The standard model of Mercator projection shows a cylinder wrapped around a spherical earth eg Wiki.

Many sites 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?