REGRESSION
, or Retrogradation of Curves, &c. See Retrogradation.
REGULAR Figure, in Geometry, is a figure that is both equilateral and equiangular, or having all its sides and angles equal to one another.
For the dimensions, properties, &c, of regular figures, see Polygon.
Regular Body, called also Platonic Body, is a body or solid comprehended by like, equal, and regular plane figures, and whose solid angles are all equal.
The plane figures by which the solid is contain- ed, are the faces of the solid. And the sides of the plane figures are the edges, or linear sides of the solid.
There are only five Regular Solids, viz,
The tetraedron, or regular triangular pyramid, having 4 triangular faces;
The hexaedron, or cube, having 6 square faces;
The octaedron, having 8 triangular faces;
The dodecaedron, having 12 pentagonal faces;
The icosaedron, having 20 triangular faces.
Besides these five, there can be no other Regular Bodies in nature.
Prob. 1. To construct or form the Regular Solids.— See the method of describing these figures under the article Body.
2. To find either the Surface or the Solid Content of any of the Regular Bodies.—Multiply the proper tabular area or surface (taken from the following Table) by the square of the linear edge of the solid, for the superficies. And
Multiply the tabular solidity, in the last column of the Table, by the cube of the linear edge, for the solid content.
| Surfaces and Solidities of Regular Bodies, the side being | |||
| unity or 1. | |||
| No. of | Name. | Surface. | Solidity. |
| sides. | |||
| 4 | Tetraedron | 1.7320508 | 0.1178513 |
| 6 | Hexaedron | 6.0000000 | 1.0000000 |
| 8 | Octaedron | 3.4641016 | 0.4714045 |
| 12 | Dodecaedron | 20.6457788 | 7.6631189 |
| 20 | Icosaedron | 8.6602540 | 2.1816950 |
3. The Diameter of a Sphere being given, to find the side of any of the Platonic bodies, that may be either inscribed in the sphere, or circumscribed about the sphere, or that is equal to the sphere.
Multiply the given diameter of the sphere by the proper or corresponding number, in the following Table, answering to the thing sought, and the product will be the side of the Platonic body required.
| The diam. of a | That may be | That may be cir- | That is equal |
| sphere being 1, | inscribed in the | cumscribed about | to the sphere, |
| the side of a | sphere, is | the sphere, is | is |
| Tetraedron | 0.816497 | 2.44948 | 1.64417 |
| Hexaedron | 0.577350 | 1.00000 | 0.88610 |
| Octaedron | 0.707107 | 1.22474 | 1.03576 |
| Dodecaedron | 0.525731 | 0.66158 | 0.62153 |
| Icosaedron | 0.356822 | 0.44903 | 0.40883 |
4. The side of any of the five Platonic bodies being given, to find the diameter of a sphere, that may either be inscribed in that body, or circumscribed about it, or that is equal to it.—As the respective number in the Table above, under the title, inscribed, circumscribed, or equal, is to 1, so is the side of the given Platonic | body, to the diameter of its inscribed, circumscribed, or equal sphere.
5. The side of any one of the five Platonic bodies being given; to find the side of any of the other four bodies, that may be equal in solidity to that of the given body.—As the number under the title equal in the last column of the table above, against the given Platonic body, is to the number under the same title, against the body whose side is sought, so is the side of the given Platonic body, to the side of the body sought.
See demonstrations of many other properties of the Platonic bodies, in my Mensuration, part 3 sect. 2 pa. 249, &c, 2d edition.
Regular Curve. See Curve.
REGULATOR of a Watch, is a small spring belonging to the balance, serving to adjust the going, and to make it go either faster or slower.
