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 II: Complex triangle coordinates

Summary This paper is the second in a series of three examining Euclidean triangle geometry via complex cross ratios. In the first paper of the series, we examined triangle shapes. In this paper, we coordinatize the Euclidean plane using cross ratios, and use these triangle coordinates to prove theorems about triangles. We develop a complex version of Ceva's theorem, and apply it to proofs of several new theorems. The remaining paper of this series will deal with 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.     

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.     

Miquel Sätze in Minkowski—Ebenen I

Summary For the coordinization of axiomatically given nonlinear incidence structures, the theorem of Miquel acts as a powerful tool. In the literature there exist many different versions of this theorem, especially in the most complicated area of Minkowski geometries. In three papers (this one and the following parts II, III) all possibilities to formulate Miquelian theorems are discussed. This part I and the first half of part II mainly deal with conditions which are necessary to get interesting results. Having done this troublesome job we are able to present a big number of Miquelian theorems which can be used to coordinize Minkowski geometries, too. At the end of part III we present a peculiar version of a nontrivial theorem of Miquelian type. Up to now this is the only one which is not equivalent to the other interesting theorems of Miquel.

     

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 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).     

A combinatorial theory of Grünbaum's new regular polyhedra,part I: Grünbaum's new regular polyhedra and their automorphism group

The new regular polyhedra, defined and investigated by Branko Grünbaum in [4], and theirn-dimensional generalizations are classified in terms of their symmetry group.     

Polar Graphs and Corresponding Involution Sets, Loops and Steiner Triple Systems

A 1-factorization (or parallelism) of the complete graph with loops is called polar if each 1-factor (parallel class) contains exactly one loop and for any three distinct vertices x₁, x₂, x₃, if {x₁} and {x₂, x₃} belong to a 1-factor then the same holds for any permutation of the set {1, 2, 3}. To a polar graph there corresponds a polar involution set , an idempotent totally symmetric quasigroup (P, *), a commutative, weak inverse property loop (P, + ) of exponent 3 and a Steiner triple system . We have: satisfies the trapezium axiom is self-distributive ⇔ (P, + ) is a Moufang loop is an affine triple system; and: satisfies the quadrangle axiom is a group is an affine space.     

Angles between Euclidean subspaces

The angle between two subspaces of dimensions p and q in a Euclidean space is considered by using exterior algebra. Some properties of angles are obtained. The relation between such a higher dimensional angle and the usual principal angles is also given. And finally, an application to Grassmann manifolds is briefly discussed.Supported by the National Natural Science Fund of P.R. China.