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
YIG sphere out of the sphere; the loops are half-turns, positioned at right-angles to each other to prevent direct electromagnetic coupling between them and each is grounded at one end. The field from an electromagnet changes the resonance frequency of the sphere and hence the frequency it will allow to pass. The advantage of this type of filter is that the garnet can be tuned over a very wide frequency range by varying the strength of the magnetic field. Some filters can be tuned from 3 GHz up to 50 GHz. YIG filters usually consist of several coupled stages, each stage consisting