Equilateral Dimension¶

Measure how large an exactly pairwise-equidistant subset a metric space can support, with the distance scale and attainment convention stated explicitly.

Version: v2 · 2026-08-30 · History
Domain-specific #: 1778
Origin domain: mathematics
Subdomain: metric geometry
Aliases: Equilateral number

Core Idea¶

The Equilateral Dimension of a metric space measures the largest size of an exactly equidistant configuration that the space can support. A subset S of (X,d) is lambda-equilateral, for lambda>0, when.

d(x,y)=lambda for every distinct x,y in S`.

For a fixed distance lambda, the careful convention writes e_lambda(X) for the largest cardinality when a largest set exists and writes e_lambda(X)=infinity when no finite largest size exists. In a general metric space, the distance scale matters. A scale-free version may be stated as.

Scope of Application¶

Equilateral dimension organizes extremal questions across several parts of metric and discrete geometry.

Finite-dimensional normed spaces. The unit-ball geometry determines which large exact equilateral configurations exist. Bounds for l_p^n, general Minkowski spaces, and Banach–Mazur neighborhoods are central examples.
Euclidean and convex geometry. Regular simplices realize the Euclidean value n+1; cube vertices realize the l_infinity value 2^n. Upper-bound proofs connect equilateral sets to convex hulls, packing, linear algebra, and topology.
General metric spaces. Tori, Hamming spaces, manifolds, projective spaces, and compacta support distinct equilateral-number problems because their metrics impose different exact-distance relations.
Infinite-dimensional spaces. Arbitrarily large finite equilateral sets need not assemble into one infinite equilateral set.

Clarity¶

Three questions must be separated.

Fixed scale or any scale? e_lambda(X) asks about one distance. In a normed space, homothety makes all positive scales equivalent. In an arbitrary metric space they can differ, so e(X) must declare whether it takes a supremum across lambda.

Manages Complexity¶

The invariant compresses a space's entire exact-distance feasibility landscape into a cardinal bound. It lets researchers compare metrics, state extremal theorems, separate construction problems from upper-bound problems, and quantify how a norm's unit-ball geometry supports symmetric configurations.

The signature also decomposes proofs. A lower bound exhibits an equilateral set and checks all pair distances. An upper bound shows that any set satisfying the equality constraints has limited cardinality.

Abstract Reasoning¶

Construct for lower bounds. Propose coordinates or combinatorial objects, calculate every pair distance, and count the set. A regular simplex, signed basis set, or cube vertex set supplies a certificate only for its own metric.

Prove upper bounds globally. Translate equal distances into linear independence, rank, convexity, volume, or combinatorial restrictions. A proof must cover every feasible equilateral set, not only a symmetric candidate.

Knowledge Transfer¶

The abstraction transfers literally among metric spaces when four items survive: a genuine metric, an exact common distance, a cardinality objective, and a declared attainment convention. This supports movement between normed spaces, discrete cubes, tori, manifolds, and graph metrics without changing the logical role.

What does not transfer is a numerical value or its geometric explanation. The cube construction is equilateral for l_infinity but not for Euclidean distance. A constant-distance error-correcting code and a regular simplex instantiate the same signature, yet their upper-bound tools and interpretations differ.

Relationships to Other Abstractions¶

Current abstraction Equilateral Dimension Domain-specific

Parents (2) — more general patterns this builds on

Equilateral Dimension is a kind of Cardinality Prime

Equilateral Dimension instantiates prime:cardinality by strict subsumption.
Equilateral Dimension presupposes Metric Prime

Equilateral Dimension instantiates prime:cardinality by strict subsumption.

Hierarchy paths (6) — routes to 3 parentless roots

Equilateral Dimension → Cardinality → Bijectivity → Function (Mapping)

Show alternative paths (5)

Neighborhood in Abstraction Space¶

Equilateral Dimension sits in a sparse region of the domain-specific corpus (63^rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Metric Geometry & Approximation (13 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08