International Transactions in Operational Research

1. Introduction

Distance geometry (DG) is the study of geometry with the basic entity being distance (instead of lines, planes, circles, polyhedra, conics, surfaces, and varieties). As did much of mathematics, it all began with the Greeks: specifically Heron, or Hero, of Alexandria, sometime between 150 BC and 250 AD, showed how to compute the area of a triangle given its side lengths.

After a hiatus of almost two thousand years, we reach Arthur Cayley: the first paper of volume I of his Collected Papers, dated 1841, which is about the relationships between the distances of five points in space (Cayley, 1841). The gist of what he showed is that a tetrahedron can only exist in a plane if it is flat (in fact, he discussed the situation in one more dimension). This yields algebraic relations on the side lengths of the tetrahedron.

Hilbert's influence on foundations and axiomatization was very strong in the 1930s Mitteleuropa (Hilbert, 1903). This pushed many people toward axiomatizing existing mathematical theories (Henkin et al., 1959). Karl Menger, a young professor of geometry at the University of Vienna and an attendee of the Vienna Circle, proposed in 1928 a new axiomatization of metric spaces using the concept of distance and the relation of congruence, and, using an extension of Cayley's algebraic machinery (which is now known as Cayley–Menger determinant), generalized Heron's theorem to compute the volume of arbitrary K‐dimensional simplices using their side lengths (Menger, 1928).

The Vienna Circle was a group of philosophers and mathematicians who convened in Vienna's Reichsrat café around the 1930s to discuss philosophy, mathematics, and, presumably, drink coffee. When the meetings became excessively politicized, Menger distanced himself from the Circle, and organized instead a seminar series, which ran from 1929 to 1937 (Menger, 1998). A notable name crops up in the intersection of Menger's geometry students, the Vienna Circle participants, and the speakers at Menger's Kolloquium: Kurt Gödel. Most of the papers Gödel published in the Kolloquium proceedings are about logic and foundations, but two, dated 1933, are about the geometry of distances on spheres and surfaces. The first (Menger, 1998, 18 Feb. 1932, p. 198) answers a question posed at a previous seminar by Laura Klanfer, and shows that a set X of four points in any metric space, congruent to four noncoplanar points in , can be realized on the surface of a three‐dimensional sphere using geodesic distances. The second (Menger, 1998, 17 May 1933, p. 252) shows that Cayley's relationship hold locally on certain surfaces that behave locally like Euclidean spaces (Fig. 1).

The pace quickens: in 1935, Isaac Schoenberg published some remarks on a paper (Schoenberg, 1935) by Fréchet on the Annals of Mathematics, and gave, among other things, an algebraic proof of equivalence between Euclidean distance matrices (EDM) and Gram matrices. This is the same result that is nowadays given to show the validity of the classical multidimensional scaling (MDS) technique (Borg and Groenen, 2010, § 12.1).

This brings us to the computer era, where the historical account ends and the contemporary treatment begins. Computers allow the efficient treatment of masses of data, some of which are incomplete and noisy. Many of these data concern, or can be reduced to, distances, and DG techniques are the subject of an application‐oriented renaissance (Mucherino et al., 2013; Liberti et al., 2014). Motivated by the global positioning system, for example, the old geographical concept of trilateration (a system for computing the position of a point given its distances from three known points) makes its way into DG in wireless sensor networks (Eren, 2004). Wüthrich's Nobel Prize for using nuclear magnetic resonance techniques in the study of proteins brings DG to the forefront of structural bioinformatics research (Havel and Wüthrich, 1985). The massive use of robotics in mechanical production lines requires mathematical methods based on DG (Rojas and Thomas, 2013).

DG is also tightly connected with graph rigidity (Graver et al., 1993). This is an abstract mathematical formulation of statics, the study of structures under the action of balanced forces (Maxwell, 1864), which is on the basis of architecture (Varignon, 1725). Rigidity of polyhedra gave rise to a conjecture of Euler's (1736) about closed polyhedral surfaces, which was proved correct only for some polyhedra: strictly convex (Cauchy, 1813), convex and higher dimensional (Alexandrov, 2005), and generic (a polyhedron is generic if no algebraic relations on hold on the components of the vectors that represent its vertices) (Gluck , 1975). It was however disproved in general by means of a very special, nongeneric nonconvex polyhedron (Connelly, 1879).

The rest of this paper will focus on the following results, listed here in a chronological order: Heron's theorem (Section 2.), Euler's conjecture and Cauchy's proof for strictly convex polyhedra (Section 3.), Cayley–Menger determinants (Section 4.), Menger's axiomatization of geometry by means of distances (Section 5.), a result by Gödel's concerning DG on the sphere (Section 6.), and Schoenberg's equivalence (Section 7.) between EDM and positive semidefinite matrices (PSD). There are many more results in DG: this is simply a choice of our favorite results among those we know best.

2. Heron's formula

Heron's formula, which is usually taught at school, relates the area of a triangle to the length of its sides and its semiperimeter as follows:

(1)

There are many ways to prove its validity. Shannon Umberger, a student of the “Foundations of Geometry I” course given at the University of Georgia in the fall of 2000, proposes, as part of his final project (http://jwilson.coe.uga.edu/emt668/emat6680.2000/umberger/MATH7200/HeronFormulaProject/finalproject.html), three detailed proofs: an algebraic one, a geometric one, and a trigonometric one. John Conway and Peter Doyle discuss Heron's formula proofs in a publicly available e‐mail exchange (https://math.dartmouth.edu/∼doyle/docs/heron/heron.txt) from 1997 to 2001.

Our favorite proof is based on complex numbers, and was submitted (http://www.artofproblemsolving.com/Resources/Papers/Heron.pdf) to the “Art of Problem Solving” online school for gifted mathematics students by Miles Edwards (also see http://newsinfo.iu.edu/news/page/normal/13885.html and http://www.jstor.org/stable/10.4169/amer.math.monthly.121.02.149 for more recent career achievements of this gifted student) when he was studying at Lassiter High School in Marietta, Georgia.

Proof.(Edwards, 2011) Consider a triangle with sides (opposite to the vertices , respectively) and its inscribed circle centered at O with radius r. The perpendiculars from O to the triangle sides split a into , b into , and c into as shown in Fig. 2. Let be the segments joining O with , respectively.

First, we note that , which implies . Next, the following complex identities are easy to verify geometrically in Fig. 2:

These imply:

where the last step uses Euler's identity (Euler, 1922, I–VIII, § 138–140, p. 148). Since is real, the imaginary part of must be zero. Expanding the product and rearranging terms, we get . Solving for r, we have the nonnegative root

(2)

We can write the semiperimeter of the triangle as . Moreover,

so , which implies that Equation 2 becomes

We now write the area of the triangle by summing it over the areas of the three triangles , , and , which yields

as claimed.

3. Euler's conjecture and the rigidity of polyhedra

Consider a square with unit sides, in the plane. One can shrink two opposite angles and correspondingly widen the other two to obtain a rhombus (see Fig. 3), which has the same side lengths but a different shape: no sequence of rotations, translations, or reflections can turn one into the other. In other words, a square is flexible. By contrast, a triangle is not flexible, i.e., it is rigid.

Euler conjectured in 1766 (Euler, 1862) that all three‐dimensional polyhedra are rigid. The conjecture appears at the end of the discussion about the problem Invenire duas superficies, quarum alteram in alteram transformare liceat, ita ut in utraque singula puncta homologa easdem inter se teneat distantias, i.e.:

To find two surfaces for which it is possible to transform one into the other, in such a way that corresponding points on either keep the same pairwise distance. (†)

Toward the end of the paper, Euler writes Statim enim atque figura undique est clausa, nullam amplius mutationem patitur, which means “As soon as the shape is everywhere closed, it can no longer be transformed.” Although the wording appears ambiguous by today's standards, scholars of Euler and rigidity agree: what Euler really meant is that 3D polyhedra are rigid (Gluck, 1975).

To better understand this statement, we borrow from Alexandrov (1956) the precise definition of a polyhedron: a family of points, open segments and open triangles is a triangulation if (a) no two elements of have common points, and (b) all sides and vertices of the closure of any triangle of , and both extreme points of the closure of any segment of are all in themselves. Note that the definition we are about to give is different from the usual definition employed in convex analysis, i.e.,  a polyhedron is an intersection of half‐spaces; however, a convex polyhedron in the sense given below is the same as a polytope in the sense of convex analysis. Here is the definition: given a triangulation in (where ), the union of all points of and all the points belonging to the segments and triangles of is called a polyhedron. Note that several triangular faces can belong to the same affine space, thereby forming polygonal faces.

Each polyhedron has an incidence structure of points on segments and segments on polygonal (not necessarily triangular) faces, which induces a partial order (p.o.) based on set inclusion. For example, the closure of the square contains the closures of the segments , , , , each of which contains the corresponding adjacent points , , , . Accordingly, the p.o. is ; ; ; ; . Since this p.o. has a bottom element (the empty set) and a top element (the whole polyhedron), it is a lattice. A lattice isomorphism is a bijective mapping between two lattices, which preserves the p.o. Two polyhedra are combinatorially equivalent if their triangulations are lattice isomorphic. If, moreover, all the lattice isomorphic polygonal faces of are exactly equal, the polyhedra are said to be facewise equal.

Under the above definition, nothing prevents a polyhedron from being nonconvex (see Fig. 4). It is known that every closed surface, independently of the convexity of its interior, is homeomorphic (intuitively: smoothly deformable in) to some polyhedron (again Alexandrov, 1956, § 2.2). This is why we can replace “closed shape” with “polyhedron” in Euler's conjecture.

The “rigidity” implicit in Euler's conjecture should be taken to mean that no point of the polyhedron can undergo a continuous motion under the constraint that the shape be the same at each point of the motion. As for the concept of “shape,” it is linked to that of distance, as appears clear from (†). The following is therefore a formal restatement of Euler's conjecture: two combinatorially equivalent facewise equal polyhedra must be isometric under the Euclidean distance, i.e., each pair of points in one polyhedron is equidistant with the corresponding pair in the other.

A natural question about the Euler conjecture stems from generalizing the example in Fig. 3 to 3D (see Fig. 5). Does this not disprove the conjecture?

The answer is no: all the polygonal faces in the cube are squares, but this does not hold in the rhomboid. The question is more complicated than it looks at first sight, which is why it took 211 years to disprove it.

3.1. Strictly convex polyhedra: Cauchy's proof

Although Euler's conjecture is false in general, it is true for many important subclasses of polyhedra. Cauchy proved it true for strictly convex polyhedra (in fact Cauchy's proof contained two mistakes, corrected by Steinitz (Lyusternik, 1966, p. 67) and Lebesgue). There are many accounts of Cauchy's proof: Cauchy's original text, still readable today (Cauchy, 1813); Alexandrov's book (Alexandrov, 2005), Lyusternik's book (Lyusternik, 1966, § 20), Stoker's paper (Stoker, 1968), Connelly's chapter (Connelly, 1993) just to name a few. Here we follow the treatment given by Pak (2010).

We consider two combinatorially equivalent, facewise equal strictly convex polyhedra , and aim to show that P and Q are isometric.

For a polyhedron P we consider its associated graph , where V are the points of P and E its segments. Note that only depends on the incidence structure of the polygonal faces, segments, and points of P. Since are combinatorially equivalent, . Consider the dihedral angles (i.e., the angle smaller than π formed by two half‐planes in intersecting on a line L) (resp., ) in P (resp., Q) induced by the line containing the segment corresponding to the edge . We assign to each edge a label (so ), and consider, for each , the edge sequence , where is the set of nodes adjacent to v. The order of the edges in is given by any circuit around the polygon obtained by intersecting P with a plane γ that separates v from the other vertices in V (this is possible by strict convexity, see Fig. 6).

It is easy to see in Fig. 6 that every edge corresponds to a vertex of . Therefore, a circuit over defines an order over . We also assume that this order is periodic, i.e., its last element precedes the first one. Any such sequence naturally induces a sign sequence ; we let be the sequence without the zeros, and we count the number of sign changes in , including the sign change occurring between the last and first elements.

Lemma 3.1.For all , is even.

Proof.Suppose is odd, and proceed by induction on : if , then there is only one sign change. So, the first edge in to be labeled with has the property that, going around the periodic sequence with only one sign change, is also labeled with , which yields , a contradiction. A trivial induction step yields the same contradiction for all odd .

We now state a fundamental technical lemma, and provide what is essentially Cauchy's proof, rephrased as in Lemma 2 in § 20 of Lyusternik (1966).

Lemma 3.2.If P is strictly convex, then for each we have either or .

Proof.By Lemma 3.1, for each we have , so we aim to show that . Suppose, to get a contradiction, that , and consider the polygon as in Fig. 6. By the correspondence between edges in and vertices of , the labels are vertex labels in . Since there are only two sign changes, the sequence of vertex labels can be partitioned in two contiguous sets of +1 and −1 (possibly interspersed by zeros). By convexity, there exists a line L separating the +1 and the −1 vertices (see Fig. 6). Since all of the angles marked +1 strictly increase, the segment also strictly increases (this statement was also proved in Cauchy's paper (Cauchy, 1813), but this proof contained a serious flaw, later corrected by Steinitz); but, at the same time, all of the angles marked −1 strictly decrease, so the segment also strictly decreases, which means that the same segment both strictly increases and decreases, which is a contradiction (see Fig. 7).

Theorem 3.3. ([Cauchy's Theorem Cauchy, 1813])If two closed convex polyhedra are combinatorially equivalent and facewise equal, they are isometric.

We only present the proof of the base case where

(3)

and is a connected graph, and refer the reader to p. 251 of Pak (2010) for the other cases (which are mostly variations of the ideas given in the proof below).

Proof.If for all , it means that all of the dihedral angles in P are equals to those of Q, which implies isometry. So we assume the alternative w.r.t. Equation 3 above: , and aim for a contradiction. Let : by Lemma 3.2 and because for each v, we have , a lower bound for M. We now construct a contradicting upper bound for M. For every , we let be the number of polygonal faces of P with h sides (or edges). The total number of polygonal faces in P (or Q) is , and the total number of edges is therefore (we divide by 2 since each edge is counted twice in the sum—one per adjacent face—given that are closed). A simple term by term comparison of and yields . Since each polygonal face f of P is itself closed, the number of sign changes of the quantities over all edges adjacent to the face f is even, by the same argument given in Lemma 3.1. It follows that if the number h of edges adjacent to the face f is even, then , and if h is odd. This allows us to compute an upper bound on M:

The middle step follows by simply increasing each coefficient. The last step is based on Euler's characteristic (Euler, 1843): . Hence we have , which is a contradiction.

3.2. Euler was wrong: Connelly's counterexample

Proofs behind counterexamples can rarely be termed “beautiful” since they usually lack generality (as they are applied to one particular example). Counterexamples can nonetheless be dazzling by themselves. Connelly's counterexample (Connelly, 1879) to the Euler's conjecture consists in a very special nongeneric nonconvex polyhedron that flexes, while keeping combinatorial equivalence and facewise equality with all polyhedra in the flex. Some years later, Klaus Steffen produced a much simpler polyhedron with the same properties (see http://demonstrations.wolfram.com/SteffensFlexiblePolyhedron/). It is this polyhedron we exhibit in Fig. 8.

4. Cayley–Menger determinants and the simplex volume

The foundation of modern DG, as investigated by Menger (1931) and Blumenthal (1953), rests on the fact that:

the four‐dimensional volume of a four‐dimensional simplex embedded in three‐dimensional space is zero, (*)

which we could also informally state as “flat simplices have zero volume.” This is related to DG because the volume of a simplex can be expressed in terms of the lengths of the simplex sides, which yields a polynomial in the length of the simplex side lengths that can be equated to zero. If these lengths are expressed in function of the vertex positions as , this yields a polynomial equation in the positions of the simplex vertices in terms of its side lengths. Thus, if we know the positions of , we can compute the unknown position of x₅ or prove that no such position exists, through a process called trilateration (Lavor et al., 2012).

The proof of (*) was published by Arthur Cayley in 1841, during his undergraduate studies. It is based on the following well‐known lemma about determinants (stated without proof in Cayley's paper).

Theorem 4.2. ([Cayley, 1841])Given five points all belonging to an affine 3D subspace of , let for each . Then

(4)

We note that Cayley's theorem is expressed for points in , but it also holds for any points in (Blumenthal, 1953). Cayley explicitly remarks that it holds for the cases and (see VIII, § 5 of Sommerville (1958) for the proof of general n). The determinant on the right‐hand side of Equation 4 is called Cayley–Menger determinant, denoted by Δ. We remark that in the proof below is the kth component of , for each .

Proof.We follow Cayley's treatment. He pulls the following two matrices

out of a magic hat. He performs the product , rearranging and collecting terms, and obtains a 6 × 6 matrix where the last row and column are (1, 1, 1, 1, 1, 0), and the ‐th component is for every . To see this, it suffices to carry out the computations using Mathematica (2014); by way of an example, the first diagonal component of is , and the component on the first row, second column of is . In other words, is the Cayley–Menger determinant in Equation 4. On the other hand, if we set for each , effectively projecting the five four‐dimensional points in three‐dimensional space, it is easy to show that since the fifth columns of both A and the fifth row of B are zero. Hence we have by Lemma 4.1, and is precisely Equation 4 as claimed.

The missing link is the relationship of the Cayley–Menger determinant with the volume of an n‐simplex. Since this is not part of Cayley's paper, we only establish the relationship for . Let , , . Then

The beauty of Cayley's proof is in its extreme compactness: it uses determinants to hide all the details of elimination theory that would be necessary otherwise. His paper also shows some of these details for the simplest case . The starting equations, as well as the symbolic manipulation steps, depend on n. Although Cayley's proof is only given for , Cayley's treatment goes through essentially unchanged for any number n of points in dimension .

5. Menger's characterization of abstract metric spaces

At a time where mathematicians were heeding Hilbert's call to formalization and axiomatization, Menger presented new axioms for geometry based on the notion of distance, and provided conditions for arbitrary sets to “look like” Euclidean spaces, at least distancewise (Menger, 1928, 1931). Menger's system allows a formal treatment of geometry based on distances as “internal coordinates.” The starting point is to consider the relations of geometrical figures having proportional distances between pairs of corresponding points, i.e., congruence. Menger's definition of a congruence system is defined axiomatically, and the resulting characterization of abstract distance spaces with respect to subsets of Euclidean spaces (possibly his most important result) transforms a possibly infinite verification procedure (any subset of any number of points) into a finitistic one (any subset of points, where n is the dimension of the Euclidean space).

It is remarkable that almost none of the results mentioned next offers an intuitive geometrical grasp, such as the proofs of Heron's formula and Cayley's theorem do. As formal mathematics has it, part of the beauty in Menger's work consists in turning the “visual” geometrical proofs based on intuition into formal symbolic arguments based on sets and relations. On the other hand, Menger himself gave a geometric intuition of his results on p. 335 of Menger (1935), which we comment in Section 5.4..

5.3. A finitistic characterization of semimetric spaces

Two sets are congruent if there is a map (called congruence map) , such that for all . We denote this relation by , dropping the ϕ if it is clear from the context.

Lemma 5.1.Any congruence map is injective.

Proof.Suppose, to get a contradiction, that with and : then and so, by Section 5.1., against assumption.

If S is congruent to a subset of T, then we say that S is congruently embeddable in T.

5.3.1. Congruence order

Now consider a set and an integer with the following property: for any , if all n‐point subsets of T are congruent to an n‐point subset of S, then T is congruently embeddable in S. If this property holds, then S is said to have congruence order n. Formally, the property is written as follows:

(5)

If for some positive integer n, then S can have congruence order n, since the definition is vacuously satisfied. So we assume in the following that .

Proposition 5.2.If S has congruence order n in , then it also has congruence order m for each .

Proof.By hypothesis, for every , if every n‐point subset of T is congruent to an n‐point subset of S, then there is a subset R of S such that . Now any m‐point subset of S is mapped by ϕ to a congruent m‐point subset of S, and again , so Equation 5 is satisfied for S and m.

In view of Proposition 5.2, it becomes important to find the minimum congruence order of a given metric space.

Proposition 5.3. (i.e., the Euclidean space that simply consists of the origin) has minimum congruence order 2 in .

Proof.Pick any with . None of its two‐point subsets is congruent to any two‐point subset of , since none exists. Moreover, T itself cannot be congruently embedded in , since and no injective congruence map can be defined, against Lemma 5.1. So the integer 2 certainly (vacuously) satisfies Equation 5 for , which means that has congruence order 2. In view of Proposition 5.2, it also has congruence order m for each . Hence we have to show next that the integer 1 cannot be a congruence order for . To reach a contradiction, suppose the contrary, and let T be as above. By Axiom 3, every singleton subset of T is congruent to a subset of , namely the subset containing the origin. Thus, by Equation 5, T must be congruent to a subset of ; but, again, contradicts Lemma 5.1: so T cannot be congruently embedded in , which negates Equation 5. Hence the integer 1 cannot be a congruence order for , as claimed.

5.3.2. Menger's fundamental result

The fundamental result proved by Menger (1928) is that the Euclidean space has congruence order but not for each in the family of all semimetric spaces. The important implication of Menger's result is that in order to verify whether an abstract semimetric space is congruent to a subset of a Euclidean space, we only need to verify congruence of each of its point subsets.

We follow Blumenthal's treatment (Blumenthal, 1953), based on the following preliminary definitions and properties, which we will not prove:

• 1. A congruent mapping of a semimetric space onto itself is called a motion;
• 2. points in are independent if they are not affinely dependent (i.e., if they do not all belong to a single hyperplane in );
• 3. two congruent ‐point subsets of are either both independent or both dependent;
• 4. there is at most one point of with given distances from an independent ‐point subset;
• 5. any congruence between any two subsets of can be extended to a motion;
• 6. any congruence between any two independent ‐point subsets of can be extended to a unique motion.

Theorem 5.4. ([Menger, 1928])A nonempty semimetric space S is congruently embeddable in (but not in any for ) if and only if: (a) S contains an ‐point subset , which is congruent with an independent ‐point subset of ; and (b) each ‐point subset U of S containing is congruent to an ‐point subset of .

The proof of Menger's theorem is very formal (see next) and somewhat difficult to follow. It is nonetheless a good example of a proof in an axiomatic setting, where logical reasoning is based on syntactical transformations induced by inference rules on the given axioms. An intuitive discussion is provided in Section 5.4..

Proof.(⇒) Assume first that , where the affine closure of T has dimension n. Then T must contain an independent subset with , which we can map back to a subset using . Since are injective, , and by Axiom 3 we have , so , which establishes (a). Now take any with and : this can be mapped via ϕ to a subset : Lemma 5.1 ensures injectivity of ϕ and hence , establishing (b).

(⇐) Conversely, assume (a) and (b) hold. By (a), let with and , with independent and . We claim that ϕ can be extended to a mapping of S into . Take any : by (b), with . Note that by Section 5.1., which implies that for any , we have . Moreover, by Property 5.3 above, is independent and has cardinality , which by Property 5.3 above implies that ω can be extended to a unique motion in . So the action of ω is extended to q, and we can define . We now show that this extension of ϕ is a congruence. Let : we aim to prove that . Consider the set : since , by (b) there is with such that . As above, we note that there is a subset such that and , that for each , and that ω is a motion of . Hence , as claimed.

6. Gödel on spherical distances

Kurt Gödel's name is attached to what is possibly the most revolutionary result in all of mathematics, i.e., Gödel's incompleteness theorem, according to which any formal axiomatic system sufficient to encode the integers is either inconsistent (it proves A and ) or incomplete (there is some true statement A which the system cannot prove). This shattered Hilbert's dream of a formal system in which every true mathematical statement could be proved. Few people know that Gödel, who attended the Vienna Circle, Menger's course in geometry, and Menger's seminar, also contributed two results that are completely outside the domain of logic. These results only appeared in the proceedings of Menger's seminar (Menger, 1998), and concern DG on a spherical surface.

6.1. Four points on the surface of a sphere

The result we discuss here is a proof to the following theorem, conjectured at a previous seminar session by Laura Klanfer. We remark that a sphere in is a semimetric space whenever it is endowed with a distance corresponding to the length of a geodesic curve joining two points.

Theorem 6.1. ([Gödel, 1986])Given a semimetric space S of four points, congruently embeddable in but not , is S also congruently embeddable on the surface of a sphere in .

Gödel's proof looks at the circumscribed sphere around a tetrahedron in , and analyzes the relationship of the geodesics, their corresponding chords, and the sphere radius. It then uses a fixed point argument to find the radius that corresponds to geodesics that are as long as the given sides.

Proof.The congruence embedding of S in defines a tetrahedron T having six (straight) sides with lengths . Let r be the radius of the sphere circumscribed around T (i.e., the smallest sphere containing T). We will now consider a family of tetrahedra , parametrized on a scalar , defined as follows: is the tetrahedron in having side lengths , where is the length of the chord subtending a geodesic having length α on a sphere of radius . As x tends toward zero, each tends toward (for each ), since the radius of the sphere tends toward infinity and each geodesic length tends toward the length of the subtending chord. This means that tends toward T, since T is precisely the tetrahedron having side lengths . For each , let be the inverse of the radius of the sphere circumscribed about . Since as , and the radius circumscribed about T is r, it follows that as . Also, since T exists by hypothesis, we can define and . Also note that it is well known by elementary spherical geometry that:

(6)

Claim.if then ϕ has a fixed point in the open interval .

Proof of the claim.First of all notice that τ(0) exists, and is a continuous function for for each α (by Equation 6). Since is defined by the chord lengths , this also means that varies continuously for x in some open interval (for some constant ). In turn, this implies that exists by continuity. There are two cases: either is at the upper extremum of I, or it is not.

• (i)

  If , then exists, its longest edge has length , so, by elementary spherical geometry, the radius of the sphere circumscribed around is greater than , i.e., greater than . Thus . We also have, however, that , so by the intermediate value theorem there must be some with .

• (ii)

  Assume now and suppose is nonplanar. Then for each y in an arbitrary small neighborhood around , must exist by continuity: in particular, there must be some where exists, which contradicts the definition of . So is planar: this means that each geodesic is contained in the same plane, which implies that the geodesics are linear segments. It follows that the circumscribed sphere has infinite radius, or, equivalently, that . Again, by and the intermediate value theorem, there must be some with .

This concludes the proof of the claim.

So now let y be the fixed point of ϕ. The tetrahedron has side lengths for each , and is circumscribed by a sphere σ with radius . It follows that, on the sphere σ, the geodesics corresponding to the chords given by the tetrahedron sides have lengths (for ), as claimed.

6.2. Gödel's devilish genius

Gödel's proof exhibits an unusual peak of devilish genius. At first sight, it is a one‐dimensional fixed‐point argument that employs a couple of elementary notions in spherical geometry. Underneath the surface, the fixed‐point argument eschews a misleading visual intuition.

T is a given tetrahedron in that is assumed to be nonplanar and circumscribed by a sphere of finite positive radius r (see Fig. 9, left).

The map τ sends a scalar x to the tetrahedron having as side lengths the chords subtending the geodesics of length () on a sphere of radius (see Fig. 9, right). The map τ is such that since for the radius is infinite, which means that the geodesics are equal to their chords. Moreover, the map ϕ sends x to the inverse of the radius of the sphere circumscribing . Since every geodesic on the sphere is a portion of a great circle, it would appear from Fig. 9 (right) that the radius used to compute () is the same as the radius of the sphere circumscribing , which would immediately yield for every x—making the proof trivial. There is something inconsistent, however, in the visual interpretation of Fig. 9: the given tetrahedron T corresponds to the case , which happens when , i.e., the radius of the sphere circumscribed around T is ∞. But this would yield T to be a planar tetrahedron, which is a contradiction with an assumption of the theorem. Moreover, if were equal to x for each x, this would yield , another contradiction.

The misleading concept is hidden in Fig. 9 (right). It shows a tetrahedron inscribed in a sphere, and a spherical tetrahedron on the same vertices. This is not true in general, i.e., the spherical tetrahedron with the given curved side lengths cannot, in general, be embedded in the surface of a sphere of any radius. For example, the case yields geodesics with infinite curvatures (i.e., straight lines laying in a plane), but , and there is no flat tetrahedron with the same distances as those of T. The sense of Gödel's proof is that the function simply transforms a set of geodesic distances into a set of linear distances, i.e., it maps scalars to scalars rather than geodesics to segments, whereas Fig. 9 (right) shows the special case where the geodesics are mapped to the corresponding segments, with intersections at the same points (namely the distances can be embedded on the particular sphere shown in the picture). More specifically, the geodesic curves may or may not be realizable on a sphere of radius . Gödel's proof shows exactly that there must be some x for which , i.e., the geodesic curves become realizable.

7. The equivalence of EDM and PSD matrices

Many fundamental innovations stem from what are essentially footnotes to apparently deeper or more important work. Isaac Schoenberg, better known as the inventor of splines (Schoenberg, 1946), published a paper in 1935 titled Remarks to Maurice Fréchet's article “Sur la définition axiomatique d'une classe d'espace distanciés vectoriellement applicable sur l'espace de Hilbert” (Schoenberg, 1935). The impact of Schoenberg's remarks far exceeds that of the original paper: these remarks encode what amounts to the basis of the well‐known MDS techniques for visualizing high‐dimensional data (Cox and Cox, 2001), as well as all the solution techniques for DG problems based on semidefinite programming (Alfakih et al., 1999; Man‐Cho So and Ye, 2007) (Fig. 10).

7.2. The proof of Schoenberg's theorem

Given a set of points in , we can write x as an matrix having as ith column. The matrix having the scalar product as its ‐th component is called the Gram matrix or Gramian of x. The proof of Theorem 7.1 works by exhibiting a 1‐1 correspondence between squared EDMs and Gram matrices, and then by proving that a matrix is Gram if and only if it is PSD.

Without loss of generality, we can assume that the barycenter of the points in x is at the origin:

(7)

Now we remark that, for each , we have

(8)

7.2.1. The Gram matrix in function of the EDM

We “invert” Equation 8 to compute the matrix in function of the matrix . We sum Equation 8 over all values of , obtaining

(9)

By Equation 7, the negative term in the right‐hand side of Equation 9 is zero. On dividing through by n, we have

(10)

Similarly for , we obtain

(11)

We now sum Equation 10 over all j, getting

(12)

(the last equality in Equation 12 holds because the same quantity is being summed over the same range , with the symbol k replaced by the symbol i first and j next). We then divide through by n to get

(13)

We now rearrange Equations 8, 11, 10 as follows:

(14)

(15)

(16)

and replace the left‐hand side terms of Equations 15 and 16 into Equation 14 to obtain

(17)

whence, on substituting the last term using Equation 13, we have

(18)

It turns out that Equation 18 can be written in a matrix form as

(19)

where and .

Gram matrices are PSD matrices

Any Gram matrix derived by a point sequence (also called a realization) in for some non‐negative integer K has two important properties: (i) the rank of G is equal to the rank of x; and (ii) G is PSD, i.e.,  for all . For simplicity, we only prove these properties in the case when is a matrix, i.e., , and is a scalar for all (this is the case in Schoenberg's problem above).

• (i)

  The ith column of G is the vector x multiplied by the scalar , which means that every column of G is a scalar multiple of a single column vector, and hence that ;

• (ii)

  For any vector y, .

Moreover, G is a Gram matrix only if it is PSD. Let M be a PSD matrix. By spectral decomposition there is a unitary matrix Y such that , where Λ is diagonal. By positive semidefiniteness, for each i, so exists. Hence , which makes M the Gram matrix of the vector . This concludes the proof of Theorem 7.1.

7.3. Finding the realization of a Gramian

Having computed the Gram matrix G from the EDM D in Section 7.2., we obtain the corresponding realization x as follows. This is essentially the same reasoning used above to show the equivalence of Gramians and PSD matrices, but we give a few more details.

Let be the matrix with the eigenvalues along the diagonal and zeroes everywhere else, and let Y be the matrix having the eigenvector corresponding to the eigenvalue as its jth column (for ), chosen so that Y consists of orthogonal columns. Then . Since Λ is a diagonal matrix and all its diagonal entries are nonnegative (by positive semidefiniteness of G), we can write Λ as , where . Now, since ,

which implies that

(20)

is a realization of G in .

8. Conclusion

We presented what we feel are the most important and/or beautiful theorems in DG (Heron's, Cauchy's, Cayley's, Menger's, Gödel's, and Schoenberg's). Three of them (Heron's, Cayley's, and Menger's) are concerned with the volume of simplices given its side lengths, which appears to be the central concept in DG. We think Cauchy's proof is as beautiful as a piece of classical art, whereas Gödel's proof, though less important, is stunning. Last but not least, Schoenberg's theorem is the fundamental link between the history of DG and its contemporary treatment.

Acknowledgments

The first author (L.L.) worked on this paper while employed at IBM TJ Watson Research Center, and is very grateful to IBM for the freedom he was afforded. The second author (C.L.) is grateful to the Brazilian research agencies FAPESP and CNPq.