The Gnomonic projection is one which has a greater expansion, away from the origin, than a conformal projection. It gives quite a distorted apperance, therefore:
However, it has one very useful special property.
It is constructed by projecting every point on the globe appearing in the map onto the plane of the map from an imagined light source in the center of the globe. Great circles on the globe are all bands like the Equator, which are defined by planes cutting the globe that intersect the center of the globe.
Any plane intersecting a plane produces a straight line.
Hence, on the gnomonic projection, all great circles are represented by straight lines, making it very useful in plotting great circle routes between arbitrary destinations. As has doubtless often been said, it is the navigational chart for the air age much as the Mercator was the navigational chart for the age of sail.
It can also be employed fairly simply for projecting the world onto the surface of various polyhedra.
Here is a simple example, the world projected onto a cube using the Gnomonic projection.