hy·per·sphere (h??p?r-sfîr?)n. Any of a set of objects resulting from the generalization of a two-dimensional circle and a three-dimensional sphere to n dimensions. In n-dimensional space, a hypersphere is the set of all points that are a given distance, called the radius, from a given point, called the center.American Heritage® Dictionary of the English Language, Fifth Edition. Copyright © 2016 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company. All rights reserved.