This map, published by mathematician Steve Waterman in 2013, depicts the world using Waterman's butterfly projection, which he devised himself in 1996.
This projection is influenced by Cahill's butterfly projection, first defined by Bernard Cahill in 1909. Both projections are based on an octahedron, but despite having many similarities, they differ in the mathematics behind them. This leads to Cahill's projection being conformal (it maintains shapes), while Waterman's seeks a compromise between shapes, angles and surfaces.
This 2012 map also truncates the southern part of the planet, creating a space dedicated to Antarctica. In this way, Waterman manages to represent all contiguous land masses, establishing the cuts only in maritime areas.
You can read more about map projections in this article: Cartographic projections: distorting the Earth in order to represent it
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