This is the hemisphere model of the hyperbolic plane, patterned with the (7,3,2) triangle group. Using a flashlight in the appropriate positions, the Poincaré disk, Klein and upper half space models of the hyperbolic plane can be obtained as projections. For best results, use parallel light rays (e.g. the sun) for the Klein model, and a point light source (e.g. http://maglite.com/shop/mini-maglite-led-2-cell-aaa-flashlight.html
in "candle mode") for the Poincaré disk and upper half space models. There is a "socket" on the model for the upper half space model projection that the Mini Maglite® AAA LED flashlight fits into.
Note: this is a smaller, less detailed version than the one shown in the video here. Larger version