- Klein bottle (Wikipedia)
In mathematics, the Klein bottle (/ˈklaɪn/) is an example of a non-orientable surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. More formally, the Klein bottle is a two-dimensional manifold on which one cannot define a normal vector at each point that varies continuously over the whole manifold. Other related non-orientable surfaces include the Möbius strip and the real projective plane. While a Möbius strip is a surface with a boundary, a Klein bottle has no boundary. For comparison, a sphere is an orientable surface with no boundary.
- Cathedral (Wikipedia)
A cathedral is a church that contains the cathedra (Latin for ‘seat’) of a bishop, thus serving as the central church of a diocese, conference, or episcopate. Churches with the function of “cathedral” are usually specific to those Christian denominations with an episcopal hierarchy, such as the Catholic, Eastern Orthodox, Anglican, and some Lutheran churches. Church buildings embodying the functions of a cathedral first appeared in Italy, Gaul, Spain, and North Africa in the 4th century, but cathedrals did not become universal within the Western Catholic Church until the 12th century, by which time they had developed architectural forms, institutional structures, and legal identities distinct from parish churches, monastic churches, and episcopal residences. The cathedral is more important in the hierarchy than the church because it is from the cathedral that the bishop governs the area under his or her administrative authority
