As a matter of fact each of these types does exist :

(a) (i) In the regular tetrahedron (Fig. 561), bounded by four equilateral triangles, each solid angle is contained by three plane angles.

FIG. 561.

(ii) In the regular octahedron (Fig. 562), bounded by eight equilateral triangles, each solid angle is contained by four plane angles.

This figure is equivalent to two square pyramids placed base to base.

FIG. 563.

A model of it can be made by cutting out Fig. 563 in cardboard, and folding at the dotted lines as edges.

(iii) In the regular icosahedron (Fig. 564), bounded by twenty equilateral triangles, each solid angle is contained by five plane angles.

(b) In the cube, bounded by six squares, each solid angle is contained by three plane angles.

(c) In the dodecahedron (Fig. 566), bounded by twelve regular pentagons, each solid angle is contained by three plane angles.

A model of this figure can be made by cutting out two figures like Fig. 567 in cardboard, and fitting them together. It is very difficult to prove logically that this figure is geometrically possible.

431. EULER'S THEOREM.-If F, E, C are the number of faces, edges, and corners respectively of any convex polyhedron, then

E+ 2 = F+C.

Suppose that the polyhedron is built up by starting with one face and adding new faces one by one, in such a way that at any stage of the process the faces form a continuous surface (which however is not a closed surface until the last face has been added).

In this process each new face will introduce a certain number of new edges and new corners. For example, suppose that in Fig. 568, we have the faces a, ß, y, δ ; then when the new face is added this introduces 3 new edges, and 2 new corners.

Also it is obvious that when any new face is added (except the last face which closes the figure) the number of new edges introduced is always one more than the number of new corners.

Now when there is only one face, the number of edges is equal to the number of corners; i. e. E = C. Also F = 1.

Hence in this case

E+1=F+C..

(i)

Again each time a new face is added (except the last face) the value of E is increased by one more than the value of C,

and the value of F is increased by 1.

Hence equation (i) is still true.

Proved.

When the last face is added the values of E and C are unchanged, but the value of F is increased by 1. Hence for the complete polyhedron

1. The straight line OX lies within the trihedral angle O(ABC). Prove that

L XOB + L XOC <L AOB + AOC.

2. In the figure of Question 1, prove that

L XOA +

XOB + L. XOC < & BOC +L COA + L AOB.

3. If the straight line OX lies within or without the trihedral angle O(ABC), prove that—

LXOA + L XOB + L XOC > }(≤ BOC + ▲ COA + ▲ AOB).

L

4. Two pyramids whose vertices are 0 and V respectively have the same polygon as base. If the vertex V of the one pyramid lies within the other pyramid, prove that the sum of the face angles at V is greater than the sum of the face angles at 0.

5. Draw a plane parallel to the base of a given pyramid such that the section of the pyramid so formed is half the area of the base.

6. Verify § 431 in the case of a pyramid on an irregular hexagonal Also in the case of the solid remaining if the upper portion of this pyramid is cut away by a plane cut.

base.

7. Work out the number of edges, faces, and corners in each type of regular polyhedron, and verify the Theorem of § 431 in each case.

CHAPTER XXXII.

THE SPHERE AND THE TETRAHEDRON.

432. DEFINITIONS.-A sphere is a solid figure bounded by one curved surface which is such that all points on this surface are at equal distances from a certain point within the surface, called the centre of the sphere. Any line from the centre to the surface is called a radius.

THEOREM.-Any plane section of a sphere is a circle. CASE I. When the plane passes through 0, the centre of the sphere (Fig. 569):

If L, M, and N, points on the section, then OL, OM, ON,... are radii of the sphere.

Hence OL, OM, ON, . . . are all equal, and the section is therefore a circle whose centre is 0, and whose radius is equal to that of the sphere.

M

FIG. 569.

CASE II. When the plane does not pass through the centre 0 of the sphere (Fig. 570):-

Hence the curve GHK is a circle whose centre is C.