discrete geometry Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geome ...

, an isosceles set is a set of points with the property that every three of them form an

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...

. More precisely, each three points should determine at most two distances; this also allows

degenerate Degeneracy, degenerate, or degeneration may refer to: Arts and entertainment * Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party i ...

isosceles triangles formed by three equally-spaced points on a line.

History

The problem of finding the largest isosceles set in a

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

of a given dimension was posed in 1946 by

Paul Erdős Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...

. In his statement of the problem, Erdős observed that the largest such set in the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

has six points. In his 1947 solution,

Leroy Milton Kelly Leroy Milton Kelly (May 8, 1914 – February 21, 2002Death-Record
for Leroy ...

showed more strongly that the unique six-point planar isosceles set consists of the vertices and center of a

regular pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...

. In three dimensions, Kelly found an eight-point isosceles set, six points of which are the same; the remaining two points lie on a line perpendicular to the pentagon through its center, at the same distance as the pentagon vertices from the center. This three-dimensional example was later proven to be optimal, and to be the unique optimal solution.

Decomposition into 2-distance sets

Kelly's eight-point three-dimensional isosceles set can be decomposed into two sets

X

(the three points on a line perpendicular to the pentagon) and

Y

(the five vertices of the pentagon), with the property that each point in

X

is equidistant from all points of

Y

. When such a decomposition is possible, in Euclidean spaces of any dimension,

X

and

Y

must lie in perpendicular subspaces,

X

must be an isosceles set within its subspace, and the set

Y'

formed from

Y

by adding the point at the intersection of its two subspaces must also be an isosceles set within its subspace. In this way, an isosceles set in high dimensions can sometimes be decomposed into isosceles sets in lower dimensions. On the other hand, when an isosceles set has no decomposition of this type, then it must have a stronger property than being isosceles: it has only two distances, among all pairs of points. Despite this decomposition theorem, it is possible for the largest two-distance set and the largest isosceles set in the same dimension to have different sizes. This happens, for instance, in the plane, where the largest two-distance set has five points (the vertices of a regular pentagon), while the largest isosceles set has six points. In this case, the six-point isosceles set has a decomposition where

X

is the singleton set of the central point (in a space of zero dimensions) and

Y

consists of all remaining points.

Upper bounds

d

-dimensional space, an isosceles set can have at most

\binom

points. This is tight for

d=6

and for

d=8

but not necessarily for other dimensions. The maximum number of points in a

d

-dimensional isosceles set, for

d=1,2,\dots, 8

, is known to be :3, 6, 8, 11, 17, 28, 30, 45 but these numbers are not known for higher dimensions.

Construction

Lisoněk provides the following construction of two-distance sets with

\binom

points, which also produces isosceles sets with

\binom+1

points. In

d+1

-dimensional Euclidean space, let

e_i

(for

i=1,\dots,d+1

) denote the vector a unit distance from the origin along the

i

th coordinate axis, and construct the set

S

consisting of all points

e_i+e_j

for

i\ne j

. Then

S

lies in the

d

-dimensional subspace of points with coordinate sum

2

; its

convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

is the

hypersimplex In polyhedral combinatorics, the hypersimplex \Delta_ is a convex polytope that generalizes the simplex. It is determined by two integers d and k, and is defined as the convex hull of the d-dimensional vectors whose coefficients consist of k ones ...

\Delta_

. It has only two distances: two points formed from sums of overlapping pairs of unit vectors have distance

\sqrt2

, while two points formed from disjoint pairs of unit vectors have distance

2

. Adding one more point to

S

at its centroid forms a isosceles set. For instance, for

d=3

, this construction produces a suboptimal isosceles set with seven points, the vertices and center of a

regular octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

, rather than the optimal eight-point set.

Generalization

The same problem can also be considered for other

metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...

s. For instance, for

Hamming space In statistics and coding theory, a Hamming space (named after American mathematician Richard Hamming) is usually the set of all 2^N binary strings of length ''N''. It is used in the theory of coding signals and transmission. More generally, a Ham ...

s, somewhat smaller upper bounds are known than for Euclidean spaces of the same dimension. In an

ultrametric space In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems ...

, the whole space (and any of its subsets) is an isosceles set. Therefore, ultrametric spaces are sometimes called isosceles spaces. However, not every isosceles set is ultrametric; for instance, obtuse Euclidean isosceles triangles are not ultrametric.

References

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Eindhoven University of Technology The Eindhoven University of Technology ( nl, Technische Universiteit Eindhoven), abbr. TU/e, is a public technical university in the Netherlands, located in the city of Eindhoven. In 2020–21, around 14,000 students were enrolled in its BSc a ...

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Electronic Journal of Combinatorics The ''Electronic Journal of Combinatorics'' is a peer-reviewed open access scientific journal covering research in combinatorial mathematics. The journal was established in 1994 by Herbert Wilf (University of Pennsylvania) and Neil Calkin (Georgi ...

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