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It is well known that a convex quadrilateral is a cyclic quadrilateral if, and only if, the sum of each pair of opposite angles is π. This result (which gives a necessary and sufficient condition for the existence of a circle which circumscribes a given quadrilateral) is beautifully complemented by Pitot’s theorem which says that a given convex quadrilateral has an inscribed circle if, and only if, the sum of the lengths of one pair of opposite edges is the same as the sum for the other pair. Henri Pitot, a French engineer, noticed the easy part of this result in 1725 (see Figure 1), and the converse was first proved by J-B Durrande in 1815. Accordingly, we shall say that a convex quadrilateral is a Pitot quadrilateral if, and only if, the sum of the lengths of one pair of opposite edges is the same as the sum for the other pair.
Many authors define an isometry of a metric space to be a distance-preserving map of the space onto itself. In this note, we discuss spaces for which surjectivity is a consequence of the distance-preserving property rather than an initial assumption. These spaces include, for example, the three classical (Euclidean, spherical, and hyperbolic) geometries of constant curvature that are usually discussed independently of each other. In this partly expository paper, we explore basic ideas about the isometries of a metric space, and apply these to various familiar metric geometries.
In [1] the second author observed that it is possible to have a binary operation * on a set X with the property that two different arrangements of brackets in a given combination x1 * … * xn of elements of X yield the same outcome for all choices of the xj. For example, for the operation of subtraction on the set of real numbers, we have
for all real numbers a, b, c and d. The author then asked whether or not a similar example might hold for an n-fold vector product on three-dimensional Euclidean space3. We shall show here that no such example can exist; thus two different arrangements of brackets in a repeated vector product will, for some vectors, yield different answers.
The set of solutions to the equation xy = yx has been studied extensively over the past three centuries, including work by well known mathematicians such as Daniel Bernoulli (1700–1782), Leonhard Euler (1707–1783), and Christian Goldbach (1690–1764). Various mathematicians have focused on the integer, rational, real, and complex solutions. For example, it has been shown (see [1]) that the equality 24 = 42 gives the only distinct integer solutions. Our exposition below presents some of the key ideas behind the positive real solutions to this equation and illustrates how rational solutions can be found. To learn more about the various solutions, the reader can consult the articles listed at the end of this paper, as well as the extensive references given in these articles. It is also possible to find some of this material on the Web.
Steiner’s proof of what is now called the Steiner-Lehmus theorem was published in 1844, the same year as the book The three musketeers, written by the French author Alexandre Dumas. The motto One for all, all for one (Einer für alle, alle für einen; Un pour tous, tous pour un; Uno per tutti, tutti per uno) of the three musketeers came into widespread use in Europe in the 19th century, and its essence is that the three musketeers are inseparable; each member pledges to support the group, and the group supports each member. Now there are three classical geometries of constant curvature, namely Euclidean, spherical and hyperbolic geometries, and one can argue that, like the three musketeers, these geometries should be considered as being inseparable; that is, an idea, theorem or proof in any one of them should automatically be considered in the other two. The issue here should be not only to decide whether a particular result is true, or false, in a given geometry, but to understand which particular properties of the geometries make it so.
Mathematical Explorations follows on from the author's previous book, Creative Mathematics, in the same series, and gives the reader experience in working on problems requiring a little more mathematical maturity. The author's main aim is to show that problems are often solved by using mathematics that is not obviously connected to the problem, and readers are encouraged to consider as wide a variety of mathematical ideas as possible. In each case, the emphasis is placed on the important underlying ideas rather than on the solutions for their own sake. To enhance understanding of how mathematical research is conducted, each problem has been chosen not for its mathematical importance, but because it provides a good illustration of how arguments can be developed. While the reader does not require a deep mathematical background to tackle these problems, they will find their mathematical understanding is enriched by attempting to solve them.
This book follows on from the book Creative Mathematics in this series which began with three essays (on research into, on writing about and on presenting mathematics) and then continued with a series of problems, each of which was divided into three parts. Part I provided an introduction to the problem followed by some elementary questions about the problem. Part II contained an answer to these questions, as well as a deeper discussion and a generalisation of the problem. This led to more advanced questions which were discussed in Part III.
This book is a natural development of this approach, the main purpose of which is to give the reader experience in working on (as far as the reader is concerned) unsolved problems. The problems in this book are, generally speaking, more difficult than those in Creative Mathematics, and we assume a greater level of maturity of the reader.
One of the main purposes of this book is to show that mathematical problems are often solved using mathematics that is not, at first sight, connected to the problem, and readers are encouraged (and even urged) to consider as wide a variety of mathematical ideas as possible when trying to solve a problem. The reader will no doubt have met problems in what might be called ‘recreational mathematics’ where problems are solved for amusement, without necessarily understanding or investigating the key mathematical ideas that lie behind the solution. Here, by contrast, we focus on the important underlying ideas rather than on the solution itself.
This book is written to help the reader learn how to do research in mathematics. Each chapter contains a project that has been chosen not because of its mathematical importance but because (in the view of the author) it provides a good illustration of how arguments develop, and how new questions arise once some progress is made. These projects have also been chosen because they do not require a deep mathematical background in order to understand the problem and start investigation. Nevertheless, the reader will probably have to learn some more mathematics in order to solve the problems. Some of the problems do not have easy answers, and some are not yet completely solved.
Each chapter focuses on one topic, and although some results and proofs are given in the discussion, many steps are omitted, and it is the responsibility of the reader to locate and fill these gaps. The general rule is that the reader should check every step and provide as much extra material as is necessary to ensure their complete understanding of each step. As we progress through a project, specific questions are asked, and the reader will need to interpret, or clarify, some of these before a solution is attempted. It hardly needs saying that the whole purpose of the book is that the reader should fully engage with these problems and fill in the (many) missing steps in the text itself. Although some theorems, and their proofs, are given, these do not have quite the same role as in most textbooks. The theorems given here serve the purpose of making further progress in order that we can ask yet more questions, for this is the real nature of research.