Hyper is a process, not a result. Specifically, it means "over", and refers to increasing dimensions. That is, three dimensions is hyperspace to two dimensions. 4-space should be called "teraspace".
"diamond" means many things to many people. For example, i gloss t-diamond in my polygloss as a packing of spheres in n dimensions. This is how the diamond atoms are arranged.
Alternately, you can consider the octahedral or tegum-product, or assorted lace-prisms and lace-tegums, many of which resemble cut diamonds. The antitegums are very much like the typical gem stone.
The PG can be found at:
http://www.geocities.com/os2fan2/gloss/index.html