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  A toroid using a square.

  

   

  A torus is a type of toroid.

  In mathematics, a toroid is a surface of revolution with a hole in the middle, like a doughnut, forming a solid body. The axis of revolution passes through the hole and so does not intersect the surface.^[1] For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow rectangle-section ring is produced. If the revolved figure is a circle, then the object is called a torus.

  The term toroid is also used to describe a toroidal polyhedron. In this context a toroid need not be circular and may have any number of holes. A g-holed toroid can be seen as approximating the surface of a torus having a topological genus, g, of 1 or greater. The Euler characteristic χ of a g holed toroid is 2(1-g).^[2]

  A toroid is specified by the radius of revolution R measured from the center of the section rotated. For symmetrical sections volume and surface of the body may be computed (with circumference C and area A of the section):

  {\displaystyle V=2\pi RA}

  {\displaystyle S=2\pi RC}

  Which gives for square and circular sections:

  The volume and surface of a toroid with square section of side a are given by

  {\displaystyle V=a^{2}2\pi R}

  {\displaystyle S=8a\pi R}

  The volume and surface of a toroid with circular section of radius r (torus) are given by

  {\displaystyle V=2r^{2}\pi ^{2}R}

  {\displaystyle S=4r\pi ^{2}R}

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  Further advances in the instrument were made by Danish astronomer Tycho Brahe (1546–1601), whose elaborate armillary spheres passing into astrolabes are figured in his Astronomiae Instauratae Mechanica.^[1]

  Armillary spheres were among the first complex mechanical devices. Their development led to many improvements in techniques and design of all mechanical devices. Renaissance scientists and public figures often had their portraits painted showing them with one hand on an armillary sphere, which represented the height of wisdom and knowledge.

  The armillary sphere survives as useful for teaching, and may be described as a skeleton celestial globe, the series of rings representing the great circles of the heavens, and revolving on an axis within a horizon. With the earth as center such a sphere is known as Ptolemaic; with the sun as center, as Copernican.^[1]

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    Sculpture of 13th-century English scientist Roger Bacon holding an armillary sphere, Oxford University Museum of Natural History

   
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    Young girl with an astronomical instrument, by Jan Gossaert, c. 1520-1540

   
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    Portarait in the frontispiece of Antoine Crespin's Propheties par l'astrologue du treschrestien Roy de France et de Madame la Duchesse de Savoye, Lyon, France, 1572

   
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    Allegory of the Arts, by Francesco de Mura, c. 1750

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