Astrophysics and Four Dimensional Geometry
We can’t visualize higher dimensions, but they are developed mathematically from axioms that are not controversial. These descriptions can be used to accurately predict astrophysical phenomena.
The tesseract is the four-dimensional analogue of a cube. A 3-d cube has square faces; a 4-d cube has tetrahedral facets.
The simplest four-dimensional shape is a 5-cell, the polytope with five tetrahedral facets. Its wireframe resembles a 2D cube with full insides.
Alicia Boole Stott
The fourth dimension of space is a new way to look at geometry, one that extends out along an axis—the w-axis—that is mutually perpendicular to the three familiar x-, y-, and z-axes. The w-axis cuts through three-dimensional space, forming a sort of hypercube or tesseract.
Alicia Boole Stott was an Irish woman who made significant contributions to four-dimensional geometry. Although she never studied mathematics, she taught herself to “see” the fourth dimension and developed a new method for visualizing four-dimensional polytopes.
She built cardboard models of regular four-dimensional polyhedra and sent them to Groningen University professor Pieter Hendrik Schoute. He approached her problems by analytical means, but she soon began to outpace him in her understanding of how these shapes worked. She published her results in 1900 and 1910, but after Schoute’s death in 1913 she set her mathematical work aside and concentrated on domestic life. She resumed her work in 1930 after her nephew introduced her to geometer Harold Scott MacDonald Coxeter.
Charles Howard Hinton
As early as the 1840s, people were beginning to consider that space might have more dimensions than they could visualize. Mobius showed that symmetrical figures could coincide if a fourth dimension were introduced, and Cayley studied n-dimensional geometry using both synthetic and analytic concepts.
Hinton was one of the first to write extensively on four-dimensional geometry. He invented a way of thinking about the higher dimensions that is still in use today, and he also coined the term “tesseract” for a four-dimensional analogue of a cube.
The idea that spaces might have more dimensions than we can see enjoyed considerable vogue in late Victorian times. The physicist Hermann von Helmholtz wrote about it, and it was even seized on by Spiritualists as a scientifically-validated means of communicating with departed souls in seances. But Hinton’s legacy has been largely forgotten. He is mentioned as a footnote in two of Iain Sinclair’s novels, and his ideas are considered in Alan Moore’s cult graphic novel From Hell.
Ludwig Schlafli (1814-1895) wrote an article in 1852 in which he discussed the four dimensions. He had already done some remarkable work with regular polyhedra such as the Platonic solids but this was to be his breakthrough.
Schlafli used a geometric method called spinorial construction. This method works for any three dimensional symmetry (Coxeter) group and allows the analogues of the Platonic solids to be constructed in four dimensions.
His article was influential because it gave the first example of a higher dimension geometry being built from the ground up and not just derived from three dimensional Euclidean geometry. This is the way that modern mathematics and physics are developed, although the non-Euclidean space of Einstein’s theory of relativity uses different methods.
His article also introduced the concept of a cross-polytope, one of the six regular convex polychora of four dimensions. These are objects which exist in four dimensions like the tesseract and can be arranged to form a cube, hexadecachoron or octahedron.
The main challenge is to project a four-dimensional object into two dimensions, with the same level of complexity that one would encounter in displaying a 3D wireframe. The resulting image must be capable of being interactively rotated, depthcued and parallel- or perspective-projected.
To accomplish this, the vertices of the 4D object are projected into the intermediate 3D region using a regular 3-dimensional projection matrix (To, From, Up and Over). Then, each 3D edge is clipped against the 2D viewport to avoid drawing any overlapping edges.
Parallel projection maintains the parallelism of lines in the original 4D shape, which is useful for accurate measurements and architectural blueprints. However, it does not provide the sense of perspective and depth commonly associated with 3D viewing that is often helpful in interpreting the structure of objects. Therefore, it may be desirable to allow the user to switch between parallel and perspective projection. This will give them a better sense of the object’s structure.