Geometries As Expressions Of Pj Problems
Peter Oye Sagay
Space, space everywhere; not a single point is known. The first primitive humans must have found the awareness of space very perplexing (imagine being thrown into a space about which you know nothing). Modern humans have become a lot less nervous about space because they now know a lot about space. For example, they are able to determine the coordinates and characteristics of spaces on earth and some spaces beyond earth. Geometry is the key knowledge that allowed humans to successfully journey from a point in time when no single point in Space was known in the context of a coordinate system, to a time when almost all points in Space are known.
What is geometry? Partitions of Space abound, so it is not surprising that there are various geometries (Eulidean, Non-Euclidian, Affine, Projective, Spherical, Inversive, etc) and as a result, several definitions of geometry. Nonetheless, there is a commonality about what geometry does: it represents a space and the possible transformations in the space. The representation is usually depicted by one or more number lines. The number of number lines indicate the dimension (usually denoted by the letter n) of the space. For example, there is the Euclidean line (a 1-dimensional Euclidean space where n =1); the Euclidean plane (a 2-dimensional Euclidean space where n=2) and the 3-dimensional Euclidean space (where n = 3). The properties of the space that are unaltered by any of the possible transformations in the space constitute its geometric properties. For example, in the Euclidean space, the possible transformations preserve lengths.
The containership property of space is common to all geometries. In fact, geometries exist for the purpose of analyzing and effecting containerships and the possible transformations of the containerships.
The number lines that depict geometries; the containership of the spaces the geometries represent; and the possible transformations in the spaces are expressions of Pj Problems. Consequently, geometries are expressions of Pj Problems