The platonic solid built with 4 triangles to every vertex is the Octahedron (8 faces). If you put a vertex on every face and connect them, you get its dual, a Cube. 

If you imagine the structure as of its faces and unfold it, you get its net. 

As part of the Platonic Solids series, this model has clearly defined colors outlining the solid itself, scaffolding in black, and its dual in white.