The Breather Equation

Credit photo: Breather & Unsplash

Credit photo: Breather & Unsplash 

Breather is a cool service where you can rent a space by the hour with an app and unlock the door with your phone! How cool is that!

As I write this post, they have more than 150 awesome locations in 6 cities and distributed in 101 commercial buildings.

Recently, CEO Julien Smith ask publicly (on Facebook it’s pretty much public): how many time it would take them to reach 800 buildings.

Challenge accepted Julien. So here is my answer.

Data Gathering

First, I scrapped trough your insightfull blog and all your newsletters that I received since 2013 (Yes, I am a keeper) for information on launch dates for breather spaces. I only found out 56 of them. Is that because some are kept secret?

Anyways, I made some fair estimations on real growth time to reach 101 buildings in 27 months.

Breather Solution for the nonlienear Schrödinger equation

Wikipedia will teach us that:

In physics, a breather is a nonlinearwave in which energy concentrates in a localized and oscillatory fashion. This contradicts with the expectations derived from the corresponding linear system for infinitesimalamplitudes, which tends towards an even distribution of initially localized energy.

A discrete breather is a breather solution on a nonlinear lattice.

 u = \left( \frac{2\, b^2 \cosh(\theta) + 2\, i\, b\, \sqrt{2-b^2}\; \sinh(\theta)} {2\, \cosh(\theta)-\sqrt{2}\,\sqrt{2-b^2} \cos(a\, b\, x)} - 1 \right)\; a\; \exp(i\, a^2\, t) \quad\text{with}\quad \theta=a^2\,b\,\sqrt{2-b^2}\;t,

Unfortunately, although it seems pretty much interesting, this has nothing to do with our actual problem.

Bacteria Growth Analogy with a Sigmoid Function

This time is for real. Now let’s assume for a moment that Julien’s plan to conquer the world will react as a bacteria population!

Because as people tend to forget is that at a certain point in time, Breather Spaces will start to disappear: an end of a lease, a fire, a company moving out, a less profitable location, competition for the food (or the locations) and so on. So as time goes by, growth will become harder and harder.

Bacteria growth will follow a sigmoid function similar to this one: f(x)=\frac{1}{1 + e^{-\lambda x}}


The general form of this equation is:



By setting some parameters in the equation, I ended up with this equation:


where A = 4, B = 200, C = 1, D = -8. x=t



[UPDATE 2017-02-22]

Still working!

[UPDATE 2017-05-24]

Slighlty behind schedule. According to the model, they should have reach 415 location by now. They are at 381. The next inflexion point should be at Month 50 meaing by the end of 2017.

[UPDATE 2017-07-02]

I made a correction in my month calculation. It seems that I wasa bit « off ». Here a the latest results:

[UPDATE 2018-03-08]

It’s the first time that my calculation is wrong. What’s happening! Let’s wait another 6 month to see if it is just a glitch.

Google Spread Sheet Available HERE

Actual Growth Rate

Later on, you said that:

We are currently adding a unit a day

Surprinsingly, (or not?), it fits this model. I predict that for the next year (between month 28 and 40) , Breather will launch 240 locations (there are 240 weekdays in a year!)


So with this empirical model, my prediction is that Breather will reach 800 commercial buildings within 100 months from the initial launch which I dated December 2013. So that should brings us to:

March 2022

Let’s face it, Microbes ruled the world and they will outlast us all one day!

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