Angle Spaces and their conformal embeddability in E n
AAngle Spaces and their conformal embeddability in E n L. Felipe Prieto-Mart´ınezNovember 30, 2020
Abstract
In this article we provide a definition of the concept of angle space. Unlike previous works onthis topic, we do it for sets with a notion of betweenness that are not necessarily metric spaces.After this, we solve two problems proposed by K. Menger in the Annals of Mathematics in 1931.The first one consists in characterizing angle spaces that can be embedded in the euclidean plane E . We answer this question in a general way characterizing those that can be embedded in E n .And the second one concerns a characterization of those angle spaces admitting a distance functioncompatible with the angle space structure. Let ( X , d ) be a metric or semi-metric space. Menger explored in various papers (see [8], for example)the notion of angle for these spaces. Let us denote by X ∗ = { ( a, b, c ) ∈ X : a (cid:54) = b, b (cid:54) = c } . K. Theauthor defined angle functions to be maps ∠ : X ∗ → [0 , π ] such that(1.1) ∠ ( A, B, C ) = ∠ ( C, B, A ) = d ( B, A ) + d ( A, C ) = d ( B, C ) or d ( B, C ) + d ( C, A ) = d ( B, A ) π if d ( A, C ) = d ( A, B ) + d ( B, C ) > angle space was then a triple ( X , d, ∠ ), where X is a set, d a distance or semi-distance and ∠ an angle function .Menger already realized that for his triples ( X , d, ∠ ) it is interesting to discuss if the distance functionand the angle function are compatible, in the sense that they are in the euclidean space, that is, forevery ( B, A, C ) ∈ X ∗ we have:(1.2) d ( B, C ) = d ( B, C ) + d ( A, B ) + 2 · d ( B, C ) · d ( A, B ) · cos ∠ ( B, A, C )In this article we use the term euclidean-compatible for a distance and angle functions satisfying thecondition above.
Euclidean-compatiblity is necessary for the triple ( X , d, ∠ ) to be, both isometricallyand conformally, embedded in the euclidean space E n .After this, other attempts have been done to provide a useful definition of angle space . Always inthe setting of metric spaces. • Some years after the article by Menger W. A. Wilson in [13] studied the concept of angle forconvex complete metric spaces X satisfying, what he called, the “four point property”: everyset of four points { A, B, C, D } ⊂ X can be embedded in E n for some n . In this setting, thereis a notion of line, ray and segment, and it is possible to define angles between them from theeuclidean-compatibility property. 1 a r X i v : . [ m a t h . M G ] N ov Of course, the more complicated the structure of the metric space is and the more axioms thatare asked for the angle function, the more properties are satisfied by ( X , d, ∠ ). This point of viewlead to the authors of [12] to restrict theirselves to studying angles in linear normal spaces. • On the other hand, the recent paper [9] makes a great effort to offer a definition of angle betweenthree points for general metric spaces. The motivation for this approach is, as the author explains,that the definition suggested for the angle of p, x, q in the Euclidean plane gives the standard anglebetween three points and in a Riemannian manifold coincides with the usual angle between thegeodesics, if x is not in the cut locus of p or q . In his paper [8], Menger stated the following two problems:
Problem 1 (copied literally from page 750 in [8] ) It would be desirable to know relations betweenthe angles of an angle-space, characteristic for those angle-spaces which are conformal to a subset ofthe plane and to the plane itself.
Problem 2 (copied literally from page 750 in [8] ) In a metrical space (...) each three points arecongruent to three points of the plane and, therefore, the definition of distances induces a definition ofangles (viz. the angles in the corresponding plane triangles). A problem (connected, of course, with thefirst mentioned) is to find conditions which are necessary and sufficient in order that in an angle-space,which at the same time is a metrical space, the angles are identical with the angles induced by thedefinition of distance.
In this paper, we provide a simple and general axiomatic definition of angle function for spaces X that are not necessary metric or semi-metric (Definition 7). This eliminates the hassle of discussing therelation between the distance and angle functions a priori. We defend that the minimum structure thatshould be required for X in order to have a not artificial notion of angle is a relation of betweenness B . There exists recent bibliography studying this kind of spaces (see [1, 3]) and it is reviewed in thefollowing section. Not only seems to us that this context is more natural to define angles, but it is alsomore general since every metric space ( X , d ) has a natural notion of betweenness :(1.3) B is between A, C ⇐⇒ d ( A, B ) + d ( B, C ) = d ( A, C )This metric betweenness is already captured in the definition by Menger (1.1). Our definition has, inaddition, one more requirement than the obvious generalization of (1.1) for spaces with a notion of betweenness : the
First Axiom of Collinearity . This new axiom seems to us to be basic if we expect toobtain a reasonable notion of angle. So, for us, angle spaces are triples ( X , B , ∠ ) where B is a relationof betweenness and ∠ is an angle function satisfying certain properties with respect to it. As usual, wesay that a function between two angle spaces is conformal if it preserves the angle.Once we have settled our framework, the rest of the article is devoted to solving the two problemsproposed by Menger. Concerning the second one, since we are not restricted to the case in which theangle space is a metric space, we now split the problem in two parts: Problem 3
Characterize metric spaces ( X, d ) that admits an angle function ∠ which is euclidean-compatible to d . Problem 4
Characterize angle spaces ( X , B , ∠ ) that admits a distance function d euclidean-compatibleto ∠ . The first one is easier, and it is solved in Section 3. The candidate to be the angle function is obvious( ∠ ( A, B, C ) can be obtained from (1.2)). This problem was trivial in the setting of Menger, but recallthat ours is slightly more restrictive and we need to check this new
First Axiom of Collinearity . The2econd problem is more complicated. We need to answer the question first for angle spaces of three andfour points (sections 4 and 5) and then we take care of the general case in Section 6 (see Theorem 17).After this, in Section 7, we solve the strong version of Problem 1 (Theorem 20):
Problem 5
Find relations, for each n ∈ N , between the angles of an angle-space, characteristic forthose angle-spaces which are conformal to a subset of E n or to E n itself. To do so, see that if ( X , B , ∠ ) can be conformally embedded in E n , then it admits a distance function d euclidean-compatible to ∠ . In this case, checking that ( X , B , ∠ ) can be conformally embedded into E n is equivalent to seeing that ( X , d ) can be isometrically embedded in E n . Menger, in his article [6]dated in 1928, had already provided a characterization of those metric spaces X that are isometricallyembeddable in E n . We recommend also the recent article [2] updating the notation and explaining indetail the results in [6]. This characterization is stated in terms of a condition that must satisfy all thefinite metric subspaces of n + 2 points of X . So our solution for Problem 5 is also given in terms of acondition for all the finite angle subspace of n + 2 points of X (but stated only in terms of the anglefunction, of course).In Section 8 we provide three open problems and some lines for future work. We also explain howsome angle spaces are related to important old and unsolved problem in Plane Geometry ( The InscribedSquare Problem ). In [4] some axiomatic approaches for the concept of betweeness for geometric use are discussed. Amongall the possibilities, we are interested in the, not very restrictive, following one:
Definition 6 (pseumetric betweenness)
A relation of betweennes in a set X is a ternary relation B satisfying the condition listed below. If ( A, B, C ) ∈ B we say that B is between A and C . And ifeither ( B, A, C ) ∈ B , ( A, B, C ) ∈ B or ( A, C, B ) ∈ B then we say that A, B, C are collinear.(B1) If ( A, B, C ) ∈ B , then A, B, C are distinct.(B2) If ( A, B, C ) ∈ B , then ( C, B, A ) ∈ B .(B3) If ( A, B, C ) ∈ B , then ( B, A, C ) / ∈ B .(B4) (Strong Transitivity) If ( P , P , P ) , ( Q , Q , Q ) ∈ B and these two triples of points have twopoints in common, then any possible set of three different points in { P , P , P } ∪ { Q , Q , Q } iscollinear. This definition provide us a framework for our concept of angle:
Definition 7 (angle function)
Let X be a space with a notion of betweenness B . A function ∠ : X ∗ → [0 , π ] is an angle function if:(i) (required in the original definition by Menger) It is symmetric in the first and third coordinates ∠ ( A, B, C ) = ∠ ( C, B, A ) .(ii) (required in the original definition by Menger) ∠ ( A, B, C ) = if A = C, ( B, A, C ) ∈ B or ( B, C, A ) ∈ B π if ( A, B, C ) ∈ B > in other case iii) (First Axiom of Collinearity) If ( B, C, D ) ∈ B , then for all A (cid:54) = B , ∠ ( A, B, C ) = ∠ ( A, B, D ) . We have some comments to do about these definitions:
Remark 8 In [1] (in relation to some results in [3] ) the concept of pseudometric betweenness is intro-duced. A relation of pseudometric betweeness is this one that satisfies ( B ) , ( B ) , ( B ) (appearing inthe previous definition) and(B4)’ (Transitivity) If ( A, B, C ) , ( A, C, D ) ∈ B , then ( A, B, D ) , ( B, C, D ) ∈ B ,instead of ( B . The reason of the name “pseudometric” is discussed there. According to this article, itseems that was also Menger in [6] the first one to prove that every metric space has a natural relation B of betweenness (the one given by (1.3) ).We can see that, if a space X has a notion of pseudometric betweenness and it is an angle space,then it satisfies the axiom ( B . So we have preferred to, directly, include it in our definition.In [1] , using the notion of collinearity, lines and segments are defined. But we prefer not to introducemore notation in this article. Instead, our condition of Strong Transitivity allow us to give a definitionof collinearity for sets of three or more points. We say that such a set is collinear if any subset of ofthree points in X is collinear. Remark 9
It is possible to define angle spaces for sets with no notion of betweenness. In this case,such a relation B would be induced by ( A, B, C ) ∈ B ⇐⇒ ∠ ( A, B, C ) = π . But in this case, axioms ( ii ) , ( iii ) would be stated in a more artificial way. Remark 10
Menger proposed a definition of angle function removing condition (iii). This may allowmore generality, but we feel that a minimum structure for the space X is needed. Also we think that apossible reason for this author to do it this way is that he was mainly interested in angle functions formetric spaces. Once we have settled the definition of betweenness relation and of angle function , an angle space is atriple ( X , B , ∠ ). We say that two angle spaces are conformal if there exists a one to one correspondencethat preserves the notion of betweeness and the angles.An angle space is called trivial if all of its triples of points are collinear. Trivial angle spaces areobviously embeddable in E and admit infinitely many distances euclidean-compatible with their anglefunction.See that if an angle space ( X , B , ∠ ) admits an euclidean-compatible distance function d then thenatural definition of betweenness for ( X , d ) (see (1.3)) and the one for ( X , ∠ ) coincide. As we have already discussed, given any metric space ( X , d ) we have a betweenness relation in X and aunique candidate for an angle function euclidean compatible with d . But this unique candidate may failto satisfy the axiom of collinearity. For example consider a metric space ( X , d ) consisting in four points { A, B, C, D } . Suppose that d ( A, B ) = d ( A, C ) = d ( B, C ) = 4, d ( B, D ) = d ( C, D ) = 2 and d ( A, D ) = 5.If we want to define a distance function ∠ euclidean-compatible with d see thatcos ∠ ( D, A, C ) = 2 + 4 − · · − , cos ∠ ( B, A, C ) = 4 + 4 − · · First Axiom of Collinearity .So we have the following: 4
A BBCC DD
Figure 1: D is between C and D and A is not collinear to any pair of points in { B, C, D } . Proposition 11
A metric space ( X , d ) admits an angle function ∠ euclidean-compatible with d if andonly if for every set of four different points { A, B, C, D } ⊂ X such that D is between B and C and A is not collinear to any pair of points in { B, C, D } , the relation of Stewart’s Theorem holds, that is: (3.1) d ( A, C ) · d ( B, D ) + d ( A, B ) · d ( C, D ) = d ( B, C ) · ( d ( A, D ) + d ( B, D ) · d ( C, D )) Proof:
The angle function is defined as: ∠ ( B, A, C ) = arccos (cid:18) d ( A, B ) + d ( A, C ) − d ( B, C ) · d ( A, B ) · d ( A, C ) (cid:19) It is well defined since d satisfies the triangular inequality. It satisfies axioms ( i ) and ( ii ) in thedefinition of angle function. Finally, in order to satisfy the First Axiom of Collinearity we only needto check that for every four different points
A, B, C, D such that D is between B, C , ∠ ( A, B, C ) = ∠ ( A, B, D ). And this occurs if and only if (3.1) holds. (cid:50)
For a space X with a relation of betweenness, a (non-degenerated) n -gon is a set of n points, such thatno set of three of them is collinear. For n = 3 , , , trigons, tetragons, pentagons and hexagons . A set of n points where three or more of them are collinear is called degenerated n -gon .Obviously, if ( X , B , ∠ ) admits an euclidean compatible distance, all of its trigon also do. So thefollowing results shows a necessary condition for an angle space to admit an euclidean-compatibledistance function: Proposition 12
Consider the angle space ( X , B , ∠ ) and X (cid:48) = { A, B, C } ⊂ X to be a non-degeneratedtrigon. If X admits an euclidean-compatible distance function then ∠ ( B, A, C ) + ∠ ( A, B, C ) + ∠ ( A, C, B ) = π Proof:
Let us follow the usual convention:(4.1) a = d ( B, C ) , b = d ( A, C ) , c = d ( A, B ) , α = ∠ ( C, A, B ) , β = ∠ ( A, B, C ) , γ = ∠ ( B, C, A )The conditions for euclidean-compatibility are(4.2) a = b + c − bc cos αb = a + c − ac cos βc = a + b − ab cos γ α is the biggest angle between α, β, γ . From the last equation we obtain, since c > c = (cid:112) a + b − ab cos γ . Let us substitute this value in the other two equation to obtain: (cid:40) b − ab cos γ = b ( (cid:112) a + b − ab cos γ ) cos αa − ab cos γ = a ( (cid:112) a + b − ab cos γ ) cos β Since
A, B, C are different, a, b (cid:54) = 0. Also, since the points are not collinear, sinα, sinβ (cid:54) = 0. So we candivide both equations by ab to obtain:(4.3) (cid:40) ba − cosγ = ( (cid:113) ( ba − cosγ ) + sin γ ) cosα ab − cosγ = ( (cid:112) ( ab − cosγ ) + sin β ) cosβ −→ (cid:40) ba = sin ( α ± γ ) sinαab = sin ( β ± γ ) sinβ In each of these last two equations we have introduced an incorrect solution. Since α is the biggestangle, then cosα, cosβ >
0. And the sign is + in both equations. So we have that: sin ( α + γ ) sin ( β + γ ) = sinαsinβ −→ − cos ( α + β + 2 γ ) + cos ( α − β ) = − cos ( α + β ) + cos ( α − β ) −→−→ cos ( α + β + 2 γ ) = cos ( α + β )which, taking into account that α, β, γ ∈ [0 , π ], implies π = α + β + γ . (cid:50) If an angle space satisfies that for every trigon (degenerated or not) { A, B, C } we have that ∠ ( ABC ) + ∠ ( BCA ) + ∠ ( CAB ) = π , then we say that it is an euclidean angle space. See that if { A, B, C } is a degenerated trigon, then automatically ∠ ( ABC ) + ∠ ( BCA ) + ∠ ( CAB ) = π .As a consequence of the proposition above, we have the following result for trigons: Theorem 13 (Law of Sines for Euclidean Angle Spaces)
Let ( X , B , ∠ ) be an euclidean angle spacewhere X = { A, B, C } is a trigon. Then there is a unique distance function d , up to multiplication by aconstant, euclidean-compatible with ∠ . And we have that: (4.4) d ( B, C )sin α = d ( A, C )sin β = d ( A, B )sin γ Proof:
We have to show that (4.2) has a unique solution in the indeterminates a, b, c , up to multi-plication by a constant, with all its elements greater than 0. We continue from (4.3). Doing the sameas in the previous proof for the indeterminate c we reach the following: a = a ba = sin ( α + γ ) sinαca = sin ( α + β ) sinα −→ a = a ba = sin ( β ) sinαca = sin ( γ ) sinα (4.4) is a direct consequence of this expression. (cid:50) See also that if ( X, B , ∠ ) is an angle space, X is a trigon, and ∠ ( A, B, C )+ ∠ ( B, C, A )+ ∠ ( C, A, B ) = π , then it can be conformally embedded in E . The condition of being euclidean is not sufficient to ensure that an angle space ( X , B , ∠ ) admits aneuclidean compatible distance function. See for instante the tetragon in the following image, for which6 A BBCC DD C DAB BA we have drawn also a “plane diagram” (analogous to the net of a polyhedron, used for papercraftmodels).None of the possible sets of three points are collinear. All the angles in grey equal to π/
3. Suppose that d ( C, D ) = 1. It is easy to see (Law of Sines) that the, the distances d ( A, C ) = d ( A, D ) = d ( B, C ) = d ( B, D ) = 1. Finally see that if ∠ ( CAB ) = ∠ ( ABD ) = π/ ∠ ( ACB ) = π/ ∠ ( ADB ) = π/ AB would have a different value according to the Law of Sines in the trigons { A, B, C } and { A, B, D } .So to characterize angle spaces that admit euclidean-compatible distance functions, we are goingto begin with tetragons. We have already discussed, as trivial, the case in which all the points in thespace, four in this case, are collinear. So two more cases are left: the one in which three points arecollinear but the four of them are not, and the one in which no set of three points is collinear.We say that an angle space satisfies the Second Axiom of Collinearity if for every tetragon { A, B, C, D } such that ( A, B, C ) ∈ B and D is not collinear to any of them, we have that:(5.1) ∠ ( ADC ) = ∠ ( ADB ) + ∠ ( BDC ) , and ∠ ( ABD ) + ∠ ( CBD ) = π We have the following:
Lemma 14
Let an euclidean angle space ( X , B , ∠ ) where X = { A, B, C, D } is a tetragon such that ( A, B, C ) ∈ B and D not collinear to any of them. It admits an euclidean-compatible distance function d if and only if it satisfies the Second Axiom of Collinearity .Proof:
We are in the following situation:
AA BB CCDD
Denote by x = d ( A, B ), x (cid:48) = d ( B, C ), a = d ( A, D ), a (cid:48) = d ( C, D ), h = d ( B, D ) and α = ∠ ( ABD ). d ( A, C ) = x + x (cid:48) , since ( A, B, C ) ∈ B . Denote also by α (cid:48) = ∠ ( C, B, D ), β = ∠ ( BAD ) = ∠ ( CAD ), β (cid:48) = ∠ ( BCD ) = ∠ ( ACD ), γ = ∠ ( ADB ), γ (cid:48) = ∠ ( BDC ), δ = ∠ ( A, D, C ). As a consequence of the Lawof Sines, we have:(5.2) a sin α = h sin β = x sin γa (cid:48) sin α (cid:48) = h sin β (cid:48) = x (cid:48) sin γ (cid:48) x + x (cid:48) sin δ = a sin β (cid:48) = a (cid:48) sin β (cid:40) sin γ sin α a + sin γ (cid:48) sin α (cid:48) a (cid:48) = sin δ sin β a (cid:48) sin γ sin α a + sin γ (cid:48) sin α (cid:48) a (cid:48) = sin δ sin β (cid:48) a This system must have a non-zero solution. So the determinant of the matrix of coefficients (oncethe equations are suitably fixed) must be zero thus we obtain that:sin γ (cid:48) sin α (cid:48) sin β (cid:48) + sin γ sin α sin β = sin δ sin β sin β (cid:48) and then, since sin γ = sin( α + β ), sin γ (cid:48) = sin( α (cid:48) + β (cid:48) ) this yields:sin β sin β (cid:48) ( cotanα + cotanα (cid:48) ) + sin( β + β (cid:48) ) = sin δ but, since δ + β + β (cid:48) = π , we have that sin ( β + β (cid:48) ) = sinδ and so cotanα = − cotaα (cid:48) and α = π − α (cid:48) .Now since: α + β + γ = πα (cid:48) + β (cid:48) + γ (cid:48) = πβ + β (cid:48) + δ = π we can conclude that δ = β + β (cid:48) . Finally, assuming that (5.1) holds we have that (5.2) is equivalent to: a sin α = h sin β = x sin γa (cid:48) sin α (cid:48) = h sin β (cid:48) = x (cid:48) sin γ (cid:48) a sin β (cid:48) = a (cid:48) sin β which has a unique solution up to multiples in the variables a, a (cid:48) , x, x (cid:48) , h . (cid:50) Proposition 15 (metrizability condition for tetragons)
Consider an euclidean angle space ( X , B , ∠ ) where X = { A, B, C, D } is a tetragon. There is a unique distance function d in X which is euclidean-compatible with ∠ if and only if it satisfies the metrizability conditions : (5.3) sin ∠ ( DBA ) · sin ∠ ( BCA ) · sin ∠ ( CDA ) = sin ∠ ( CBA ) · sin ∠ ( DCA ) · sin ∠ ( BDA )sin ∠ ( CAB ) · sin ∠ ( DCB ) · sin ∠ ( ADB ) = sin ∠ ( DAB ) · sin ∠ ( ACB ) · sin ∠ ( CDB )sin ∠ ( BAC ) · sin ∠ ( DBC ) · sin ∠ ( ADC ) = sin ∠ ( DAC ) · sin ∠ ( ABC ) · sin ∠ ( BDC )sin ∠ ( BAD ) · sin ∠ ( CBD ) · sin ∠ ( ACD ) = sin ∠ ( CAD ) · sin ∠ ( ABD ) · sin ∠ ( BCD ) Proof:
Consider the point D and the following convention: AABB CCDD bc x c x b x a The distances x a , x b , x c are determined, up to multiplication by a constant, by the angles in thetriangles { A, B, D } , { A, C, D } and { B, C, D } according to: x a sin ∠ ( ABD ) = x b sin ∠ ( BAD ) x a sin ∠ ( ACD ) = x c sin ∠ ( DAC ) x b sin ∠ ( BCD ) = x c sin ∠ ( CBD )
8f and only if sin ∠ ( ACD ) · sin ∠ ( BAD ) · sin ∠ ( CBD ) = sin ∠ ( BCD ) · sin ∠ ( CAD ) · sin ∠ ( ABD ).On one hand, looking at these triangles we have that: x a sin ∠ ( ABD )sin ∠ ( ABD ) = x b sin ∠ ( ABD )sin ∠ ( BAD ) = c x a sin ∠ ( ADC )sin ∠ ( ACD ) = x c sin ∠ ( ADC )sin ∠ ( DAC ) = b x b sin ∠ ( BDC )sin ∠ ( BCD ) = x c sin ∠ ( BDC )sin ∠ ( CBD ) = a On the other hand and according to this, the distance is well defined if and only if: a sin ∠ ( BAC ) = b sin ∠ ( ABC ) = c sin ∠ ( ACB )which happens as a consequence of (5.3). Repeat the same argument for all the vertices. (cid:50)
Let ( X , B , ∠ ) be an euclidean angle space that satisfies the Second Axiom of Collinearity and thetetragon metrizability condition. Let
A, B, C, D such that not all of them are collinear. Define:(6.1) d A,B,λ ( C, D ) = λ if { A, B } = { C, D } sin ∠ ( ABD )sin ∠ ( ADB ) · λ if A = C sin ∠ ( ABC )sin ∠ ( ACB ) · λ if A = D sin ∠ ( BAD )sin ∠ ( ADB ) · λ if B = C sin ∠ ( BAC )sin ∠ ( ACB ) · λ if B = D sin ∠ ( CAB )sin ∠ ( ADC ) · d ( A, C ) = sin ∠ ( CBD )sin ∠ ( BCD ) · d ( B, D ) = if { A, B } ∩ {
C, D } (cid:54) = ∅ = sin ∠ ( CAB )sin ∠ ( ACD ) · d ( A, D ) = sin ∠ ( CBD )sin ∠ ( BDC ) · d ( B, C ) A=CD B A=D BC A B=CD A B=DC A B CD
Figure 2: All the possibilities listed in the definition of d A,B,λ (except the first one).See that if this angle space admits an euclidean-compatible distance d such that d ( A, B ) = λ , then d ( C, D ) = d A,B,λ ( C, D ) for every
C, D such that
A, B, C, D are not all of them collinear.
Lemma 16
Let ( X , B , d ) be an euclidean angle space that satisfies the Second Axiom of Collinearity and the tetragon metrizability condition. The following two statements are equivalent, and both will bereferred as the
Global Compatiblity Condition :(i) For every
A, B, A (cid:48) , B (cid:48) , C, D such that { A, B, C, D } , { A, B, A (cid:48) , B (cid:48) } , { A (cid:48) , B (cid:48) , C, D } have at leastthree different points that are not collinear. Then d A,B,λ ( A (cid:48) , B (cid:48) ) = λ (cid:48) = ⇒ d A (cid:48) ,B (cid:48) ,λ (cid:48) ( C, D ) = d A,B,λ (cid:48) ( C, D )9 ii) For every non-degenerated pentagon { A, B, C, D, E } ⊂ X (6.2) sin ∠ ( DAE ) · sin ∠ ( ABD ) · sin ∠ ( ACB ) · sin ∠ ( ADC ) · sin ∠ ( CED ) == sin ∠ ( CAD ) · sin ∠ ( ABC ) · sin ∠ ( DCE ) · sin ∠ ( ADB ) · sin ∠ ( AED ) and for every non-degenerated hexagon { A, B, C, D, E, F } ⊂ X (6.3) sin ∠ ( EAF ) · sin ∠ ( ABE ) · sin ∠ ( ACB ) · sin ∠ ( CDA ) · sin ∠ ( CED ) · sin ∠ ( EF C ) == sin ∠ ( CAD ) · sin ∠ ( ABC ) · sin ∠ ( ECF ) · sin ∠ ( CDE ) · sin ∠ ( AEB ) · sin ∠ ( EF A ) Proof:
Condition ( i ) is equivalent to saying that any hexagon in X , degenerated or not, admits awell defined euclidean-compatible distance function. The angle space being euclidean, the Second Axiomof Collinearity and the tetragon metrizability conditions, ensure that trigons and tetragons admit aneuclidean-compatible distance function. So we only need to care about non-degenerated pentagons andhexagons. The argument is very similar to the one in the proof of Proposition 15.Let us begin with hexagons. Every non-degenerated hexagon admits an euclidean-compatible dis-tance function if and only if for every labeling of its points { A, B, C, D, E, F } we have d A,B,λ ( C, D ) = λ (cid:48) = ⇒ d A,B,λ ( E, F ) = d C,D,λ (cid:48) ( E, F )and, according to (6.1), we can see that this is equivalent to (6.3).We can proceed in a similar way to show that every pentagon admits a well defined euclidean-compatible distance function d if and only if for every labeling of its points { A, B, C, D, E } we have(6.3). (cid:50) Theorem 17 (angle spaces that admit an euclidean-compatible distance)
For ( X , B , ∠ ) be anangle space with 4 our more points. Then it admits an euclidean-compatible distance function d if andonly if it satisfies the following conditions:(A) it is euclidean,(B) it satisfies the Second Axiom of Collinearity,(C) it satisfies the tetragon metrizability condition and(D) it satisfies the Global Compatibility Condition.In this case, the distance function d is unique up to multiplication by a constant.Proof: Fix two points
A, B ∈ X . We are going to construct the euclidean-compatible distancefunction d . Impose, that d ( P, P ) = 0 and that d ( A, B ) = λ . Let d A,B,λ be the function defined in (6.1).Consider any pair of distinct points
P, Q ∈ X . If
A, B, P, Q are not all of them collinear, then wecan define d ( P, Q ) from (6.1). See that, in this case, d ( P, Q ) = d ( Q, P ) and that d ( P, Q ) > P, Q simultaneously collinear to
A, B then, by hypothesis, we can choosea fifth point R which is not collinear to A, B , neither to
C, D . Define µ = d A,Bλ ( A, R ) and then define d ( P, Q ) = d A,R,µ ( P, Q ). d ( P, Q ) is, then, greater than 0. And it is well defined (it does not dependon the choice of R ) according to the Global Compatibility Condition . The symmetry condition for thedistance function is also guaranteed.Suppose that d does not satisfy the condition of euclidean-compatibility, that is, there exists anon-degenerated trigon { A (cid:48) , B (cid:48) , D } such that: d ( A (cid:48) , B (cid:48) ) sin ∠ ( A (cid:48) DB (cid:48) ) (cid:54) = d ( A (cid:48) , D ) sin ∠ ( A (cid:48) B (cid:48) D )Then this would contradict the Global Compatibility Condition in the pentagon { A, B, A (cid:48) , B (cid:48) , D } .The fact that d must satisfy the triangular inequality is a consequence of the euclidean-compatibility. (cid:50) Solution to Problems 1 and 5
C. L. Morgan provides in [10] a characterization of the metric spaces ( X , d ) that can be embedded in E n which is, to our pourposes, more suitable than the one given by Menger in [7]. To explain his resultwe need to introduce some notation.Let ( X , d ) be a metric space. Define, for every A, B, C ∈ X the quantity (cid:104)
A, B, C (cid:105) = 12 (cid:0) d ( A, C ) + d ( B, C ) − d ( A, B ) (cid:1) In the context of this article, our ( n + 1)-gons are called n -simplices. For each ( n + 1)-gon or n -simplex { X , . . . , X n } ⊂ X define: Vol n ( { X , . . . , X n } ) = 1 n ! (cid:112) D n ( X , . . . , X n )where D n ( X , . . . , X n ) is the determinant of the n × n matrix which i, j entry equals (cid:104) X i , X j , X (cid:105) . Thisfunction Vol n is well defined and is the so called simplicial n -volume. For general metric spaces, it canbe a complex number.We say that a space ( X , d ) is n - flat if the simplicial n -volume of every ( n + 1)-gon is a real number.In this case, its dimension is the largest natural number (if it exists) for which there is an ( n + 1)-gonwith positive simplicial volume. See that if, for some n , there is no ( n + 1)-gon with positive simplicialvolume then there is no ( k + 1)-gon with positive simplicial volume for k ≥ n . We have that: Theorem 18 (proved by Morgan in [10] ) A metric space ( X , d ) can be embedded in E n if and onlyif it is k -flat for every ≤ k ≥ n , and of dimension less than or equal to n . If the metric space ( X , d ) has also an angle function euclidean compatible with d , then (cid:104) A, B, C (cid:105) = d ( A, C ) · d ( B, C ) · cos ( ∠ ( A, C, B ))and so D n ( x , . . . , x n ) = (Π nj =1 d ( x , x j ) ) cos ∠ ( x , x , x ) . . . cos ∠ ( x , x , x n )... ... cos ∠ ( x n , x , x ) . . . cos ∠ ( x n , x , x n ) (cid:105) Let ( X , B , ∠ ) be an angle space that admits an euclidean compatible distance function d . It can beconformally embedded in E n if and only if ( X , d ) can be isometrically embedded in E n . So everythingtogether proves that: Corollary 19 (isometrical embeddabililty in terms of angles)
Let ( X , B , ∠ ) be an angle spacethat admits an euclidean compatible distance function. This angle space can be conformally embeddedin E n if and only if the two following properties hold:(1) Forall ≤ k ≤ n , { X , . . . , X k } ⊂ X , cos ∠ ( X , X , X ) . . . cos ∠ ( X , X , X k ) ... ... cos ∠ ( X k , X , X ) . . . cos ∠ ( X k , X , X n ) (cid:105) ≥ .(2) Forall subset { X , . . . , X n +1 } ⊂ X , cos ∠ ( X , X , X ) . . . cos ∠ ( X , X , X n +1 ) ... ... cos ∠ ( X n +1 , X , X ) . . . cos ∠ ( X n +1 , X , X n +1 ) (cid:105) = 0 .
11s a consequence of Theorem 17 and of the previous corollary we finally obtain:
Theorem 20 (solution for Problem 5)
Let ( X , B , ∠ ) be an angle space. It can be conformally em-bedded in E if and only if it can be endowed with an euclidean-compatible distance function, that is:(A) it is euclidean,(B) it satisfies the tetragon compatibility condition,(C) it satisfies the Second Axiom of Collinearity,(D) it satisfies the Global Compatibility Condition,and the resultant metric space can be isometrically embedded in E n , that is:(1) Forall ≤ k ≤ n , { X , . . . , X k } ⊂ X , cos ∠ ( X , X , X ) . . . cos ∠ ( X , X , X k ) ... ... cos ∠ ( X k , X , X ) . . . cos ∠ ( X k , X , X n ) (cid:105) ≥ .(2) Forall subset { X , . . . , X n +1 } ⊂ X , cos ∠ ( X , X , X ) . . . cos ∠ ( X , X , X n +1 ) ... ... cos ∠ ( X n +1 , X , X ) . . . cos ∠ ( X n +1 , X , X n +1 ) (cid:105) = 0 . Our result of conformal embeddability of angle spaces ( X , B , ∠ ), as the one by Menger for isometricalembedabbility of metric spaces ( X , d ), is more useful when X is a finite set. If X has an infinite numberof points, conditions (A), (B), (C), (D), (1) and (2) in Theorem 20 may not be easily checked. Sosome work is still left to be done. It would be desirable to find easily verifiable criterions for conformalembeddability for angle spaces with an infinite set of points.Consider the class T of angle spaces ( X , B , ∠ ) such that X is also a topological space homeomorphicto the unit circle S . There is a correspondence between Jordan curves an elements in T and betweenknots and elements in T . This new viewpoints for these objects may be of interest. See for examplethat a well known unsolved and old conjecture, the Inscribed Square Problem (see [5, 11]), can be statedin this terms, via this correspondence:
Conjecture 21
Let an angle space ( X , B , ∠ ) which is conformal to a Jordan curve. Then there existsa tetragon { A, B, C, D } such that: ∠ A, B, C = π/ , ∠ B, C, D = π/ , ∠ C, D, A = π/ , ∠ D, A, B = π/ and the rest of agles in the tetragon equal π/ . This lead to the following:
Problem 22
Find necessary and sufficient conditions for an angle space ( X , B , ∠ ) to be conformal toa Jordan curve (in terms of easily verifiable properties of the angle function ∠ ). Finally, it is also possible to consider problems of conformal embeddability of angle spaces in otherangle spaces. Also for non-euclidean geometries. For example:
Problem 23
Characterize those angle spaces ( X , B , ∠ ) that can be conformally embedded in the unitsphere S . Problem 24
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