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, then reconnects them with more edges
 * Cantellation (rhombi-): expands faces outwards, then reconnects them with squares
 * Runcination (prismato-): expands cells outwards, then reconnects them with polygonal prisms
 * Sterication (celli-): expands tera outwards, then reconnects them with polyhedral prisms
 * Quasi-: expands inwards instead of outwards
 * For example, a "quasirhombicuboctahedron" is created by expanding the faces of either the cube or the octahedron inwards.
 * Rectification: contracts edges to points while retaining their positions

Bowers-style acronyms
Bowers-style acronyms attempt to shorten the names of polytopes down to a more pronounceable size, which becomes more and more necessary as the names become longer. This is typically done by taking a few letters from each of the words in the generic name and concatenating them in the order they were found in, adding vowels in between if pronunciation would be difficult otherwise. For example, "tesseract" becomes "tes" and "hexadecachoron" becomes "hex," while the Truncation modification is denoted with a "t," as seen in the truncated hexadecachoron becoming "thex."

While this method produces short, relatively pronounceable names, the choices behind them can seem arbitrary, and some information is lost when shortening the generic name. Some of the names are also very similar (such as giffophi, gifophi, and gifophix).

Conway polyhedron notation
This notation, invented by John Conway, represents the "starting" polyhedron with an uppercase letter (such as T for tetrahedron, O for octahedron, C for cube), and uses single lowercase letters preceding it to represent modifications to it. These modifications may be as simple as truncation or taking the dual, or they might split existing faces into many new faces, edges, and vertices. For example, "rectified truncated icosahedron" is written as atI.

This nomenclature has trouble describing polyhedra that aren't highly symmetric modifications of relatively simple polyhedra. It has also not been adapted to higher dimensions, and thus is not widely used. A visualization tool that makes use of it can be found here.

Stewart's toroid notation
In his exploration of toroidal polyhedra, Bonnie Stewart created a shorthand for the regular-faced polyhedra that he used as "building blocks," with a number and letter representing each used polyhedron. The letter is based on what "family" the polyhedron is a part of (for example, P for prisms, Y for pyramids, or J for Johnson solids). The number specifies the polyhedron further, usually representing the polyhedron's symmetry group or position in a list. Many of these shorthands, along with syntax that details how they're connected, are used to describe the structure of a more complicated polyhedron.

Like the Conway polyhedron notation above, this notation was made to tackle a specific problem, and ignores most others. It has also not been adapted to higher dimensions, and thus is not widely used.

Elemental naming scheme
This method assigns prefixes to polytopes based on the family of symmetry groups that the polytope belongs to. The prefixes correspond to the associations of Platonic solids with classical elements. A dimensional suffix is assigned as described above. Truncated polytopes are notated by adding an infix, such as "-runcicanti-," "-peri-," or "-meso-." It avoids confusion between unrelated polytopes whose names would otherwise sound similar, such as the dodecahedron and dodecateron, the hexahedron (cube) and hexateron, or the octahedron and octachoron (tesseract).

This nomenclature is not widely used outside of the Hi.gher.space community.

Tapertopic notation
This notation, originating in the Hi.gher.space community, describes the higher-dimensional analogues of cubes, spheres, and cones. Numbers represent hyperspheres (1 being a dyad, 2 being a circle, and so on). If there are multiple such numbers, then the prism product of them all is taken (so a cube would be "111" and a cylinder would be "21"). Superscripts represent "taperings" (the operation that makes cone-like shapes - for example, 21 represents a cone.)

Toratopic notation
This notation, originating in the Hi.gher.space community, describes the curved objects that are the higher-dimensional analogues of the torus. The capital letter "I" represents a digon. Parentheses around a set of "I"s represent a "spheration."

The number of "I"s shows the dimension of the object. A square is II, a circle is (II), a cylinder is (II)I, and a torus is ((II)I).

Bracket notation
This notation, originating in the Hi.gher.space community, describes the higher-dimensional analogues of cubes, spheres, and tegums (or bipyramids). It uses different kinds of brackets around groups of "I"s and other brackets to denote the products that produce these shapes.

Square brackets around things make a prism product of them. This is the equivalent of writing them in sequence in the tapertopic notation above. A cube is written as [III] here.

Parentheses around things make a sphere-like product of them. Parentheses around a group of $n$ "I"s is the equivalent of writing a number greater than 1 in tapertopic notation. A cylinder is written as [(II)I] here. When tegum products aren't used, this notation resembles the toratopic notation above.

Chevrons around things make a tegum product of them. For example,  represents the octahedron, which can be viewed as a point that the tegum product is applied to thrice.