Solid of revolution in the context of Three-dimensional object

A lathe (/leɪð/) is a machine tool that rotates a workpiece about an axis of rotation to perform various operations such as cutting, sanding, knurling, drilling, deformation, facing, threading and turning, with tools that are applied to the workpiece to create an object with symmetry about that axis.

Lathes are used in woodturning, metalworking, metal spinning, thermal spraying, reclamation, and glass-working. Lathes can be used to shape pottery, the best-known such design being the potter's wheel. Most suitably equipped metalworking lathes can be used to produce most solids of revolution, plane surfaces, and screw threads or helices. Ornamental lathes can produce more complex three-dimensional solids. The workpiece is usually held in place by either one or two centers, at least one of which can typically be moved horizontally to accommodate varying workpiece lengths. Other work-holding methods include clamping the work about the axis of rotation using a chuck or collet, or attaching it to a faceplate using clamps or dog clutch. Lathes equipped with special lathe milling fixtures can be used to complete milling operations.

Solid geometry or stereometry is the geometry of three-dimensional Euclidean space (3D space).A solid figure is the region of 3D space bounded by a two-dimensional closed surface; for example, a solid ball consists of a sphere and its interior.

Solid geometry deals with the measurements of volumes of various solids, including pyramids, prisms, cubes (and other polyhedrons), cylinders, cones (including truncated) and other solids of revolution.

A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) one full revolution around an axis of rotation (normally not intersecting the generatrix, except at its endpoints).The volume bounded by the surface created by this revolution is the solid of revolution.

Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on whether or not the line is parallel to the axis. A circle that is rotated around any diameter generates a sphere of which it is then a great circle, and if the circle is rotated around an axis that does not intersect the interior of a circle, then it generates a torus which does not intersect itself (a ring torus).

In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution.

The theorems are attributed to Pappus of Alexandria and Paul Guldin. Pappus's statement of this theorem appears in print for the first time in 1659, but it was known before, by Kepler in 1615 and by Guldin in 1640.

Because of the problem of the aerodynamic design of the nose cone section of any vehicle or body meant to travel through a compressible fluid medium (such as a rocket or aircraft, missile, shell or bullet), an important problem is the determination of the nose cone geometrical shape for optimum performance. For many applications, such a task requires the definition of a solid of revolution shape that experiences minimal resistance to rapid motion through such a fluid medium.

Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts.

Mathematics and art have a long historical relationship. Artists have used mathematics since the 4th century BC when the Greek sculptor Polykleitos wrote his Canon, prescribing proportions conjectured to have been based on the ratio 1:√2 for the ideal male nude. Persistent popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. In the Italian Renaissance, Luca Pacioli wrote the influential treatise De divina proportione (1509), illustrated with woodcuts by Leonardo da Vinci, on the use of the golden ratio in art. Another Italian painter, Piero della Francesca, developed Euclid's ideas on perspective in treatises such as De Prospectiva Pingendi, and in his paintings. The engraver Albrecht Dürer made many references to mathematics in his work Melencolia I. In modern times, the graphic artist M. C. Escher made intensive use of tessellation and hyperbolic geometry, with the help of the mathematician H. S. M. Coxeter, while the De Stijl movement led by Theo van Doesburg and Piet Mondrian explicitly embraced geometrical forms. Mathematics has inspired textile arts such as quilting, knitting, cross-stitch, crochet, embroidery, weaving, Turkish and other carpet-making, as well as kilim. In Islamic art, symmetries are evident in forms as varied as Persian girih and Moroccan zellige tilework, Mughal jali pierced stone screens, and widespread muqarnas vaulting.