1.7: Regular Polygons and Regular Polyhedra
Regular polygons have already featured rather often (e.g. in Problems 3 , 12 , 19 , 27 , 28 , 29 ). This is a general feature of elementary mathematics; so the neglect of the geometry of regular polygons, and their 3D companions, the regular polyhedra, is all the more unfortunate. We end this first chapter with a first brief look at polygons and polyhedra.
1.7.1: Regular Polygons are Cyclic
A polygon \(ABCDE...\) consists of \(n\) vertices \(A, B, C, D, E, ...,\) and \(n\) sides \(AB\), \(BC\), \(CD\), \(DE\)... which are disjoint except that successive pairs meet at their common endpoint (as when \(AB\), \(BC\) meet at \(B\)). A polygon is regular if any two sides are congruent (or equal), and any two angles are congruent (or equal). Can a regular \(n\)-gon \(ABCDE...\) always be inscribed in a circle? In other words, does a regular polygon automatically have a “centre”, which is equidistant from all \(n\) vertices?
1.7.2: Regular Polyhedra
(a) You are given a regular tetrahedron with edges of length \(2\). Is it possible to choose positive real numbers \(a\) and \(b\) so that an a by \(b\) rectangular sheet of paper can be used to “wrap”, or cover, the regular tetrahedron without leaving any gaps or overlaps?
(b) Given a cube with edges of length \(2\), what is the smallest sized rectangle that can be used to wrap the cube in the same way without cutting the paper? (In other words, if we want to completely cover the cube, what is the smallest area of overlap needed? How small a fraction of the paper do we have to waste?
Can a cross-section of a cube be:
(i) an equilateral triangle?
(ii) a square?
(iii) a polygon with more than six sides?
(iv) a regular hexagon?
(v) a regular pentagon
Can one use the Sun’s rays to produce a plane shadow of a cube:
(i) in the form of an equilateral triangle?
(ii) in the form of a square?
(iii) in the form of a pentagon?
(iv) in the form of a regular hexagon?
(v) in the form of a polygon with more than six sides?
The imparting of factual knowledge is for us a secondary consideration. Above all we aim to promote in the reader a correct attitude, a certain discipline of thought, which would appear to be of even more essential importance in mathematics than in other scientific disciplines. ...
General rules which could prescribe in detail the most useful discipline of thought are not known to us. Even if such rules could be formulated, they would not be very useful. Rather than knowing the correct rules of thought theoretically, one must have assimilated them into one’s flesh and blood ready for instant and instinctive use. Therefore for the schooling of one’s powers of thought only the practice of thinking is really useful.
G. Pólya (1887–1985) and G. Szegö (1895–1985)