Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. The set of all points in a given hyperspace that are at a given distance from a given point.
 n. A zorb; the act of zorbing, entering a zorbing ball, strapping into a harness, and rolling down a hill.
from the GNU version of the Collaborative International Dictionary of English
 n. A mathematical object existing in more than three dimensions, analogous to the sphere in that all points on the surface are equidistant from the central point; a generalization of a sphere in more than three dimensions.
from The Century Dictionary and Cyclopedia
 n. Equidistantial surface.
 n. In fourdimensional space, the threedimensional quadric spaces through the intersection of X5 = 0 and where is a threedimensional space which is met by any line of S4 in two points.
Etymologies
Examples

Sometimes the term hypersphere is used to denote the hypersolid, the portion of fourspace inclosed by this locus, which is then called the boundary or hypersurface of the hypersphere.

Either that or it has something to do with the hypersphere as described by Poincaré, making the universe finite but unbounded.

Its probably more like a finite hypersphere 4dimensional spherelikething.
Astronomers Find Black Holes Do Not Absorb Dark Matter  Universe Today

It is relevant to discussions between Sam, myself, Lawrence, John, and a few others earlier in the thread, considering a hypersphere universe and the merging of GR/SR/QM.

Make arbitrary Earthstyle coordinates for a hypersphere.

~ Also from Joe: Kosmic Coordinates on the Kronos mandala or, the birth of a hypersphere.

She never heard the ringing in a locked safe of the hypersphere shaking from the shock of the music of a siren.

If the mansion stood on a hypersphere slightly greater in diameter than the mansion grounds, a person could move from any point to any other with what, in threespace, would seem to be right angles.

For a moment, I gave off a shaking shock wave similar to what the hypersphere had done; but something was carried with it.

The shock waves the hypersphere gave off were only "sound" along one surface (hypersurface?) of the concentrically expanding ripples of pressure.
john commented on the word hypersphere
Reminds me of Rounder Records.
July 19, 2009
sionnach commented on the word hypersphere
in contrast to the hyposphere, which is manifestly deficient in sphericity, a bit like a hedgehog being stretched on the rack by some tziganoporcaphobic Torquemada.
July 19, 2009
bilby commented on the word hypersphere
Extremely spherical.
July 19, 2009