiar landmarks with each bit of progress, giving me a satisfying sense of movement.
[The following are panels detailing completion percentages, correlated time periods, and notable events from this time period.]
7.308% December 18, 2011 Around this time: Kim Jong-Il dies. US leaves Iraq.
31.12% February 1995 Around this time: Windows 95 debuts. OJ found not guilty.
47.91% 1844 Around this time: Rubber vulcanized, bicycle invented, wrench patented.
70.33% 24,000 years ago Around this time: Caves painted, ceramic art made. Neanderthals extinct.
90.42% 68 million years ago Around this time: First flowering plants.  Chicxulub impact kills off most dinosaurs.
100% 13.76 billion years ago Around this time: Universe begins. First stars ignite.
Download complete.
[Cueball watches a download progress on a laptop in amazement and happiness.  Megan stands nearby and looks at Cueball with a bemused posture.] Cueball: Swoosh! Watching all that time blur past is such a rush! Megan: So...you've tried to make an extreme sport out of... waiting . Cueball: Swoosh!
