Problem 97
Determine centers, axes, and planes of symmetry for the figure consisting of two intersecting lines.
Problem 98
Prove that a prism has a center of symmetry if and only if its base does.
Problem 99
Determine the number of planes of symmetry of a regular prism with \(n\) lateral faces.
Problem 100
Determine the number of planes of symmetry of a regular pyramid with \(n\) lateral faces. 131\. Let three figures \(\Phi, \Phi^{\prime}\), and \(\Phi^{\prime \prime}\) be symmetric: \(\Phi\) and \(\Phi^{\prime}\) about a plane \(P\), and \(\Phi^{\prime}\) and \(\Phi^{\prime \prime}\) about a plane \(Q\) perpendicular to \(P\). Prove that \(\Phi\) and \(\Phi^{\prime \prime}\) are symmetric about the intersection line of \(P\) and \(Q\). 132\. What can be said about the figures \(\Phi\) and \(\Phi^{\prime \prime}\) of the previous problem if the planes \(P\) and \(Q\) make the angle: (a) \(60^{\circ} ?\) (b) \(45^{\circ} ?\)
Problem 101
Determine the number of planes of symmetry of a regular pyramid with \(n\) lateral faces.
Problem 102
Let three figures \(\Phi, \Phi^{\prime}\), and \(\Phi^{\prime \prime}\) be symmetric: \(\Phi\) and \(\Phi^{\prime}\) about a plane \(P\), and \(\Phi^{\prime}\) and \(\Phi^{\prime \prime}\) about a plane \(Q\) perpendicular to \(P\). Prove that \(\Phi\) and \(\Phi^{\prime \prime}\) are symmetric about the intersection line of \(P\) and \(Q\).
Problem 104
Prove that if a figure has two symmetry planes making an angle \(180^{\circ} / n\), then their intersection line is an axis of symmetry of the \(n\)th order.
Problem 105
Describe the cross section of a cube by the plane perpendicular to one of the diagonals at its midpoint.
Problem 106
Show that the 24 ways of superimposing the cube onto itself correspond to 24 different ways (including the trivial one) of permuting its four diagonals.
Problem 108
Classification of regular polyhedra. Let us take into account that a convex polyhedral angle has at least three plane angles, and that their sum has to be smaller than \(4 d(\S 48)\). Since in a regular triangle, every angle is \(\frac{2}{3} d\), repeating it 3,4 , or 5 times, we obtain the angle sum smaller than \(4 d\), but repeating it 6 or more times, we get the angle sum equal to or greater than \(4 d\). Therefore convex polyhedral angles whose faces are angles of regular triangles can be of only three types: trihedral, tetrahedral, or pentahedral. Angles of squares and regular pentagons are respectively \(d\) and \(\frac{6}{5} d\). Repeating these angles three times, we get the sums smaller than \(4 d\), but repeating them four or more times, we get the sums equal to or greater than \(4 d\). Therefore from angles of squares or regular pentagons, only trihedral convex angles can be formed. The angles of regular hexagons are \(\frac{4}{3} d\), and of regular polygons with more than 6 sides even greater. The sum of three or more of such angles will be equal to or greater than \(4 d\). Therefore no convex polyhedral angles can be formed from such angles. It follows that only the following five types of regular polyhedra can occur: those whose faces are regular triangles, meeting by three, four or five triangles at each vertex, or those whose faces are either squares, or regular pentagons, meeting by three faces at each vertex.
