Circle Limit IV – M. C. Escher – an example of hyperbolic geometry

Dear reader,

sometimes I look at the spectrum of activities I performed during my working day, and the time of day I devoted to them. More often than not, the result of mapping my schedule is something similar to figure 1 (each colour corresponds to time dedicated to a single task/project).


I envision that a schedule like that will be familiar to many of you.

What conclusions can we draw from that picture? First of all, we notice that there isn’t a single recurring predominant task, but many different ones. I suspect this is a common plague afflicting knowledge workers. However, in each day there is a job that apparently gets more attention than the others, and it does not show up in later days. Is it a sign of a deadline? Maybe we spend so much time on it because it must be finished on time? Moreover, every day it takes up more and more time on our schedule, starting slowly and getting momentum as time goes on, another sign of upcoming deadlines. Lastly, often the execution of this supposedly-important-last-day activity gets interrupted by other brief tasks: maybe they are urgent annoyances? Or is it a sign of poor planning?

To analyse general aspects we can remove the planning aspect from the map by ordering the tasks by time spent on each day, as in figure 2:



Well, now things are getting interesting! I am sure that you spot a pattern in those colours. Let me now take a digression, and talk for a while about a scientific approach to procrastination.

I happened to stumble online on a beautiful little article in the American Institute of Physics publication Physics Today. The author, Tomasz Durakiewicz, begins the piece with this question:

Throughout our lives we all are under pressure to deliver on deadlines. Yet we often have a substantial time window in which to complete the given tasks—for example, term papers, book chapters, tax returns. When during such a window do we deliver? And to what extent do we procrastinate?

Long story short: the answer can be, with some approximation, summed up in a formula that states that we procrastinate with a hyperbolic behaviour. If half the time has passed between inception and the deadline, we double our effort. A quarter of time left, we quadruple it. One day before deadline? We are all in. You can read the whole story, and the simple mathematic behind it, at the AIP Scitation site.

A hyperbola is a simple shape, we all are, conscious or not, familiar with it. Let me know if you recognise the shape in the next picture and link it to our common behaviour:



I shifted the unit of measure on the y-axis from “time of day” to “cumulative percentage of task done”. Recognise it now? Every day we add more and more effort until we get to the end at the deadline: it happens to any one of us!

Sometimes we can use this to work to our advantage: we could set (seemingly) impossible deadlines for us and then work to meet them. We will find a way, we can be more efficient, we can have the idea in one minute that solves a problem that was lurking there unattended in the year before we firmly committed to addressing it. This effect is called the Parkinson’s Law (Wikipedia article here): “Work contracts to fit in the time we give it.”

You may now ask what this is all about: did I just show you how messy my days are and what kind of bad procrastinator I am? Maybe, and maybe not. My point is different: let assume that we are all natural procrastinators (not difficult to believe) and let’s also assume that our days are planned the best way possible (given all otherwise immovable constraints). Now, under these tenets, would a situation like picture 3 look so bad after all? Take a look at what an ideal(ized) schedule may look like:

Procrastination 4.png


Power working at his best! Full focus on a single task until completion! Bang bang bang, delivering like no one before! …is it really that good?

This may be fitting if you work at a forge, in a restaurant, at a laundry, or generally are performing relatively brief tasks: you clearly want to do one thing at a time, start to finish. That was my favourite trick for surviving through winter Sundays in my mum’s patisserie in my hometown in the Alps: with a room overcrowded with customers, the only possible working style is to get an order, seal your ears and eyes until you prepare and deliver it, then take another one. Repeat according to entry order until everyone got satisfied. However, if you are involved in creative endeavours (artistic, invention, marketing, big projects…), this may not be the best approach. I’m an advocate for letting new concepts sit in my back brain for a while before working on them, I’ve got the feeling that my brain performs a good chunk of the work in autopilot mode. Like putting a cake in the oven: you are not really “working on the cake” while it cooks, but you can do something else and then still deliver a good cake (of course you need to put the finishing touches on it, packaging and the likes).

Probably the truth it’s in the middle: you have to choose the right approach for every different type of work, and you must give yourself a deadline in order to deliver. Otherwise, you could forever fiddle around with mental constructions and never get to actually do anything.

I’ll be surely thinking more about my procrastination problems, let’s see where it would lead me.

Until next time, you can procrastinate playing with the attached file: an Excel spreadsheet in which the formula in the article by Durakiewicz is shown working. Try changing the number of days before the deadline, look at the “daily percentage of work done” column, and let me know if it resonates with you or not. Enjoy my procrastination-calculator!