Planar sets with a fixed number of distances

affine·mess n points in the plane with exactly d distinct distances. Click a cell for its catalog.

Planar sets with a fixed number of distances

Take n points in the plane and count the distinct distances between them, d. The table shows, for each pair (n, d), how many configurations there are up to similarity. Click a cell to see them.

Reading a cell

Each line of a cell is one dimension of the solution set. 0d: 34 means 34 rigid configurations; 1d: 27 means 27 one-parameter families; 2d: [3, 20] means between 3 and 20 two-parameter families; ≥ 3 means no upper bound is known. A family is a realizable pattern of which pairs of points are at equal distance (an edge colouring of K_n up to symmetry), and its dimension is that of its solution set modulo similarity, measured in the full configuration space. Solid cells are known exactly; dashed cells carry bounds from constructed examples, each verified numerically to be isolated or to lie on a family of the stated dimension; a dimension that is not listed simply has no known example (and is excluded where a cell is marked finite). Blank cells are impossible.

Marks on the solid cells say where the proof of completeness comes from. No mark: a published classification (the configurations were reconstructed here from the paper, and where the paper lists sets rather than a count — Shinohara’s six-point sets, Lan–Wei’s and Shinohara’s eight- and seven-point lists — the number shown is ours). †: a published classification that we re-verified independently by exact computation. ‡: our exact computation only; we found nothing published, so these cells are the novel ones. “Exact computation” means: for every colouring type the polynomial system was solved symbolically, so unrealizable types are proved unrealizable, rigid configurations are isolated as real algebraic numbers, and each positive-dimensional component was decomposed and tested for real points. Dashed cells carry numerical bounds only.

Why the dimensions

Modulo similarity, n points have 2n − 4 degrees of freedom and C(n,2) distances. Asking for exactly d values imposes C(n,2) − d coincidences, so one expects a solution set of dimension 2n − 4 − C(n,2) + d. No component can be smaller than that, but symmetric configurations give bigger ones, which is why some cells mix rigid configurations with families (three and four points are the only cases where that never happens).

What is known

The maximum number of points for d distances is g(d) = 3, 5, 7, 9, 12, 13 for d = 1, …, 6 (Erdős–Fishburn 1996 for d ≤ 4, Shinohara 2008 for 5, Wei 2012 for 6), with the extremal configurations classified. For d = 7, g(7) = 16: Amit and Astra (2026) give a computer-assisted proof that no seventeen-point seven-distance set exists (a diameter-graph reduction to eleven-point cores with at most six distances, an exact interval certificate against a thirteen-cycle of diameters, and exact extension calculations; the one computational input taken from the literature is Szöllősi–Östergård’s exhaustive candidate list at eleven points). We re-derived its lattice part independently — the eleven twelve-point and fifty-two eleven-point lattice sets with at most six distances, and the extension bound 16 for every core. There are exactly two 16-point sets, Erdős–Fishburn’s two lattice pieces: we proved that only finitely many exist (a diameter-graph argument reduces a one-parameter family of sixteen-point sets to a ten-point core with at most six distances, and every such core is rigid, by the regenerated Szöllősi–Östergård candidate list and exact certification), then enumerated them by extending each of the 257 ten-point sets with at most six distances by six points. The same enumeration classifies the ten-, eleven- and twelve-point sets with five or six distances (the ‡ cells at the top of those columns) and shows that the lists for thirteen points with six distances and twelve with five, and the maximum sets in every column up to seven distances, are the only possibilities. Two-distance sets were classified by Einhorn and Schoenberg (1966), three-distance sets with at least five points by Shinohara (2004).

For the dashed cells the lower bounds come from explicit constructions: subsets of regular polygons, pieces of the triangular lattice, one-point extensions of the smaller complete lists, a systematic search over configurations with a cyclic or dihedral symmetry (orbit structures with up to five parameters and up to two extra coincidences), and — for the top rows of the five-, six- and seven-distance columns (8–12 points with five distances, 9–13 with six, 13–16 with seven) — an exhaustive sweep of the subsets of the twelvefold set R₁₂ + R₁₂, the 73 sums of two twelfth roots of unity (points on circles of radii 2, 2cos 15°, √3, √2, 1, 2sin 15° and the centre, at angles that are multiples of 15°): it contains pieces of the triangular lattice, the 12-gon with its centre, and twelvefold configurations that are neither, among them the three ten-point six-distance patterns that no previously known family of examples realizes. Configurations are counted up to similarity by an exact congruence test (a vertex bijection matching all distances; the sweep contains many homometric pairs, non-congruent sets with the same multiset of distances). Every configuration in a catalog was checked numerically to have exactly the stated number of distances and the stated local dimension. Upper bounds are shown only where a classification or a finiteness argument exists: Lan and Wei (2013) count 42 seven-point four-distance sets; eight-point four-distance sets are finite because they extend a seven-point set with three or four distances by a point with finitely many positions; Wei (2011) classifies eleven-point five-distance sets; the cells (10,5), (10,6), (11,6), (12,6) and (16,7) are classified here from the complete candidate-pattern lists (every eight-, nine- and ten-point pattern with at most five or six distances was also certified: the only flexible ones are the three-triangle family T + h·R₉₀(T) and its two one-vertex deletions).

The four-point row, and the five-point row with two to seven distances, are certified by exact computation. For each colouring type of K₅ the squared-distance equations, with the scale fixed and the inequations “all values distinct and nonzero” imposed through a Rabinowitsch variable, were handed to msolve: an empty Gröbner basis proves the type unrealizable, a zero-dimensional system yields its real solutions with certified isolating intervals, and a positive-dimensional one is decomposed into minimal primes with Singular, each prime checked for degeneracy and probed for real points by random hyperplane slices; real-isolated points of the curves were excluded by checking that every curve is smooth. For 21 of the 768 types with five to seven distances the primary decomposition did not finish; there the certificate is the dimension of the saturated ideal (a Gröbner basis computation, which shows every component has the expected dimension), certified real points on random slices, and full rank of the Jacobian at those points, so they lie on smooth components of that dimension. Rigid configurations are counted up to similarity by an exact orbit computation on the certified roots (interval arithmetic), with the real-root counts cross-checked by Sturm sequences; every unrealizable verdict was re-verified independently, and the components on which random slices found no real point were settled one by one (twelve are empty, by a positive-definite generator; nine have real points on bounded pieces the slices missed). For five to seven distances a separate search looked for hidden rigid configurations — real points of a positive-dimensional type that are isolated in the real locus, invisible to the dimension count: wherever the Jacobian of the ten equations has full rank the real locus is a manifold, and at the finitely many real rank-deficient points each component was shown to be smooth (after eliminating the linearly occurring variables, by the minors of its Jacobian) or its singular locus to have no real points; none was found; for four types at d = 5 whose decomposition never finishes, the twenty candidate points were settled by a certified local test (a smooth surface through the point containing the curve, and an indefinite intrinsic Hessian, so the point is a node of two real branches by the Morse lemma). This corrected the numerical search in two places: for five points and four distances it had reported two rigid types as families, and had called the equally spaced collinear quintuple rigid when it in fact lies on a one-parameter family.

The cells for four points, and the counts 34 and 9 for five and six points with three distances, come from a numerical computation made for this applet (October 2026): every colouring of the edges of K_n with d colours up to symmetry was tested for realizability by random-start least squares, the dimension of the solution set was measured at each solution, and rigid solutions were clustered by congruence. For five points with four distances the run was repeated with a second random seed, and the 34 five-point three-distance sets were also obtained by a second, independent method (extending the four-point families). The fourth nine-point four-distance set, which Erdős and Fishburn count but which we could not find described anywhere accessible, was recovered by a search over mirror-symmetric configurations. These counts are numerical and have not been proved by hand; the lists are exhaustive only up to the reliability of a random search.

Flexing

Tinted cards are families. Hovering (or tapping, on a touch screen) makes the configuration move along its family: at each frame the tangent space of the solution set is computed from the Jacobian of the equal-distance equations, the configuration takes a small step along it and is projected back onto the constraint set by a few Newton iterations, so the equalities hold exactly throughout. One-parameter families glide back and forth along their curve, turning around before two points merge or two distance values coincide, so the motion loops; higher-dimensional families wander within the same bounds. Only one card moves at a time; tapping a moving card restarts it from its resting shape. Where branches of a family cross, the motion may switch branches.

Coordinates

Each catalog entry has a coordinate list in a natural frame: a regular-polygon frame (circumradius 1) or a lattice frame when the configuration has one, otherwise the shortest distance is 1 with the first point at the origin. Exact expressions are given where the frame allows; clicking the list copies the decimal coordinates as JSON.

Code and data

Everything behind this page is public: the build pipeline, every certificate and result file, and the write-ups are in the folder tools/fixed-distances of the site’s GitHub repository (README). In particular: REPRODUCING.md, a step-by-step walkthrough, with commands and expected outputs, for verifying that exactly two sixteen-point seven-distance sets exist; g7count/REPORT.md, the account of that computation; g7fin/FINITENESS.md, the finiteness argument and the F(5), F(6), F(7) notes; g7fin/gen, the candidate-pattern generator that re-derives the Szöllősi–Östergård counts.

Sources

S. J. Einhorn, I. J. Schoenberg, Indag. Math. 28 (1966). · P. Erdős, P. Fishburn, Discrete Math. 160 (1996) 115–125. · M. Shinohara, European J. Combin. 25 (2004) 1039–1058; Discrete Math. 308 (2008) 3048–3055. · X. Wei, Ars Combin. 102 (2011); Electron. J. Combin. 19(4) (2012) P38. · W. Lan, X. Wei, Math. Notes (2013). · F. Szöllősi, P. R. J. Östergård, Electron. J. Combin. 27(1) (2020) P1.23. · E. Bannai, E. Bannai, D. Stanton (1983); A. Blokhuis (1984). · L.-R. Bao, W.-H. Yu, arXiv:2509.00880 (2025). · K. Balaji et al., arXiv:1911.11688.