Nomenclature

Historically, polytopes have been given names using Ancient Greek roots. As the study of polytopes has widened in scope, various naming schemes have been devised for naming new types of polytopes.

Generic names
The most general method of naming a polytope is according to their number of facets (edges for polygons, faces for polyhedra, cells for polychora, etc.) and their dimensionality. This is the most common naming scheme for convex regular polytopes. Examples of names formed in this way include pentagon, dodecahedron, and hexacosichoron.

Numeral prefixes
The numeral prefixes used for naming polytopes are not entirely consistent. There are multiple competing systems which coexist.

Wikipedian system
Articles about polygons on Wikipedia generally follow this system., which is in accordance with the consensus among most mathematicians. Prefixes for larger numbers of facets generally list digits in order from most to least significant. This is done by combining the prefixes for each "digit" in the decimal representation of the number of facets individually. Larger multiples of one thousand are named using multiplicative prefixes derived with -kis-; this is potentially problematic for generating polytope names as -kis- has a separate meaning.

Wikipedia also refers to a one million sided polygon as a megagon, which uses the S.I. prefix mega- to mean one million.

Modern Greek based system
One option for deriving new numeral prefixes is to use Modern Greek numerals, rather than Ancient Greek. This has the benefit of being more consistent, as Ancient Greek numerals could typically be represented in more ways than their Modern Greek counterparts. Prefixes may be derived from Modern Greek words for numbers, altered to reflect their Ancient Greek origins. As in the Wikipedian system, Modern Greek speakers say numbers with their digits in order from most to least significant. For example, twenty one is είκοσι ένα (eíkosi éna), which may be turned into the prefix icosihena-. For multiples of one thousand, the number one thousand, χίλια (chília) is written in the plural, χιλιάδες (chiliádes). For plurals of larger powers of ten, replace -ιο (-io) with -ια (-ia). The Greek word for million, εκατομμύριο (ekatommýrio), literally means "hundred myriad". Greek speakers generally say "one hundred myriad" ένα εκατομμύριο (éna ekatommýrio), and not just "hundred myriad", but for prefixes the "one" can be removed.

Dimensional suffixes
The suffix used for a polytope indicates its dimensionality.

Modifications of polytopes
Truncation: expands edges outwards Cantellation (rhombi-): expands faces outwards

Runcination (prismato-): expands cells outwards

Sterication (celli-): expands tera outwards

Rectification: contracts edges to points while retaining their positions

Quasi-: expands inwards instead of outwards

For example, a "quasirhombicuboctahedron" is created by expanding the faces of either the cube or the octahedron inwards.