ADDENDA for Glimpses of Algebra and Geometry For comments/suggestions, please write to G. Toth, Department of Mathematics, Rutgers University, Camden, New Jersey, 08102, U.S.A., or send a message through e-mail: gtoth@crab.rutgers.edu There is a new solutions manual for selected problems under Glimpses_Solutions.tex (in LaTeX), Glimpses_Solutions.ps (in Postscript), and an HTML file. The notations used here are adapted from Pctex commands, for example, _ = lower index, ^ = upper index, \leq = less than or equal, \geq = greater than equal, etc. page line 5 -6 Replace `prime' by `number'. 6 9 ... stated by Albert Girard in 1632. 7 -3 The notion of perpendicular coordinate axes can be traced back to Archimedes and Apollonius. Both Descartes and Fermat used coordinates but only with nonnegative values; the idea that coordinates can also take negative values is due to Newton. 30 -5 This means that for any cubic curve a new coordinate system can be introduced, the new coordinates depend rationally on the old ones, and in terms of the new coordinates, the cubic is given by a Weierstrass form. Because of the rational dependence of the old and new coordinates, the rational points on the original curve correspond to rational points on the new curve. 57 8 This geometric representation of complex numbers is named after Jean Robert Argand in 1806, although nine years before it has been announced by the Norwegian cartographer Caspar Wessell before the Danish Academy of Sciences. 62 -3 To be precise, Gauss did not state explicitly that the condition that the Fermat primes are distinct is necessary; this gap was filled by Wantzel in 1837. In the footnote, it is obvious that the statement can be reduced to the case when n is a product of of Fermat primes. Indeed, as in Problem 3 of Section 9, once a regular polygon is constructed, its sides can be doubled. 69 -16 (Actually, m\leq n because a degree n polynomial can have at most n roots, but we do not need this additional fact.) For ... open neighborhood of q. Finally, we delete from V_0 the closed set f(S^2-(V_1 \cup \ldots \cup V_m)) to obtain an open neighborhood V of q. (Notice that S^2-(V_1 \cup \ldots \cup V_m) is closed in S^2, hence it is compact. Its f-image is also compact and thus closed in S^2.) Clearly, # is constant (=m) on V. The claim follows. (I am indebted to Hillel Gauchman to bring out this point.) 81 -10 Theorem 4 asserts in particular that the only possible groups of central symmetries in two dimensions are: C_1,C_2,C_3,\ldots and D_1,D_2,D_3,\ldots Central symmetry frequently occurs in Nature. Most of us observed in childhood that snowflakes have sixfold (some threefold) symmetries. Flowers usually have fivefold symmetries and, depending on whether the petals are bilaterally symmetric or not, their symmety group is $D_5$ or only $C_5$. We finish this section with a quote from Hermann Weyl's Symmetry regarding Theorem 4: "Leonardo da Vinci engaged in systematically determining the possible symmetries of a central building and how to attach chapels and niches without destroying the symmetry of the nucleus. In abstract modern terminology, his result is essentially our above table of the possible finite groups of rotations (proper and improper) in two dimensions." 88 17 That the point group {\bar G} is discrete can be seen directly from Theorem 6. 92 4 For an algebraic proof of the Crystallographic Restriction, consider the trace tr(R_{\theta}) of R_{\theta} \in {\bar G}, 0<\theta \leq \pi. With respect to a basis in L_G, the matrix of R_{\theta} has integral entries (Theorem 6). Thus, tr(R_{\theta}) is an integer. On the other hand, with respect to an orthonormal basis, the matrix of R_{\theta} has diagonal entries both equal to \cos (\theta ), in particular, tr(R_{\theta}) = 2\cos (\theta ). Thus, 2\cos (\theta ) is an integer and this is possible only for n=2,3,4 or 6. 196 2 ... reflection. Compare this with the second statement in Theorem 10.) 121 2 The hyperbolic distance formula was derived by the requirement that the linear fractional transformations with real coefficients must be hyperbolic isometries. The reason why the hyperbolic cosine function is used in the definition of d_H is apparent from the formula in line 2 of the next page; d_H becomes additive on 3 points along a hyperbolic line: d_H(z_1,z_2)+d_H(z_2,z_3)=d_H(z_1,z_3) for $z_1,z_2,z_3 consecutive points on a hyperbolic line. (Once again, this point was brought out by Hillel Gauchman.) 124 9 The area element dxdy/y^2 follows from the hypebolic distance formula. In fact, the line element is ds=|dz|/\Im(z), z=x+iy, as follows from the logarithmic growth formula on the top of p. 122; the metric is ds^2=(1/y^2)dx^2+(1/y^2)dy^2 so that the area element becomes \sqrt{(1/y^2)(1/y^2)}dxdy= dxdy/y^2. Some authors use the angular defect formula on the previous page as the definition of the hyperbolic area for triangles. An analogy that substantiates this approach is Girard's spherical excess formula treated in Problem 10(a) of Section 17. 158 7 (Another proof is based on Schwarz's lemma asserting that the conformal self-maps of D^2 are linear fractional transformations, and thereby they have the form given in Problem 1 of Section 13. They can be parametrized by 3 real parameters: \Re (w), \Im (w) and \theta. On the other hand, the linear transformations z\mapsto az+b, a,b \in C, form a 4-parameter family of conformal self-maps of the complex plane.) 168 -3 Theorem 10 is essentially due to Euler. 177 6 triangular antiprism. (The pentagonal antiprism appeared about 100 years earlier as octaedron elevatum in Fra Luca Pacioli's Da Divina Proportione printed in 1509. This classic is famous for its elaborate drawings of models made by Leonardo da Vinci.) 201 21 The symmetry group of a Platonic solid acts simply transitively on the bisecting edges of the solid and its reciprocal. Thus the order of the symmetry group is 2E. 207 6 We just noted that a golden rectangle has the property that if a square is sliced off, the remaining rectangle is similar to the original rectangle. Used recursively, a circular quadrant can be inserted into each sliced off square to obtain an approximation of an Archimedes' spiral. The most commonly known phenomenon in Nature that patterns this is the Nautilus shell, where the shell grows in a spiral for structural harmony in weight and strength. 213 -10 An excellent article about the buckyball: "Mathematics and the Buckyball" written by Chung F. and Sternberg S. in the American Scientist, Vol. 81, 1993). 215 13 This argument proving the existence of the dodecahedron was called by my students the `roof-proof'. It is essentially contained in Euclid's treatise on the dodecahedron in Book XIII of the Elements. 236 18 In Section 17 we described an algorithm to color the faces of the icosahedron with 5 colors such that the faces with the same color were disjoint. To obtain a 4-coloring we pick the four faces of a specific color group and recolor them using the remaining 4 colors such that no two faces with the same color meet at a common edge. (This can be done because each face has 3 adjacent faces and we have 4 colors.) This way we obtain a coloring of the faces of the icosahedron with 4 colors subject to the condition that faces with the same color can only touch each other at vertices. By reciprocity, this gives a coloring of the Schlegel diagram of the reciprocal dodecahedron with 4 colors in the sense discussed above. 292 4 G acts transitively on G(a) in the sense that any two points in the orbit G(a) can be carried into each other by suitable elements of G. 295 -1 A closed subspace in a compact Hausdorff space is compact. The continuous image of a compact topological space is compact.