The term “B8 polytotope” does not correspond to a widely recognized, singular geometric object in the mathematical literature. No standard reference defines a specific polytope uniquely identified by this name. Consequently, the expression is ambiguous and may be used informally to denote any polytope that possesses the symmetry of the Coxeter group B₈, or to refer to members of the families of uniform polytopes generated by that group.
Possible Interpretations
| Interpretation | Description |
|---|---|
| 8‑cube (also called the octeract) | The regular convex 8‑dimensional hypercube. It is one of the two regular polytopes associated with the Coxeter group B₈, having Schläfli symbol {4,3⁶}. |
| 8‑orthoplex (also called the 8‑cross‑polytope) | The regular convex 8‑dimensional cross‑polytope, dual to the 8‑cube. It is the other regular polytope in the B₈ family, with Schläfli symbol {3⁷,4}. |
| Uniform B₈ polytopes | A series of uniform (vertex‑transitive) polytopes derived from the B₈ Coxeter group by various Wythoff constructions. This includes the 8‑cube, 8‑orthoplex, several truncations, cantellations, and other derived forms. |
| Other context‑specific uses | In some texts, “B8 polytope” might be shorthand for a specific member of the B₈ family chosen by the author (e.g., a particular truncation). Without explicit definition, the meaning remains unclear. |
Relation to Coxeter Group B₈
The Coxeter group B₈ (also denoted C₈) is a reflection group acting in eight‑dimensional Euclidean space. Its Coxeter‑Dynkin diagram consists of eight nodes arranged linearly, with one branch labeled “4” indicating a dihedral angle of 45°. Polytopes invariant under this group’s reflections are often catalogued under the “B₈ family.” The regular members—8‑cube and 8‑orthoplex—serve as the fundamental examples.
Lack of Established Usage
Because “B8 polytope” is not a standard term, scholarly sources do not provide a dedicated definition, historical background, or unique properties tied to that exact phrase. Researchers typically refer to the specific polytopes (e.g., “8‑cube”) rather than using the generic label.
Note: The information above reflects common interpretations of the phrase based on established mathematical concepts related to the B₈ Coxeter group. Accurate information about a distinct object named “B8 polytope” is not confirmed.