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 geom ...

, 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 Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...

. More precisely, each three points should determine at most two distances; this also allows degenerate 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, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

of a given dimension was posed in 1946 by

Paul Erdős Paul Erdős ( ; 26March 191320September 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 discrete mathematics, g ...

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

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

has six points. In his 1947 solution, Leroy Milton Kelly 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 () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...

. 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, 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

\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, a regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Regular octahedra occur in nature as crystal structures. An octahedron, more generally, can be any eight-sided polyh ...

, 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 (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

s. For instance, for Hamming spaces, 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\ for all x, y, and z. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Formal d ...

, 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

{{reflist, refs= {{citation , last = Blokhuis , first = A. , contribution = Chapter 7: Isosceles point sets , doi = 10.6100/IR53747 , pages = 46–49 , publisher =

Eindhoven University of Technology The Eindhoven University of Technology (), Abbreviation, abbr. TU/e, is a public university, public technical university in the Netherlands, situated in Eindhoven. In 2020–21, around 14,000 students were enrolled in its Bachelor of Science, BS ...

, type = Ph.D. thesis , title = Few-Distance Sets , year = 1983 , zbl = 0516.05017 {{citation , last = Croft , first = H. T. , doi = 10.1112/plms/s3-12.1.400 , journal = Proceedings of the London Mathematical Society , mr = 0155230 , pages = 400–424 , series = Third Series , title = 9-point and 7-point configurations in 3-space , volume = 12 , year = 1962 {{citation , last1 = Grossman , first1 = Howard , last2 = Thebault , first2 = Victor , last3 = Schell , first3 = E. D. , last4 = Scheffe , first4 = Henry , last5 = Erdős , first5 = Paul , author5-link = Paul Erdős , date = August 1946 , doi = 10.2307/2305860 , issue = 7 , journal = The American Mathematical Monthly , page = 394 , title = Problems for Solution: E731–E735 , volume = 53, jstor = 2305860 . See in particular problem E735. {{citation , last = Fiedler , first = Miroslav , doi = 10.13001/1081-3810.1012 , journal = Electronic Journal of Linear Algebra , mr = 1615350 , pages = 23–30 , title = Ultrametric sets in Euclidean point spaces , volume = 3 , year = 1998, doi-access = free {{citation , last = Ionin , first = Yury J. , issue = 1 , journal = Electronic Journal of Combinatorics , mr = 2577309 , page = Research Paper 141, 24 , title = Isosceles sets , url = https://www.combinatorics.org/Volume_16/Abstracts/v16i1r141.html , volume = 16 , year = 2009, doi = 10.37236/230 , doi-access = free {{citation , last1 = Erdős , first1 = Paul , author1-link = Erdős , last2 = Kelly , first2 = L. M. , author2-link = Leroy Milton Kelly , date = April 1947 , doi = 10.2307/2304710 , issue = 4 , journal = The American Mathematical Monthly , page = 227 , title = E735 , volume = 54, jstor = 2304710 {{citation , last = Kido , first = Hiroaki , doi = 10.1016/j.ejc.2005.01.003 , issue = 3 , journal = Electronic Journal of Combinatorics , mr = 2206471 , pages = 329–341 , title = Classification of isosceles eight-point sets in three-dimensional Euclidean space , volume = 27 , year = 2006, doi-access = free ?{{citation , last = Lisoněk , first = Petr , doi = 10.1006/jcta.1997.2749 , doi-access = free , issue = 2 , journal = Journal of Combinatorial Theory , mr = 1429084 , pages = 318–338 , series = Series A , title = New maximal two-distance sets , volume = 77 , year = 1997 Discrete geometry