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Peter R. Polyhedra have cropped up in many different guises throughout recorded history. In modern times, polyhedra and their symmetries have been cast in a new light by combinatorics an d group theory. This book comprehensively documents the many and varied ways that polyhedra have come to the fore throughout the development of mathematics.
The author strikes a balance between covering the historical development of the theory surrounding polyhedra, and presenting a rigorous treatment of the mathematics involved. It is attractively illustrated with dozens of diagrams to illustrate ideas that might otherwise prove difficult to grasp.
Historians of mathematics, as well as those more interested in the mathematics itself, will find this unique book fascinating. Egyptian geometry. Babylonian geometry. Chinese geometry. A common origin for oriental mathematics. Greek mathematics and the discovery of incommensurability. The nature of space. Democritus dilemma. What is a polyhedron? Von Staudts proof. Complementary viewpoints. The GaussBonnet theorem. Equality Rigidity and Flexibility.
Disputed foundations. Stereoisomerism and congruence. Cauchys rigidity theorem. Liu Hui on the volume of a pyramid. Eudoxus method of exhaustion. Hilberts third problem. Rules and Regularity. The mathematical paradigm. Primitive objects and unproved theorems. The problem of existence. Constructing the Platonic solids. The discovery of the regular polyhedra. What is regularity? Bending the rules. Polyhedra with regular faces.
Decline and Rebirth of Polyhedral Geometry. Heron of Alexandria. Pappus of Alexandria. The decline of geometry. The rise of Islam. Thabit ibn Qurra. Collecting and spreading the classics. The restoration of the Elements. A new way of seeing. Early perspective artists. Leon Battista Albert. Polyhedra in woodcrafts. Piero della Francesca. Wenzel Jamnitzer. Perspective and astronomy. Polyhedra revived. Fantasy Harmony and Uniformity. A mystery unravelled.
The structure of the universe. Fitting things together. Rhombic polyhedra. The Archimedean solids. Star polygons and star polyhedra. Semisolid polyhedra. Uniform polyhedra. Surfaces Solids and Spheres. Plane angles solid angles and their measurement. Descartes theorem. The announcement of Eulers formula. The naming of parts. Consequences of Eulers formula. Exceptions which prove the rule.
Cauchys early career. Rotating rings and flexible frameworks. Are all polyhedra rigid? The Connelly sphere. Further developments. When are polyhedra equal? Stars Stellations and Skeletons. Poinsots star polyhedra. Poinsots conjecture. Cauchys enumeration of star polyhedra.
Stellations of the icosahedron. Bertrands enumeration of star polyhedra. Regular skeletons. Symmetry Shape and Structure. Systems of rotational symmetry. How many systems of rotational symmetry are there? Reflection symmetry. Prismatic symmetry types.
Compound symmetry and the S2n symmetry type. Cubic symmetry types. Icosahedral symmetry types. Determining the correct symmetry type. Groups of symmetries. Crystallography and the development of symmetry. Counting Colouring and Computing.
Colouring the Platonic solids. How many colourings are there? A counting theorem. Applications of the counting theorem. Proper colourings. How many colours are necessary?
The fourcolour problem. Combination Transformation and Decoration.
List of books about polyhedra
Would you like to tell us about a lower price? If you are a seller for this product, would you like to suggest updates through seller support? Polyhedra have cropped up in many different guises throughout recorded history. In modern times, polyhedra and their symmetries have been cast in a new light by combinatorics and group theory. This book comprehensively documents the many and varied ways that polyhedra have come to the fore throughout the development of mathematics. The author strikes a balance between covering the historical development of the theory surrounding polyhedra, and presenting a rigorous treatment of the mathematics involved. It is attractively illustrated with dozens of diagrams to illustrate ideas that might otherwise prove difficult to grasp.
Polyhedron A 3-D solid which consists of a collection of Polygons , usually joined at their Edges. The word derives from the Greek poly many plus the Indo-European hedron seat. A polyhedron is the 3-D version of the more general Polytope , which can be defined on arbitrary dimensions. A Convex Polyhedron can be defined as the set of solutions to a system of linear inequalities. A polyhedron is said to be regular if its Faces and Vertex Figures are Regular not necessarily Convex polygons Coxeter , p. A Convex polyhedron is called Semiregular if its Faces have a similar arrangement of nonintersecting regular plane Convex polygons of two or more different types about each Vertex Holden , p. These solids are more commonly called the Archimedean Solids , and there are 13 of them.
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