A note on a problem of Apollonius

Degenerate cases of the problem of Apollonius, to construct a circle tangent to each of three given circles, are discussed and exhaustively classified for proper circles (finite and non-zero radius). Singular cases are considered, and an outline of the extension of the problem to higher dimensions given. Amusing alternative interpretations of the results are obtained.     

The Apollonius contact problem and Lie contact geometry

A simple classification of triples of Lie cycles is given. The class of each triad determines the number of solutions to the associated oriented Apollonius contact problem. The classification is derived via 2-dimensional Lie contact geometry in the form of two of its subgeometries—Laguerre geometry and oriented M?bius geometry. The method of proof illustrates interactions between the two subgeometries of Lie geometry. Two models of Laguerre geometry are used: the classic model and the 3-dimensional affine Minkowski space model.     

A case of the 3-dimensional problem of Apollonius

Summary Inn-dimensions the problem of Apollonius is to determine the (n–1)-spheres tangent ton+1 given (n–1)-spheres. In case no two of the given (n–1)-spheres intersect and no three have the property that one separates the other two, the expected number of solutions is 2 ⁿ⁺¹. Whenn=2 this special problem does indeed always have 8 solutions, but for higher dimensions it turns out that the number of solutions becomes dependent on the relative size and location of the given (n–1)-spheres. We describe in detail the dependence of the number of solutions in the case of the 3-dimensional problem of Apollonius on the 6 inversively invariant parameters that describe configurations of 4 given spheres. We find that the number of solutions, if finite, can be any integer from 0 to 16 and, if infinite, can be a one-, two- or three-fold infinity where the stated multiplicity refers to the number of one-parameter families of solutions that are present.     

The Apollonius problem in four-dimensional Laguerre planes

The number of circles of a four-dimensional locally compact Laguerre plane touching three given circles or points depends only on the given geometric configuration but not on the Laguerre plane.     

On the circles of Apollonius associated with a triangle

Given a ratio , >>0, and a triangle ABC, on the sides and , using ratios , and , three circles of Apollonius are denned. In this paper, we will show that the three centers are collinear, the circles are coaxal and develop a necessary and sufficient condition that these circles intersect. J. A. Hoskins, W. D. Hoskins and R. G. Stanton obtained these results in a recent paper using algebraic computation. Our aim is to establish all these results using only results from elementary Euclidean geometry and thereby uncovering more geometric insights and avoid lengthy calculations.     

Hyers-Ulam stability of a generalized Apollonius type quadratic mapping

Let X,Y be linear spaces. It is shown that if a mapping satisfies the following functional equation:

(0.1)     

The Kissing Triangles: The Aesthetics of Mathematical Discovery

Papert's (1978) appeal to reconsider the power and possibilities of the aesthetic in mathematics learning is often ignored in mathematics education research. This paper begins with the premise, put forth by Dewey (1934), that the aesthetic structures many dimensions of inquiry and experience. In the same way that using particular paintings, musical compositions, or even everyday experiences has been instrumental to attempts by philosophers to understand the aesthetic dimensions of meaning and experience in artistic domains, I propose that analysing a particular encounter with mathematics may help reveal the nature and role of the often nebulous responses of elegance, beauty, and `fit' to which mathematicians lay claim in their mathematical activity. To achieve this, I draw on and adapt the defining features of the aesthetic character of experience set forth by the aesthetician Beardsley (1982). This, in turn, sheds light on the role thataesthetics can play in mathematical inquiry and experience, and provides initial categories and conjectures that can be used to investigate the potential roles of aesthetics in mathematics learning contexts.This revised version was published online in September 2005 with corrections to the Cover Date.     

A new hyperstability result for the Apollonius equation on a restricted domain and some applications

Let \((G,+)\) be an abelian group equipped with a complete ultrametric d that is invariant (i.e., \(d(x + z, y + z)= d(x, y\)) for \(x, y, z \in G\)), X be a normed space and \(U\subset X\setminus \{0\} \) be a nonempty subset. Under some weak natural assumptions on U and on the function \(\chi :U^3\rightarrow [0,\infty )\), we study new hyperstability results when \(f:U\rightarrow G\) satisfy the following Apollonius inequality

$$\begin{aligned}&d\Big (4f\Big (z-\frac{x+y}{2}\Big )+f(x-y),2f(x-z)+2f(y-z)\Big )\leqslant \chi (x,y,z),\\ {}&\quad x, y, z\in U,\;\;x-z,y-z,x-y,z-\frac{x+y}{2}\in U. \end{aligned}$$

Moreover, we derive some consequences from our main results.

     

The Circle Semigroup of a Ring

Let R be a ring and define x ○ y = x + y - xy, which yields a monoid (R, ○), called the circle semigroup of R. This paper investigates the relationship between the ring and its circle semigroup. Of particular interest are the cases where the semigroup is simple, 0-simple, cancellative, 0-cancellative, regular, inverse, or the union of groups, or where the ring is simple, regular, or a domain. The idempotents in R coincide with the idempotents in (R, ○) and play an important role in the theory developed. This revised version was published online in July 2006 with corrections to the Cover Date.     

A Decomposition of the Rotation of Circle

The paper concerns the dynamics-related properties of the rotation map of a circle (rotation of the plane). A self-similar structure of orbits of the rotation map is established. That is, a possibility of decomposition of orbits of a given rotation map into a finite set of orbits of other such maps is proved—it is shown that every orbit of iterates of the rotation of circle on irrational angle, after linear re-scaling of its argument can be represented as a finite set of such orbits situated on another circles. A pointwise self-similarity of classical trigonometric system is established and an application to Fourier expansion, which emphasizes a possibility of shifting of signals with respect to time, is presented. The free mechanical motion is also considered. A special dynamical spectrum of frequencies or speeds, associated with a given uniform circular or rectilinear motion, is defined. We prove that an appropriate fragmentation of time axis yields a decomposition of a given orbit of the free continuous-time motion into a set of such orbits propagating in new time and such decomposition is consistent with the decomposition of the per time unit discrete motion. Particularly, our theorems assert that due to a piecewise-linear transform of spatial and time variables the rectilinear rays change their direction.     

A Stein Conjecture for the Circle

We discuss the manner in which one might expect directional maximal functions to control the Fourier extension operator via L ² weighted inequalities. We prove a general inequality of this type for the extension operator restricted to circles in the plane.     

空间曲线的双切圆问题

Peter和Duan研究了平面曲线双切圆问题,Duau还研究了Rn中的(n-1)维闭子流形的双切球问题.本文则研究了空间曲线的双切圆问题,并利用几何分析的方法,获得了如果空间曲线是优良曲线则必定存在双切圆的结果,推广了平面曲线必定存在双切圆的结果.     

A Conformal Averaging Process on the Circle

In this paper we consider one of the simplest discrete conformally natural averaging processes. We start with a collection P of n ≥ 5 cyclically ordered points on the circle and produce a new collection P′ of n cyclically ordered points on the circle. The naturality means that (T(P))′=T(P′) for T any Möbius transformation of the circle. Let P ^(k) denote the kth iterate of P under the process. We show, for any initial choice P, that the limit lim _k→∞ P ^(K) exists and is conformally equivalent to a collection of n evenly spaced points. The convergence is exponentially fast. We also consider our process on projective 1-manifolds, which are topological circles equipped with exotic geometric structures based on the Lie group PSL₂(R).     

The Flow Lab: A Simple Activity for Generating NOS Principles

While there is near universal agreement within the science education community that a strong understanding of nature of science (NOS) ‐ both as a student goal and a teacher attribute ‐ is critical, we still struggle with how to achieve these aims. Barriers range from pragmatic logistics to fundamental curricular tensions. In this paper, we share a classroom activity designed to aid in the learning of key NOS principles that we have found constructive. We discuss the basic activity, connections to science studies literature, and opportunities for explicit reflection on nature of science.     

The Lie Algebra of Homeomorphisms of the Circle

We define and study an infinite-dimensional Lie algebrahomeo₊which is shown to be naturally associated to thetopologicalLie grouphomeo₊of all orientation-preserving homeomorphisms of the circle. Roughly, we rely on the universal decorated Teichmüller theory developed before as motivation to provide Fréchet coordinates on the homogeneous space given byhomeo₊modulo the group of real fractional linear transformations, whose corresponding vector fields on the circle we then extend by the usual Lie algebrasl₂of real traceless two-by-two matrices in order to definehomeo₊. Surprisingly,homeo₊turns out to be equal to the algebra of all vector fields on the circle which are “piecewisesl₂” in the obvious sense. It is evidently important to consider the relationship between our new Fréchet coordinates and the usual trigonometric functions on the circle, and we undertake here both natural infinitesimal calculations. We finally apply some further previous work in order to give sufficient conditions on the Fourier coefficients of a certain class of homeomorphisms of the circle which arises naturally in topology and number theory.     

Circle packings of maps —The Euclidean case

In an earlier work, the author extended the Andreev-Koebe-Thurston circle packing theorem. Additionally, a polynomial time algorithm for constructing primal-dual circle packings of arbitrary (essentially) 3-connected maps was found. In this note, additional details concerning surfaces of constant curvature 0 (with special emphasis on planar graphs where a slightly different treatment is necessary) are presented. Conferenza tenuta il 24 novembre 1997 Supported in part by the Ministry of Science and Technology of Slovenia, Research Project J1-7036 This note is a supplement to the Colloquium talk of the author at the “Seminario Matematico e Fisico di Milano” in November 1997.