Triangles III: Complex triangle functions

Summary This paper is the third in a series of three examining Euclidean triangle geometry via complex cross ratios. In the first two papers, we looked at triangle shapes and triangle coordinates. In this paper, we look at the triangle coordinates of the special points of a triangle, and show that they are functions of its shape. We then show how these functions can be used to prove theorems about triangles, and to gain some insight into what makes a special point of a triangle a centre.     

Triangles I: Shapes

Summary This paper is the first in a series of three examining Euclidean triangle geometry via complex cross ratios. In this paper we show that every triangle can be characterized up to similarity by a single complex number, called its shape. We then use shapes and two basic theorems about shapes to prove theorems about similar triangles. The remaining papers in this series will examine complex triangle coordinates and complex triangle functions.     

Reconstructing polygons from x-rays

We present strategies for interactively reconstructing polygons from carefully chosen x-ray probes, generalizing previous results for convex polygons to a significantly larger class of objects. In particular, we show that n+h+2 parallel x-ray probes are sufficient to determine an n-gon P with h vertices on its convex hull, provided no three vertices of P are collinear. If given an upper bound n on the number of vertices of P, then 2n+2 parallel probes or 3n origin probes suffice. Further, we show that lg n–2 probes are necessary. Finally, we present verification strategies for arbitrary polygons. Interactive probing strategies have the potential to minimize radiation exposure in medical imaging.     

Complex hyperbolic Fenchel-Nielsen coordinates

Let Σ be a closed, orientable surface of genus g. It is known that the representation variety of π₁(Σ) has 2g−3 components of (real) dimension 16g−16 and two components of dimension 8g−6. Of special interest are the totally loxodromic, faithful (that is quasi-Fuchsian) representations. In this paper we give global real analytic coordinates on a subset of the representation variety that contains the quasi-Fuchsian representations. These coordinates are a natural generalisation of Fenchel-Nielsen coordinates on the Teichmüller space of Σ and complex Fenchel-Nielsen coordinates on the (classical) quasi-Fuchsian space of Σ.     

Functional equations associated with triangle geometry

Summary Triangle geometry is treated in the context of functional equations of three variablesa, b, c which may be regarded as the sidelengths of a variable triangle. Trianglecenters (e.g., incenter, circumcenter, centroid), andcentral lines (e.g., the Euler line) are defined and partitioned into classes:0-centers, 1-centers, 2-centers and0-lines, 1-lines, and 2-lines. Criteria for parallelism, perpendicularity, and other geometric relations are proved in terms of these classes. The Euler line and central lines parallel or perpendicular to the Euler line serve as examples.     

Symmetric incidence groupoids

In [7] point-reflection geometries were studied which can be derived from commutative kinematic spaces without involutory elements. But the class of point-reflection geometries is larger. For example, elliptic planes with their reflections cannot be derived from commutative kinematic spaces. Here we investigate a larger class of reflection geometries.This paper was sponsored by Vigoni Program 1999.     

Vectorspacelike representation of absolute planes

The pointset E of an absolute plane can be provided with a binary operation "+" such that (E, +) becomes a loop and for each a E \ {o} the line [a] through o and a is a commutative subgroup of (E, +). Two elements a, b E \ {o} are called independent if [a] ∩ [b] = {o} and the absolute plane is called vectorspacelike if for any two independent elements we have E = [a] + [b] := {x + y | x [a], y [b]}. If is singular then (E, +) is a commutative group and is vectorspacelike iff is Euclidean. If is a hyperbolic plane then is vectorspacelike and in the continous case if a, b are independent, each point p has a unique representation as a quasilinear combination p = α · a + μ · b where α · a [a]and β · b [b] are points, α, β real numbers such that λ (o, λ · a) = |λ|· λ (o, a) and λ (o, μ · b) = |μ|. λ(o, b) and λ is the distance function. This work was partially supported by the Research Project of MIUR (Italian Ministery of Education and University) “Geometria combinatoria e sue applicazioni” and by the research group GNSAGA of INDAM. Dedicated to Walter Benz on the occasion of his 75 ^th birthday, in friendship     

Antipodes sur un tétraèdre régulier

The aim of this article is to compute the set of farthest points of an arbitrary point of the surface of a regular tetrahedron endowed with its intrinsic metric.     

On affine images of regular pentagons and pentagrams

In this paper we explore pentagons that are affine images of the regular pentagon and the regular pentagram. We obtain their characterizations in terms of two mild forms of regularity that deal with the notions of medians for a pentagon and the natural requirement that they are concurrent. Using these characterizations we show that there are various values involving the number 5 (thus related to the golden section) for which a careful selection of division points on appropriate segments determined by any pentagon will result in a pentagon that is the affine image of either a regular pentagon or a regular pentagram.     

Configurations of inscribed equilateral triangles

This paper explores equilateral triangles XYZ with vertices on sidelines of a given triangle ABC such that one side of XYZ is parallel to the corresponding side of ABC. There are six such triangles. They have many interesting properties which we investigate using trilinear coordinates. Our results improve and add to the earlier results of Blas Herrera Gómez about these configurations. We obtain new characterizations of several central points of the triangles and identify interesting pairs of triangles that are homologic (or perspective) and orthologic. The recognition of the Darboux cubic of a triangle is also accomplished in these configurations. Triples of circles intersecting in a point and six points on a conic also appear.      

On triangulations of the convex hull ofn points

A setS ofn points in Euclideand-space determines a convex hull which can be triangulated into some numberm of simplices using the points ofS as vertices. We characterize those setsS for which all triangulations minimizem. This is used to characterize sets of points maximizing the volume of the smallest non-trivial simplex. This work was supported in part by NSF Grants MCS 81-02519 and MCS 82-03347. This work supported in part by NSF Grants MCS 81-02519 and MCS 82-03347 Dedicated to Paul Erdős on his seventieth birthday     

Self-duality of chiral polytopes

We prove that self-dual chiral polytopes of odd rank possess a polarity, that is, an involutory duality, and give an example showing this is not true in even ranks. Properties of the extended groups, that is of the groups of automorphisms and dualities, of self-dual chiral polytopes are discussed in detail.     

How many atoms can be defined by boxes?

We study the functionb(n, d), the maximal number of atoms defined byn d-dimensional boxes, i.e. parallelopipeds in thed-dimensional Euclidean space with sides parallel to the coordinate axes. We characterize extremal interval families definingb(n, 1)=2n-1 atoms and we show thatb(n, 2)=2n ²-6n+7. We prove that for everyd, exists and . Moreover, we obtainb*(3)=8/9.     

A combinatorial theory of Grünbaum's new regular polyhedra,Part II: Complete enumeration

The new regular polyhedra as defined by Branko Grünbaum in 1977 (cf. [5]) are completely enumerated. By means of a theorem of Bieberbach, concerning the existence of invariant affine subspaces for discrete affine isometry groups (cf. [3], [2] or [1]) the standard crystallographic restrictions are established for the isometry groups of the non finite (Grünbaum-)polyhedra. Then, using an appropriate classification scheme which—compared with the similar, geometrically motivated scheme, used originally by Grünbaum—is suggested rather by the group theoretical investigations in [4], it turns out that the list of examples given in [5] is essentially complete except for one additional polyhedron.So altogether—up to similarity—there are two classes of planar polyhedra, each consisting of 3 individuals and each class consisting of the Petrie duals of the other class, and there are ten classes of non planar polyhedra: two mutually Petrie dual classes of finite polyhedra, each consisting of 9 individuals, two mutually Petrie dual classes of infinite polyhedra which are contained between two parallel planes with each of those two classes consisting of three one-parameter families of polyhedra, two further mutually Petrie dual classes each of which consists of three one parameter families of polyhedra whose convex span is the whole 3-space, two further mutually Petrie dual classes consisting of three individuals each of which spanE ³ and two further classes which are closed with respect to Petrie duality, each containing 3 individuals, all spanningE ³, two of which are Petrie dual to each other, the remaining one being Petrie dual to itself.In addition, a new classification scheme for regular polygons inE ⁿ is worked out in §9.     

A geometric construction of the K-loop of a hyperbolic space

It is well known that the homogeneous orthochronous proper Lorentzgroup is isomorphic to the proper motion group of the hyperbolic space. To each Lorentz boost \ {id} there corresponds in the hyperbolic space exactly one lineL such that fixes each of the two ends ofL . Furthermore has no fixed points but each plane containingL is fixed by . If we fix a pointo, then to each other pointa there is exactly one boosta ⁺ such thatL _a+ is the line joiningo anda anda ⁺(o)=a. The set P of points of the hyperbolic space is turned in a K-loop (P, +) bya+b:=a ⁺(b). Each line of the hyperbolic space has the representationa+Z(b) wherea, b P,b 0 andZ(b):= {x P |x+b=b+x}.Dedicated to H. Salzmann on the occasion of his 65th birthdaySupported by the NATO Scientific Affairs Division grant CRG 900103.     

On combinatorial properties of spheres in euclidean spaces

For λ>√2 there exists a triangle-free graphG such that for nod canG be imbedded into thed-dimensional unit sphere with two points joined if and only if their distance is >λ.     

On the products of cross-ratios on diagonals of polygons

The purpose of this paper is to establish a conjecture of B. Grünbaum, which states that in every n-polygon P in the plane, n 5, some diagonals intersect in a pattern that defines a new n-polygon (P), such that the product of the cross-rations on the diagonals of P is equal to the product of the corresponding cross-ratios on the diagonals of (P).