Mathematical Surprises

This is open access book provides plenty of pleasant mathematical surprises. There are many fascinating results that do not appear in textbooks although they are accessible with a good knowledge of secondary-school mathematics. This book presents a selection of these topics including the mathematical formalization of origami, construction with straightedge and compass (and other instruments), the five- and six-color theorems, a taste of Ramsey theory and little-known theorems proved by induction. Among the most surprising theorems are the Mohr-Mascheroni theorem that a compass alone can perform all the classical constructions with straightedge and compass, and Steiner's theorem that a straightedge alone is sufficient provided that a single circle is given. The highlight of the book is a detailed presentation of Gauss's purely algebraic proof that a regular heptadecagon (a regular polygon with seventeen sides) can be constructed with straightedge and compass. Although the mathematics used in the book is elementary (Euclidean and analytic geometry, algebra, trigonometry), students in secondary schools and colleges, teachers, and other interested readers will relish the opportunity to confront the challenge of understanding these surprising theorems

Mathematical Surprises

This is open access book provides plenty of pleasant mathematical surprises. There are many fascinating results that do not appear in textbooks although they are accessible with a good knowledge of secondary-school mathematics. This book presents a selection of these topics including the mathematical formalization of origami, construction with straightedge and compass (and other instruments), the five- and six-color theorems, a taste of Ramsey theory and little-known theorems proved by induction. Among the most surprising theorems are the Mohr-Mascheroni theorem that a compass alone can perform all the classical constructions with straightedge and compass, and Steiner's theorem that a straightedge alone is sufficient provided that a single circle is given. The highlight of the book is a detailed presentation of Gauss's purely algebraic proof that a regular heptadecagon (a regular polygon with seventeen sides) can be constructed with straightedge and compass. Although the mathematics used in the book is elementary (Euclidean and analytic geometry, algebra, trigonometry), students in secondary schools and colleges, teachers, and other interested readers will relish the opportunity to confront the challenge of understanding these surprising theorems

Exploring Structure in Origami Constructions

In origami, an artist uses paper to construct a three-dimensional object by making folds from a set of seed points. The intersections formed from the folds are then used as reference points for new folds. Abstractly, we can represent the paper with a plane. We form a subset of this plane by intersecting along angles from seed points, and we are interested in special properties of this subset. Under certain constraints, the origami construction gives rise to a subset with mathematical structure, including the topological structure of a lattice or the algebraic structure of a subring. We first explore the conditions that give rise to an origami ring that is a subset of the complex numbers. The complex plane is an example of a Euclidean space, which is constructed using the Parallel Postulate. When this axiom is altered, we work with the hyperbolic plane, where multiple parallel hyperbolic lines can intersect a particular point. In our origami constructions, new reference points are made by intersecting two lines. Since the hyperbolic plane has fundamentally different geometry due to its axiomatization, constructed points are different when we operate in the hyperbolic plane compared to when they start in the complex plane. We reach partial results showing that the origami procedure is different in the hyperbolic plane. However, we do show a new classification of origami lattices by using the classical modular group as a moduli space for complex lattices, and raise new questions about the containment of all lattices in origami lattices

Geometric Constructions, Origami, and Galois Theory

Geometric constructions using an unmarked straightedge and a compass have been studied for thousands of years. In these constructions, we can draw circles and lines starting with any two points, and we can create new points where they intersect. An n-gon is said to be constructible if can be constructed in a finite number of steps using these guidelines. We begin with constructions of several n-gons, and examine the field theory behind geometric constructions. Galois theory then provides a precise classification of which n-gons are constructible and which are not. Next is an exploration of origami construction, which examines a single-fold construction axiom, and establishes the classification of origami-constructible n-gons. For example, a heptagon is not constructible using traditional construction techniques, but it is constructible using origami. Finally, we investigate new axioms, which might allow for additional constructions, and examine their implications