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§§ 126, 127 GENERAL POLYHEDRONS 91
III. GENERAL POLYHEDRONS
126. Polyhedrons classified as to Faces. A polyhedron of four faces is called a tetrahedron; one of six faces, a heasahedron; one of eight faces, an octahedron; one of twelve faces, a dodecahedron; one of twenty faces, an icosahedron.
127. Regular Polyhedron. A polyhedron whose faces are congruent regular polygons, and whose polyhedral angles are equal, is called a regular polyhedron.
It is proved in § 130 that there cannot be more than five regular polyhedrons, and as a matter of fact there are just five. The five regular polyhedrons are shown in the above illustration in the order in which the polyhedrons are mentioned in § 126.
These regular solids occupied the attention of Pythagoras and his followers (about 550 B.C.). They were also studied so extensively in the school of Plato (about 375 B.C.) that they are often known as the Platonic Bodies. The early Greek writers connected them in a fanciful way with various phenomena of nature. For example, they assigned the tetrahedron to fire, the hexahedron to earth, the octahedron to air, the icosahedron to water, and the dodecahedron, apparently the last one discovered, to the universe.
Some of the regular polyhedrons are met in the study of crystals. Thus, the cube is found in salt crystals and the
regular octahedron in certain compounds of copper.
Pages 91–94, while important in the study of crystals, may be omitted without affecting the sequence of propositions.
128. Models of Regular Polyhedrons. In a work printed in 1525 a great artist, Albrecht Dürer, showed how to make paper models of the five regular polyhedrons. His description suggested drawing on stiff paper the diagrams shown below, and then cutting along the full lines and pasting strips of thin paper on the edges as indicated. By folding on the dotted lines and keeping the edges together by the pasted strips of paper, the models can be easily made.
129. Relation of Parts of a Polyhedron. If the number of edges of a polyhedron is represented by e, the number of vertices by v, and the number of faces by f, then e+2=v+f. This remarkable relation was possibly known to the Greek mathematician Archimedes (about 250 B.C.), but was first clearly stated by the French writer Descartes (about 1635). It was also discovered independently by Euler (1752), and is often known as Euler's Theorem.
The proof of this law is too difficult to be given at this time, but the student will be asked to verify it for certain special cases on page 94.
Proposition 16. Regular Polyhedrons 130. Theorem. There cannot be more than five regular polyhedrons.
Proof. A polyhedral Z has at least three faces. $ 78 Also, the sum of its face s is less than 360°, $ 3,2 because the polyhedral Z would flatten out into
a plane at 360°. Since each Z of an equilateral A is 60°, $ 8,4 polyhedral & may be formed with three, four, or five equilateral A as faces.
Now the sum of six face 4 of 60° is 360°. Ax. 1
Hence not more than three regular polyhedrons with equilateral A as faces are possible. Since each of a square is 90°,
$ 10, 2 a polyhedral Zmay be formed with three squares as faces.
Now the sum of four face s of 90° is 360°. Ax. 1
Hence not more than one regular polyhedron with squares as faces is possible.
Since each Z of a regular pentagon is 108°, $ 12, 3 a polyhedral Z may be formed with three regular pentagons as faces.
Now the sum of four face of 108° is 432° Ax. 1
Hence not more than one regular polyhedron with regular pentagons as faces is possible.
Now the sum of three is of a regular hexagon is 360°; of a regular heptagon, more than 360°; and so on.
Hence there cannot be more than five regular polyhedrons.
It is not of enough importance to prove that there are actually five regular polyhedrons as stated in $ 127. In elementary work the fact may be safely assumed.
Exercises. Polyhedrons 1. Count the number of edges, vertices, and faces on each of the five regular polyhedrons and then fill in the blank spaces in the following table:
2. From the above table show that in each case the law e +2=v+f holds true.
3. Assuming that the law in Ex. 2 is true for all polyhedrons, prove that a seven-edged polyhedron is impossible.
4. If the centers of the six faces of a cube are joined, what kind of polyhedron is constructed ? Draw the figure and prove that any two edges are equal.
5. If the centers of the four faces of a regular tetrahedron are joined, what kind of polyhedron is constructed ? Draw the figure and prove any two edges equal.
6. As in Ex. 4, consider the figure which results from joining the centers of the faces of a regular octahedron.
7. A quartz crystal is in the form of a hexagonal prism with a pyramid on one of its bases, as here shown. Show that the relation stated in Ex. 2 holds true.
8. A given polyhedron has six vertices and five faces. How many edges are there?
131. Cylindric Surface. A surface generated by a moving straight line which is always parallel to a fixed straight line, and touches a fixed curve not in a plane with the fixed line, is called a cylindric surface, or a cylindrical surface.
The moving line is called the generatrix and the fixed curve is called the directrix. In the first figure below, ABC' is the directrix of the cylindric surface shown. The generatrix in any position is called an element.
132. Cylinder. A solid bounded by a cylindric surface and two parallel plane surfaces cutting all the elements is called a cylinder.
It is evident that all elements of a cylinder are equal. The terms bases, lateral area, and altitude are used as with prisms.
133. Cylinders classified. If the elements of a cylinder are perpendicular to the bases, the cylinder is called a right cylinder; if oblique, it is called an oblique cylinder. A cylinder whose bases are circles is called a circular cylinder.
The second figure above shows a right circular cylinder, and the third an oblique cylinder. The straight line through the centers of the bases of a circular cylinder is called the axis of the cylinder.
Since a right circular cylinder can be generated by revolving a rectangle about one side as an axis, it is also called a cylinder of revolution.