Some geometrical aspects of 3- and 4-spaces

A couple of abstracts about the geomtery of space: Historically, there have been many attempts to produce the appropriate mathematical formalism for modeling the nature of physical space, such as Euclid's geometry, Descartes' system of Cartesian coordinates, the Argand plane, Hamilton's quaternions, Gibbs' vector system using the dot and cross products. We illustrate however, that Clifford's geometric algebra (GA) provides the most elegant description of physical space. Supporting this conclusion, we firstly show how geometric algebra subsumes the key elements of the competing formalisms and secondly we show how it provides an intuitive representation and manipulation of the basic concepts of points, lines, areas and volumes. We also provide two examples where GA has been found to provide an improved description of two key physical phenomena, electromagnetism and quantum theory, without using tensors or complex vector spaces. This paper also provides pedagogical tutorial-style coverage of the various basic applications of geometric algebra in physics.
James M. Chappell, Azhar Iqbal & Derek Abbott (2011). Geometric Algebra: A natural representation of three-space, arXiv: We indicate that Heron's formula (which relates the square of the area of a triangle to a quartic function of its edge lengths) can be interpreted as a scissors congruence in 4-dimensional space. In the process of demonstrating this, we examine a number of decompositions of hypercubes, hyper-parallelograms, and other elementary 4-dimensional solids.
J. Scott Carter & David A. Mullens (2015). Some Elementary Aspects of 4-dimensional Geometry, arXiv:
There's also a minimalistic introduction to euclidean planes.