No, no, ‘iamond’ is no typing error, these things are called like this and are nothing else than shapes composed from equilateral triangles.
Here somebody went the (for me crucial) step further and made nice lantern diagrams from.
I still must find out whether they really can be used as lanterns.
The paper lampshade on the other hand is also made from nearly equilateral triangles.
Links:
Iamond Lantern (iamonds)
The actual folding diagram (GIF file, ca. 16 KB)
About shapes made from equilateral triangles (Polyiamond)
Here at unikatissima:
Paper Lampshade