3-sphere two 3-balls. This construction is analogous to a construction of a 2-sphere, performed by gluing the boundaries of a pair of disks. A disk is a 2-ball, and the boundary of a disk is a circle (a 1-sphere). Let a pair of disks be of the same diameter. Superpose them and glue corresponding points on their boundaries. Again one may think of the third dimension as temperature. Likewise, we may inflate the 2-sphere, moving the pair of disks to become the northern and southern hemispheres. After removing a single point from the 2-sphere, what remains is homeomorphic to the Euclidean
Sublunary sphere - the idea usually going hand in hand with geocentrism and the concept of a spherical Earth. Avicenna carried forward into the Middle Ages the Aristotelian idea of generation and corruption being limited to the sublunary sphere. Medieval scholastics like Thomas Aquinas - who charted the division between celestial and sublunary spheres in his work "Summa Theologica" - also drew on Cicero and Lucan for an awareness of the great frontier between Nature and Sky, sublunary and aetheric spheres. The result for medieval/Renaissance mentalities was a pervasive awareness of the existence, at the Moon, of what C.S. Lewis called 'this