Welcome to the Polyscheme learning project at Wikiversity on Euclidean spaces of 4 or more dimensions. This article is about one of the regular polytopes or conceptual objects which resides in higher-dimensional space. It is a commentary on the Wikipedia Polyscheme article, providing learning resources that complement the encyclopedia, but do not replace its article. Some of the expanded content may be unsupported by references, and some may be opinion, not established fact, as of this date of publication. Participants should feel free to ask questions and propose corrections or additions on its discuss page.
Polyscheme is the name given to geometric objects of any number of dimensions (polytopes) by Ludwig Schläfli, the Swiss mathematician who discovered all the regular polytopes which exist in the higher dimensions of Euclidean space before 1853, at "a time when Cayley, Grassmann[b] and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."[1] The Wikiversity was hosted on very slow servers in those days, and other researchers also discovered the 4-polytopes before Schläfli's article was published posthumously in 1901, but Schläfli is the founding author of the Polyscheme learning project. H.S.M. Coxeter is its founding editor, whose 1948 book Regular Polytopes tells the whole story of the project.
The Polyscheme project is intended to be a series of wiki-format articles on the regular polytopes, the fourth spatial dimension, and the general dimensional analogy of Euclidean and spherical spaces of any number of dimensions. This series of articles expands the corresponding Wikipedia encyclopedia articles to book length, to provide textbook-like treatment of the subject in depth, additional learning resources, and a subject-wide web of cross-linked explanatory footnotes which pop-up in context.[c]
Some of what is in these companion articles is opinion, not established fact, as of this date of publication, and some of it is just commentary, not essential fact. The commentary and recent research is precisely the difference between the learning project article and the corresponding encyclopedia article; you can compare them to detect it, or just read the encyclopedia instead if you don't trust it.
Most project articles are an annotated and expanded version of the Wikipedia article which they replace for learning purposes. Some project articles, however, do not reproduce the Wikipedia article, and are only a commentary on it. Participants are directed by a banner to "See also" the Wikipedia article when reading these commentaries.
Polyschemes have been a subject of active and ongoing research since their discovery by a Swiss researcher around 1850. But for the first 50 years of its history Ludwig Schläfli's paper on the subject was unpublished, entirely inaccessible to other researchers. Even after its publication, Schläfli's paper remained obscure for another 50 years, in part because the mathematics it contained was only accessible to a few mathematicans who could read that language. H.S.M Coxeter finally made the subject widely accessible in his 1948 book Regular Polytopes, which synthesized all the research that had been published since Schläfli and added Coxeter's discoveries, including his invention of the theory of reflecting symmetry groups, the group theory mathematics that underlies geometry. Since then, Coxeter's book has been the encyclopedia of Euclidean geometry, and every polyscheme researcher has been able to begin with it instead of reinventing the wheel, and contribute new chapters to it.
↑Grassmann moved on later in life from inventing a theory of mathematics to inventing a theory of linguistics. He reached the understanding that the true origin story of human languages is found in their common symmetries, which are intrinsic properties discovered in nature, not invented, rather than in the history of our common human linguistic experience.
↑In 1844, Grassmann proposed a new foundation for all of mathematics, the idea of vector spaces. He showed that once geometry is put into the algebraic form he advocated now known as the Grassmannian, the number three has no privileged role as the number of spatial dimensions; the number of possible dimensions is in fact unbounded. Even deeper than his invention of a language of mathematics was Grassmann's foundational role in the science of all languages.[a]
↑Arthur Fry with a Post-it note on his foreheadIf you hover the cursor over a footnote it will pop up in a floating box like a post-it note, so you can quickly get a deeper explanation of a term or a sentence you can't parse. If you click on the explanatory footnote, you can read a larger note like this one where it appears in the Notes section, below, like a mini-article within this article, which may occur in other Polyscheme project articles as well in some cases. The notes are a subject-specific hypertext of polyscheme concepts, a wiki within a wiki. From the Notes section you can see all the places where this explanatory note is cited in this article, and even go there yourself if you want to understand what depends on this concept. Many of the explanatory notes contain footnote references themselves, to other explanatory notes. You can go as far down this rabbit hole as you need to go for comprehension, but beware of getting lost underground in a maze of little tunnels! At least with footnotes there is no danger of leaving the article altogether, and never coming back to finish what you started.
↑Coxeter 1973, pp. 141-144, §7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853."
