The second species of the octahedron consists of two inter secting tetrahedra, whose sur faces when produced to the third species will be found to consist of six intersecting rhombic prisms having infinite volume.
Developing the solids in the systematic order thus defined, viz., the formation of succeeding solids by producing the faces of the first till they meet again, then producing the faces of the second to form the third, etc., etc., we arrive at four regular species for the dodecahedron and eight for the icosahedron. The four species of the dodecahedron are all regular-faced polygons, the first and third being ordinary pentagons, those of the second and fourth being pentagons of the second series or pentacles. Of the eight species of icosahedron, derived in systematic order, only the first and seventh are regular, their faces being equilateral triangles. (See SOLIDS: Geometric.) In making up such models it is generally convenient to start with a model of the first species and build up or convert it into the second species by adding to each face the appropriate com plement by dowelling it to the intersecting corner, the forms of the faces of the various complements being obtained from the complete plan of the face of the polyhedron.
Technical Construction.—Symmetrical Solids and surfaces of revolution can be turned in a lathe, a templet representing a plane section containing the axis being applied to the work from time to time until the whole solid of revolution is worked up.
Surfaces which are non-symmetrical round the axis may also be turned or formed in a suitable lathe having a chuck capable of eccentric motion. Such models may attain a high order of ac curacy since micrometer measurements may be applied to the work in the machine. It is of course easy to represent many of the sur faces by means of fixed wires shaped and assembled to represent their principal axes (Pl. I. figs. 8, 9, so), but a more intriguing series of flexible models can be made up of rods or strips, pin jointed or hinged at their extremities since such provide a mechan ism whereby ruled surfaces of the hyperboloids, etc., may be dem onstrated and allow, of conversion or "reversal" into their con focal surfaces. (See figs. 2a and 2b.)
Thread Models.—Ruled surfaces, i.e., surfaces generated by the motion of a straight line, fall naturally into a class for easy modelling, since the generating line can be represented by succes sive stretched threads. (See SURFACE.) Thread models can, therefore, illustrate a wide variety of combinations as in Plate I. fig. 12, which consists of two circular discs drilled with equi distant holes closely together, supported as shown and threaded with weighted cords so that the cords may slide through the lower holes.
We have in this model a demonstration of ) a cylinder—when the discs and cords hang freely, 2) a hyperboloid of revolution—when one or other disc is rotated slightly relatively to the other, 3) the limiting position of a pair of cones upon further rotation. thus providing an interesting example of maximum and minimum values since the cylinder represents the maximum volume for a given perimeter and the cone the minimum, the circular ends being of constant value. Threads stretched as generators across the bars of a jointed quadrilateral of which the sides are movable in pairs may be used to illustrate the changes from a plane through all forms of paraboloid to double plane. P1. I. fig. 11 shows an example ; the hyperbolic paraboloid generated by a single system of right lines. It comprises two bars pierced with equi distant holes, one bar being fixed, the other capable of swinging round an axis which can also be inclined at different angles to the fixed bar.
With the bars placed parallel, the strings indicate a plane. When inclined to one another yet in the same plane they still illustrate a plane but when the bars are not in the same plane the strings assume the surface of a twisted plane; viz., the hyperbolic para boloid, a natural surface for the maximum cleavage properties of a ploughshare. It may be observed from the model that no two strings lie in the same plane and therefore no part of the surface is truly plane. Such a surface cannot be made by simply twist ing a plane sheet of metal which would show malformation on opposite sides of the axis.