Strange shapes

Dear Lazyweb:

ghoti and I were trying to explain a strange shape to xanna and Jacob in the hope that they knew the proper geometric name for it. This is the shape of the containers that the ice lollies we like come in (old but still accurate picture, search for "Jubbly"). It's the original tetrahedron shape used by Tetra Pak, now called the Tetra Classic (indeed the pack is so labelled). It's not a regular tetrahedron, though: it's constructed from four isosceles triangles. It's most easily constructed using this net (apologies for the dodgy quality of my Inkscape use):



Fold it such that edges A meet; in doing so edges B will also meet, along with edges C.

MathWorld reckons that it's a special case of an isosceles tetrahedron, but that only requires opposite pairs of the tetrahedral edges with the same length as A to have the same length, whereas in fact four edges have the same length in this shape.

Does anyone know if there's a proper name for this polyhedron? Doubly-isosceles tetrahedron or something?