This is a conformal projection of the Earth on a circle, except that angles are halved at the poles instead of being faithful there.
It might easily be mistaken for the Van der Grinten projection, but this projection is actually conformal, and is not merely a conventional projection. (One way to tell them apart is to note that the spacing between the meridians increases towards the sides of the map.)
The way in which this projection was derived can be explained in a simple way without too much mathematics.
The trick behind it is this:
Take two Mercator maps, one at exactly half the scale of the other. Copy the smaller one onto the larger one, with them aligned so that their equators coincide.
Now, using the larger map to guide the relationship between each point and the globe, copy the traced version of the smaller map to the globe. It will only fill one hemisphere of the globe, since the smaller map is half as wide as the larger map.
Make a stereographic projection of the image of the smaller map on the globe.
Since each of the mappings performed along the way was conformal, the sum total of all of them is conformal as well, because angles were preserved every step of the way.
What appears to be a somewhat similar projection, but based on reducing the starting Mercator map to a size slightly larger than half the size of the one on which it is placed, is noted on the web site of the famed musician Wendy Carlos, who is well remembered for the popular album "Switched-On Bach".
The idea of conformally mapping the globe to half of itself, the basic step which allows this projection to be produced, was referred to in a Mathematical Games column by Martin Gardner in Scientific American magazine on unusual map projections, where the final step of copying from the globe to the Stereographic projection was not taken, but instead an actual globe with the world on it with that transformation was made. It was noted that many people found it difficult to notice there was anything unusual about that globe. This is what such a globe would look like:
That many people did not notice anything wrong could be simply because conformality eliminates the most visually obvious type of error, or perhaps because of the baneful influence of the Mercator projection, which has led many to suppose that areas near the poles are larger than their actual size.