This is where the computational aspects of Real Spaces will be developed. Briefly Real Spaces differ from Euclidean and Riemann spaces, in that they alter the common notion of dimension. In Euclidean or Riemann Geometry, the dimensionality of the dimensional space is always one. So for instance in Euclidean 3-space, the dimensions consist of three linear spaces, in Euclidean 4-space, the dimensions consist of four linear spaces, etc. The change to Real Spaces is simple but has profound effects. This change was first elucidated by Dr. S.K. Kapoor, through his work on Vedic Geometry, the traditional approach to geometry associated with the traditional, or Vedic, civilisation of India. According to Dr. Kapoor's writings the difference in order between a space, or domain, and it's dimension is of order 2. Therefore Real 3-space is similar to Euclidean 3-space, in that it has dimensions of order 1, however Real 4-space has dimensions of order 2, they are planar, whilst Real 5-space has dimensions of order 3, they are solid in nature, and Real 6-space has hyper-solid dimensions. This simple change radically alters how certain properties of such spaces are developed, in particular it alters the construction of a hyper-sphere, in higher spaces, since there is no obvious way to construct a metric. My initial considerations in this regard are contained in my partially completed paper, The Axiom of Dimensionality, other considerations are posted in the Higher Geometry Forum, and the Advanced Geometry Research group. If you are unfamiliar with Real Space geometry it is worthwhile perusing these references. It is also worthwhile obtaining a copy of Dr. Kapoor's book Vedic Geometry, as it is based on the material contained within it that I will begin the process of developing the transformations required to construct the hyper-sphere in Real 4-space. These will be done computationally using Python, and also symbolically by developing and extending the standard tesor approach to Euclidean manifolds. As a prerequisit to this development I would suggest that you go through the associated topic of Computational Triples, in order to get a flavour of the development of a class based computational system in geometry.

Jumping into Computational Real Geometry

A structured point

A Dynamic Point