Definitions
from The American Heritage® Dictionary of the English Language, 5th Edition.
 noun Any of a set of objects resulting from the generalization of a twodimensional square and a threedimensional cube to n dimensions. A hypercube has 2^{n} corners, each of which is connected to its n nearest adjacent corners by edges that all have the same length.
 noun A network whose nodes have the connectivity of such an object.
from the GNU version of the Collaborative International Dictionary of English.
 noun A mathematical object existing in more than three dimensions, analogous to the cube in that each twodimensional facet of the surface is a square; a generalization of a cube in more than three dimensions.
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 noun geometry A
geometric figure in four or more dimensions, which isanalogous to a cube in threedimensions . Specifically, the ndimensional equivalent of acube for any nonnegative integer n.  noun geometry Such a figure in four dimensions; a
tesseract .  noun computing A
computer architecture in which eachprocessor is connected to n others based onanalogy to a hypercube of n dimensions.
Etymologies
from Wiktionary, Creative Commons Attribution/ShareAlike License
Examples

Abstract: A representation of the genetic code as a six – dimensional Boolean hypercube is proposed.

Hyperspace means a space with more than three dimensions, and a hypercube is a cube in more than three dimensions.

Hyperspace means a space with more than three dimensions, and a hypercube is a cube in more than three dimensions.

The hypercube is a very particular case of the hyperprism.

For example, if we want to visualize the 4dimensional equivalent of a cube (called a "hypercube", with 16 nodes and 32 edges), we can get a feel for it by "stereographically projecting" it into 3dimensions:

For example, if we want to visualize the 4dimensional equivalent of a cube (called a "hypercube", with 16 nodes and 32 edges), we can get a feel for it by "stereographically projecting" it into 3dimensions:

Twitter search for "hypercube" and you'll see the naysayers like

: P matociquala: It might be a hypercube and have LOTS of corners hawkwing_lb: (the one shape heaven shouldn't be is a pyramid)

Silently, he ushered him through a door marked with the twodimensional shadow of a hypercube.
365 tomorrows » Sam Clough : A New Free Flash Fiction SciFi Story Every Day

Gwiazda has created a kind of 4th dimensional hypercube format for some of these images, which allows him to imagine that he can simultaneously see in front, to both sides, and behind him.
whichbe commented on the word hypercube
Based on the Tesseract model. The concept of a tesseract, or four dimensional cube, is often used as an illustration of time travel, and might also give a clue to how a warp in time might occur. As the tesseract constantly moves, collapses in on itself and then expands again in a never ending loop, so does time move forward in a never ending line. If, however, an object or person was able to move from one line of this tesseract to another at any junction point, they would in essence be moving not with time, but in a straight line perpendicular to it. This movement has led many science fiction authors to juxtapose the thought not of time travel, but of being able to stop time and then start it again.
(Wikipedia)
May 21, 2008
asativum commented on the word hypercube
In mathematics, I believe a hypercube is a fourdimensional surface; in one sense it is to an ordinary cube what a cube is to a twodimensionsal square. Like other 4D objects, when rotated in 3space, it appears to change shape, but I'm pretty sure that's the only way time is involved. You can perhaps imagine your way to the timealtering speculation from here.
May 22, 2008