Solid Geometry

Solid Geometry

Solid geometry is that subsection of geometry that deals with figures in three-dimensional space. Euclidean geometry, sometimes called parabolic geometry, is divided into two subsections: plane geometry, geometry dealing with figures in a plane, and solid geometry, geometry dealing with solids in three-dimensional space. Solid geometry is sometimes called three-dimensional Euclidean geometry and deals with solids, such as polyhedra and spheres, and lines and planes in three-dimensional space. Solid geometry is concerned with the study of figures in three-dimensional Euclidean space, which is usually denoted R³.

Solid geometry is a branch of Euclidean geometry, developed by Greek mathematician Euclid in the 4th century B.C., that is governed by Euclid's five postulates as laid out in his work The Elements. More specifically the basics of solid geometry are described in Book XI of The Elements. Book XI consists of 28 definitions, in which Euclid departed from traditional definitions and instead used motion-based definitions, and 39 propositions. These motion-based definitions were formulated as revolutions of figures about a diameter. The definitions cover everything from the definition of a solid, inclination of figures in three-dimensional space, similarity of those figures, definitions of pyramids, spheres, cones, cylinders, cubes, octahedrons, icosahedrons, and dodecahedrons. There are also several definitions that deal with angles and angular relations of figures relative to each other. The first 18 propositions deal with lines and planes and their relations to figures in Euclidean space. The elementary theorems of three-dimensional geometry, solid geometry, are laid out in this book. The rest of the propositions deal with planes and solids in Euclidean space.

Book XII of Euclid's The Elements is concerned with the measurement of figures and contains 18 propositions dealing with this endeavor. Euclid used Eudoxus' method of exhaustion in this book and so many believe that most of this work is due to Eudoxus. Book XIII, the final book in Euclid's original version of The Elements, contains 18 propositions describing the construction of regular solids. It is thought that this book is based on Comparison of the Five Figures, a work by Aristaeus. The first six propositions involve the golden ratio of cut lines. The remainder of the propositions describes the construction of the five regular solids, pyramid, octahedron, cube, icosahedron, and dodecahedron, and proves that no other regular polyhedra are possible. After Euclid wrote The Elements a fourteenth and fifteenth books were added by Hypsicles in about 170 B.C. and Isadorus of Miletus in about 530 A.D.. In the Book XIV Hypsicles, working from treatises by Aristaeus and Apollonius, compares the five regular solids with respect to their faces, surface areas, and volumes. In Book XV, inferior to the previous one because of its impreciseness and inaccurateness, Isadorus deals with several topics including inscribing certain regular solids within others, determination of the number of edges and vertices on each solid, and determining the angle of inclination between adjacent faces in each solid.

Solid geometry is an extension and reinforcement of the propositions describing plane geometry. It forms the basis for the foundations of other branches of mathematics including geometry and trigonometry. The development of geometry in general was driven by three classical problems in Greek mathematics: plane, solid, and linear problems. Those problems whose solutions employ the use of one or more sections of a cone are solid problems. René Descartes advanced solid geometry in the 17th century by inventing Cartesian coordinates to express geometric relations in algebraic form. These advances led to the development of analytic geometry, descriptive geometry, differential geometry and projective geometry which are important to engineering and the natural sciences as well as mathematics.