# The Circle. A Mathematical Exploration Beyond the Line

Alfred Posamentier has (co)authored a few dozen books on mathematics and mathematics education. Several of these are intended to popularize math. In 2012 he co-authored with Ingmar Lehmann a book on triangles *The Secrets of Triangles: A Mathematical Journey*. Triangles are about the simplest mathematical objects interesting enough to prove many theorems about, and they have many applications. Think of the fact that any polygonal area can be triangulated and if the triangular net is made fine enough, they approximate and subdivide any area or surface.

In this book, the triangles are replaced by circles as the key objects. It contains, among other things about circles, a large variety of classical and less known —but nevertheless quite interesting— theorems that can be formulated concerning properties of circles. These involve often cyclic polygons (i.e., whose vertices lie on a circle) or polygons whose sides are tangent to it. It will not come as a surprise that also here triangles still play a prominent part in this game too.

In a first chapter some elementary properties and definitions are recalled. It may be a surprise for some readers that the circle is not the only 2D object that has constant breadth. This property it has to share with the Reuleaux triangle, an "inflated" equilateral triangle that is the intersection of three circles with centers at the vertices and radius equal to the side length. A slightly flattened version of it found an ingenious application in the design of the Wankel engine.

Chapters 2 and 3 discuss a first set of theorems concerning circles. In most cases this starts with a known theorem like for example Ptolemy's (in a cyclic quadrilateral the product of the diagonals equals the sum of the products of the opposite sides). In the case of a rectangle this reduces to Pythagoras' theorem. Generalizations consider n-gons whose vertices are on a circle. This kind of strategy to discuss a theorem is repeated for other theorems: a formulation of the theorem is given, mentioning its origin and sometimes a few sentences about the mathematician that is behind it, then a proof, and by considering special cases, or sometimes generalizations, seemingly unrelated theorems turn out to be included as well. Almost always, there is a further exploration of the problem involved. For example if the theorem claims the collinearity of 3 points, then it may be followed by an analysis of properties about circumferences or areas of triangles and/or circles that were involved in the proof. In this way we are guided along the theorems of Simson, Miquel, Pascal, Brianchon, Ceva, the butterfly theorem, the nine point circle theorem, the six and the seven circles theorems, the Gergonne point theorem, Poncelet's porism, the arbelos, and Ford circles. When a proof is too complicated, it is not included (notes and references are at the end of the book), and sometimes more technical stuff is moved to an appendix, but most proofs are rather easy to follow and are brought in a reader friendly way with many figures that illustrate the successive steps to be followed in the proofs. So proofs are not hard abstract manipulations of formulas, but they rely heavily on the visualization. The text merely explains what the successive steps are, where for example the equal angles are, or the similar or congruent triangles are and why this should be true. It still requires some mental flexibility and an elementary geometric knowledge to follow each step of the proof but it is easy going.

Circle packing (in a confined space like a rectangle, a circle, or a triangle) is discussed in a short chapter 4. It is explained how it is related to an application in a computer program *TreeMaker* to design origami patterns.

The next 4 chapters deal with geometric constructions. Equicircles (the 4 circles that are tangent to all 3 sides of a triangle, 1 inside and 3 outside the triangle) get their own chapter with a computation of their centers and radii.

Chapter 6 is a discussion of the Apollonius problem. That is how to construct a circle that is defined in different ways by points, lines and other circles like containing 3 given points (PPP), or two points and a tangent line (PPL), or a point and 2 tangent lines (PLL), 2 points and a tangent circle (PPC), etc. There are 10 such possibilities. All constructions should be done with straightedge and compass.

The next chapter introduces reflection in a circle. This transforms circles and lines (i.e. circles with infinite radius) into circles and lines. Some of the previous Apollonius problems can be solved in the reflected setting. It can also help in the construction of Steiner and Pappus chains.

Finally chapter 8 is discussing Mascheroni constructions. The ancient Greek tradition allowed to use only straightedge and compass to do all the geometric constructions. It is impossible to do everything with only the straightedge, but one can do without it as was proved by Mohr and independently by Mascheroni. Of course we cannot draw a straight line with a compass, but one can construct any point on a specified line whenever it is needed. Of course the constructions are more involved, but it is shown that with 5 fundamental constructions everything can be done without the straightedge.

Chapter 9 is again a short interruption from the mathematics since it gives a very brief survey of how the circle was used in arts, in shaping the landscape, and in architecture. This topic could easily be the subject of a whole mathematical picture book on its own, but I do not think the authors have the intention of being complete here. There are just a few examples and it serves to relax a bit from the mathematics in the previous chapters and in the two chapters to follow.

The remaining two chapters are indeed back into mathematics, but mainly descriptive.

Chapter 10 is for example more recreational: no proofs, but still many graphs. It's all about circles rolling along a line or a circle: the cycloids, hypocycloids, epicycloids, and related curves. Several of these come with a history like the Aristotle wheel paradox and of course the invention of the wheel itself. For the playful aspect, the Spirograph is clearly the instrument of choice. This chapter is not by the authors but it is contributed by Christian Spreitzer.

The last chapter is about spherical geometry. The circle is the only curve that fits also on a sphere. It is well known that the shortest path between two points on a sphere follows a great circle through these points. This explains the route followed during transatlantic flights. There is also the spherical triangle whose angles can sum up to just below 540° and the counterintuitive hight to which a rope around the equator will rise when its length is increased by 1 meter.

An afterword is contributed by Erwin Rauscher who gives a cultural introduction to the circle. The author is different, but it would easily blend in with Chapter 9 on the use of the circle in arts.

Like several of Posamentier's previous books, this is a book mostly about mathematics, but the "gentle" version, painted on a cultural and historic canvas. The proofs stress the importance of the visual aspect. I am afraid that much of this geometric kind of reasoning has nowadays been largely replaced by algebraic manipulations which, in my opinion, is regrettable. I am old enough to have had this geometric education in secondary school still largely influenced by the Greek tradition of Euclid's Elements, and I remember how I enjoyed solving these geometric "puzzles". It may be one of the reasons that made me decide to become a mathematician. I truly enjoyed re-living this happy experience of my youth by reading this book since during my math studies at the university and in my later career I never used or needed this kind of geometric argumentation.

**Submitted by Adhemar Bultheel |

**2 / Dec / 2016