The simplest of the five regular solids: four equilateral triangles on six edges, meeting four at a time. Plato gives it to fire, on the argument that fire is the sharpest and most penetrating of the bodies, and the tetrahedron is the sharpest solid, with the smallest volume for its surface. It is the only Platonic solid that is its own dual.
Sacred Ratio
1:3 (insphere radius to circumsphere radius)
Construction
Take an equilateral triangle and raise a fourth point above its centre until the three new edges equal the original three. Euclid constructs it inscribed in a sphere at Elements XIII.13, and the whole family at XIII.13-17. Its faces meet three at each vertex, so the vertex figure is a triangle.