This polyhedron can be made in 3 colors. Use one color for the decagon faces, a second color for the square faces and the third color for the hexagonal faces.
To see the full-sized template click on this image:
It is interesting to compare this model with the icosidodecahedron.
Notice that each of the pentagonal faces have had their vertices truncated to form decagons. Each of the triangular faces have been truncated to form hexagons. The shape of all of the vertice truncations is where the four shapes have met creating a square.
The large decagon faces receive a lot of stress since the polygon is trying to become equally spherical. It is best to make this polygon small or use very stiff paper for the decagons.
Begin with a decagon face and surround it alternately with squares and hexagons. Surround each hexagon face alternately with squares and decagons. And, surround each square face alternately with hexagons and decagons.
To complete this model you will need a total of 12 decagons, 30 squares, and 20 hexagons.