The standard model of Mercator projection shows a cylinder wrapped around a spherical earth eg [Wiki](http://en.wikipedia.org/wiki/Mercator_projection). 
[Many sites](http://docs.openlayers.org/library/spherical_mercator.html) describe the resulting square map like this:
"...spherical Mercator maps use an extent of the world from -180 to 180 longitude, and from -85.0511 to 85.0511 latitude. ... a cutoff in the north-south direction is required, and this particular cutoff results in a perfect square of projected meters."
This would result in a mapping from degrees latitude (Φ) to Y from the X axis of Y = R.tan(Φ), but this does not return 85.0511 as the angle for which the map is a square where Y= 2 Π R
The standard mapping equation provided in the literature is Y = R ln (tan( Π/4 + Φ/2)). 
I am looking for a physical interpretation of this formula, as it is certainly not the classical one of a sphere inside a cylinder.
Can anyone throw some light please?











Deriving the Mercator projection algorithm
In a [webpage](http://www.math.ubc.ca/~israel/m103/mercator/mercator.html) on Mercator's projection, [Robert Israel](https://mathoverflow.net/users/13650/robert-israel) offers the following suggestion:



> 
> If you want a physical model of
>  Mercator's projection, let the globe
>  be a spherical balloon that is blown
>  up inside the cylinder, and sticks to
>  the cylinder when it comes into
>  contact with it.
> 
> 
> 


