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I wish I had a bathroom to tile - I reckon this could be considered "in vogue" for the next 30 years or so, until they find a newer pentagon. Does anyone know if this can be coloured with 3 colours? Obviously 4 is possible due to the 4 colour theorem and 2 will not work due to to three faces sharing a corner.

> Obviously 4 is possible due to the 4 colour theorem...

Unless your bathroom includes a loop, such as all four walls (even with holes for windows and doors) or over the ceiling. Then the coloured area is no longer a plane, and so the 4 colour theorem does not apply.

Wouldn't all four walls still function as a plane, topologically? Think about looking into a cube (box) with one side open, so that you see five faces. The projection of those faces onto your retina or a photograph is a direct mapping to a plane.

Four walls plus the ceiling are equivalent to that open box. Four walls minus the ceiling are equivalent to a plane with a hole.

You have to include both the ceiling and the floor to break out of plane topology. And what you get is a sphere.

A sphere is also four-colorable, functioning equivalently to a plane for such purposes; consider stereographically projecting the sphere (minus a point) onto a plane.

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