Errata Page<\/u><\/p>\n
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This page contains all of the essential errata that I am currently aware of in books and papers that I have either authored or co-authored, listed in reverse chronological order. It does not include all inconsequential typos. It also includes additional references and results.\u00a0 I appreciate the various mathematicians and readers who have sent me corrections\/comments. Please forward to me any other corrections that you might find.<\/p>\n
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Books:<\/u><\/p>\n
<\/p>\n
(Thanks especially to Craig Kaplan and Dirk\u00a0 Frettloeh for sending corrections.)<\/p>\n
p. 29: An additional exercise for Section 1.1 is: \u201cFind an example of a monohedral tiling such that for every tie T, any tile that is a neighbor of T is also adjacent to T.\u201d<\/p>\n
p.30, Problem 11: Replace Figure 1.6 with Figure 1.3. And replace \u201cblue\u201d, \u201cyellow\u201d and \u201cpurple\u201d by\u00a0 \u201cred\u201d, \u201corange\u201d and \u201cyellow\u201d.<\/p>\n
p. 31: Bottom: Replace two occurrences of q by x.<\/p>\n
p.39: Problem 4: Replace \u201cUse the classification of isometries into five types\u201d by \u201cUse the fact every isometry is a product of 0,1,2, or 3 reflections\u201d<\/p>\n
Problem 7 (b) Add \u201cangles between -pi and pi\u201d<\/p>\n
p.43: Figure 1.26 (b): In the lowermost right, theta should be -theta.<\/p>\n
p.44: Line -8 \u201cthere would be two rotation centers (see problem 8)\u201d should be \u201c by Exercise 1.2\/9, we would have a glide reflection.\u201d<\/p>\n
p.47: Figure 1.29: One black triangle is missing.<\/p>\n
p.51: Definition 1.19: Remove stray )<\/p>\n
p.54: Caption for Figure 1.39: \u201ceight isohedral tilings\u201d should be \u201cseven isohedral tilings\u201d<\/p>\n
p.56: Problem 6: Replace \u201cFigure 1.37\u201d by \u201cFigure 1.34\u201d.<\/p>\n
p. 57: Problem 10 (b): Drop “edge-to-edge”<\/p>\n
p.69: Add triangle to figure for p31m<\/p>\n
p.70: At the top, “International Union of IUCr” should be “International Union of Crystallography (IUCr)”.<\/p>\n
p.73: Problem 10: Switch order of 1.40 and 1.37<\/p>\n
p.82: Third line: “In the top row we first see the regular hexagonal tiling and then two non-periodic tilings” should be \u201cIn the top row we first see the regular hexagonal tiling, then a tiling with translational symmetry, then a non-periodic tiling and then a periodic tiling. In the bottom row we see a countable sequence of patches, each generating a unique periodic tiling.\u201d<\/p>\n
p.84: Problem 8(a): This is false as stated. If we denote the sequence of vertices in order by A, B, C, D, E, F, and AB and DE are parallel, then we must also assume there is a translation taking A to E and B to D.<\/p>\n
Problem 10: Drop symmetry group pm.<\/p>\n
p.99: Open Question 2 asks “If there is a prototile that admits a tiling with only one aspect, must it admit such a tiling that is periodic, or also isohedral?\u201d<\/p>\n
That there is one that is periodic has been proved in the case the tiles are polyominoes. See:<\/p>\n
\u201cArbitrary versus periodic storage schemes and tessellations of the plane using one type of polyomino\u201d, H.A.G Wijshoff, J. van Leeuwen, Information and Control, Vol. 62, issue 1, July 1984, 1-25.<\/p>\n
p.104: Figure 2.19(d): The second row of squares is offset incorrectly.<\/p>\n
p.107: Figure 2.24: the number 34 in the lower left corner should be 33.<\/p>\n
p.110: Line 12: \u201cany two nonadjacent sides\u201d should be \u201cany two nonadjacent sides or a vertex and a side if there are no nonadjacent sides\u201d.<\/p>\n
p.112: Line 8: \u201cAnd once again, all vertices on the boundary of the new patch are incident to three triangles or a square and a triangle.\u201d Should be replaced by \u201c All vertices on the boundary of the new patch are incident to three triangles, a square and a triangle, or two squares. We repeat the process.\u201d<\/p>\n
p.119: Problem 1: This is the same as Problem 2 from Section 2.1.<\/p>\n
p.129: Theorem 2.8: There is a formula that applies even if the two numbers are not even. This will be substituted here in next edition.<\/p>\n
p.145: Proof of Theorem 2.15, that the Euler characteristic of any patch is 1. The proof did not consider the case of a patch with only one tile. In that case, there are no vertices, no edges and one tile so the Euler characteristic is also 1.<\/p>\n
p.148: 3 lines above exercises, fix types of T\u2019s.<\/p>\n
p.148: Problem 2. Replace \u201cand has a translational symmetry\u201d by \u201cand has only one direction of translational symmetries\u201d<\/p>\n
p.149: Problem 4: Remove \u201cUse this to show that if the original tiling uses all of the prototiles in the protoset, we can obtain a periodic tiling that uses all of the prototiles in the protoset.\u201d This is false. (Could replace with \u201cWhy is this not enough to show that if the original tiling uses all of the prototiles in the protoset, we can obtain a periodic tiling that uses all of the prototiles in the protoset.\u201d<\/p>\n
Problem 12: \u201ca a\u201d should be \u201ca\u201d.<\/p>\n
p.154 bottom: \u201cMartin Garner\u201d should be \u201cMartin Gardner\u201d.<\/p>\n
p.159: Problem 13. \u201cunit area hexagon\u201d should be \u201cunit area regular hexagon\u201d.<\/p>\n
p.165: Problem 1: (T) and (T) should be overline{v}(T) and overline{e}(T).<\/p>\n
p. 165 Problem 5 Drop sentence “Show that we can construct such tilings that are not balanced.”<\/p>\n
p.178: Problem 7: \u201cpolygonal tiles. Then show\u201d should be \u201cpolygonal tiles, such\u201d<\/p>\n
p.180: Figure 3.15: (b) should be yellow, (c) should be green, (d) should be purple and (e) should be red.<\/p>\n
p.180: Line -10: \u201cthree\u201d should be \u201climited\u201d and \u201c Either they appear as every\u2026in Figure 3.16.\u201d should be replaced by \u201c Either they appear as in Figure 3.16(a), but allowing for a shift of one in any row or rows in the figure. Or they appear as in Figure 3.15(b), but allowing for a shift of one in any column or columns in the figure. Or they appear as in Figure 3.16(c).\u201d<\/p>\n
p.181: Figure 3.16 is slightly misleading, because these are more than three configurations of cornered and cornerless tiles. \u00a0In particular, in (a) each row can be shifted horizontally by a whole tile, and likewise for the columns in (b).<\/p>\n
p.184: Figure 3.20. Add (a) , (b), (c). Figure (c) is slightly wrong.<\/p>\n
p.185: Line 1: \u201cof on eof\u201d should be \u201cof one of\u201d<\/p>\n
p.185: Regarding the Project, this is possible if you allow the use of any number of modified hexagonal prototiles. \u00a0In particular, any set of Wang tiles be converted into an equivalent set of modified hexagons. \u00a0Pick two opposite edges of a regular hexagon and decorate them with unique markings that force them to mate with each other. \u00a0Then transcribe the markings from each Wang tile to the other four edges.<\/p>\n
p.185: Problem 2 Replace \u201cRobinson protoset?\u201d by \u201cRobinson protoset appearing in Figure 3.15?\u201d<\/p>\n
Problem 8: Replace \u201cwhat fraction of the tiles are crosses?\u201d with \u201cwhat fraction of the tiles are cornered crosses?<\/p>\n
p.186: In Exercise 4, replace \u201cStarting with the uniform tiling (3.6^2) as appears in Figure 2.3\u201d by \u201cStarting with the uniform tiling (3.6.3.6) as appears in Figure 2.5\u201d<\/p>\n
p.193: It appears this open problem was solved in N. Dolbilin, \u201cThe Countability of a Tiling Family and the Periodicity of a Tiling\u201d, N. Dolbilin, Discrete and Computational Geometry, 12: 403-414 (1995).<\/p>\n
p.194: Problem 6: \u201cpatch touch a \u201c should be \u201c patch share a\u201d<\/p>\n
p.199: Line (: After the limit, it should be F_{2n-1}\/F_{2n}<\/p>\n
p.200: line 5: Replace \u201cby adding heights to the corners of the resulting 2-dimensional tiling to obtain a tiling in 3-space.\u201d should be \u00a0\u201cby projection to 3-space.\u201d<\/p>\n
p.200: – 8 “from 4-dimensional up to 12-dimensional spaces”. \u00a0Should be \u201cfrom 4-dimensional to higher dimensional spaces\u201d<\/p>\n
p.217: line -2 “snub cuboctahedron,k” should be “snub cuboctahedron,”<\/p>\n
p.220: Problem 5: Replace \u201cShow that the only positive integer solutions\u201d by \u201cShow that for p , q > 2, the only positive integer solutions\u201d.<\/p>\n
Problem 11 Assume there is a vertex that occurs along the side of a polygon.<\/p>\n
p,222: Line -2: The lower limit should be 0, not a.<\/p>\n
p.226: line -1 “Although the octahedra” should be \u00a0“Although the octagons”<\/p>\n
p.230: Exercise 5(b). Note that the (p-2)(q-2) > 4 is equivalent to 1\/p+1\/q < \u00bd, which occurs at various other points in the book.<\/p>\n
p.233: Figure 4.21: The yz-plane is missing its lower half.<\/p>\n
p.236: Figure 4.26: The caption is missing a closing parenthesis<\/p>\n
p.241: Problem 11 It should be “three square faces”<\/p>\n
p.249: Problem 5 should refer to Figure 4.47.<\/p>\n
p.258: Line 12 “we can define the unit 3-sphere to be…” The exponent above S should be a 3.<\/p>\n
p.264: “FigureA.3” should be “Figure A.3”<\/p>\n
p.269: For Project 14, see also the last few pages of Andrew Glassner’s essay about aperiodic tilings for a concise example of Wang tiles performing computation: \u201cPenrose tiling\u201d, Andrew Glassner\u2019s Notebook, IEEE Computer Graphics and Applications, July\/August 1998, p. 78-86.<\/p>\n
p.270: Line 1: “In Section 3.6, we will see” should be “In Section 3.6, we saw”<\/p>\n
p.272: Line 18: “side-length” should be “side length”<\/p>\n
p.273: Notes for Section 2.7: \u201cif a finite polygonal protoset admits a tiling\u201d should be \u201cif a finite polygonal protoset admits a tiling with a translational symmetry\u201d<\/p>\n
p.274: Notes for Section 4.3: \u201cthe article \u201cshould be \u201cThe article\u201d.<\/p>\n
p.276: 11. \u201cEdmund Harris should be \u201cEdmund Harriss\u201d<\/p>\n
p.290: [43] It should be \u201cAufbau der Ebene aus kongruenten Bereichen\u201d<\/p>\n
p.291 [64] f should be F<\/p>\n
[66] should be followed by Delone<\/p>\n
[69] Remove ,<\/p>\n
[78] \u201ctetrhedra\u201d should be \u201ctetrahedra\u201d<\/p>\n
p.292: [91] kelvin should be capitalized.<\/p>\n
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II. Calculus <\/strong>by Jon Rogowski, Colin Adams and Robert Franzosa, Macmillan Publishing 4th<\/sup> edition, 2017. This book will continue to be updated in its forthcoming electronic versions.<\/p>\n <\/p>\n <\/p>\n III. Introduction to Topology: Pure and Applied<\/strong>, by Colin Adams and Robert Franzosa, Prentice Hall\/Pearson, June, 2007.<\/p>\n <\/p>\n Page 21, Example 0.14. It should read \u201ch(0, 0) = h(0, 1), but (0, 0) = (0, 1).\u201d<\/p>\n Page 72, Theorem 2.18. The assumption that B is regularly closed is unnecessary; B can be any subset of X.<\/p>\n Page 87, Exercise 3.21. A modified, clarified version:<\/p>\n 3.21. Consider the sets A, B, and C, illustrated in the figure below. A is the disk in the plane. B is the set [\u22121, 1) \u00d7 (\u22121, 1), and C = {(x, y) | \u2212 1 \u2264 x + y < 1 and \u2212 1 < x \u2212 y < 1}. Determine whether or not each set is open, closed, both, or neither in each of the producttopologies on the plane given by R \u00d7 R, Rl<\/sub> \u00d7 R, and Rl<\/sub> \u00d7 Rl<\/sub>, where Rl<\/sub> is the real line in the lower limit topology.<\/p>\n Page 93, Example 3.18. In the last sentence \u201conly\u201d is incorrect. Replace the last sentence with,<\/p>\n \u201cDigital circles arise when both m and n are odd. Do they arise in any other cases?\u201d<\/p>\n Page 97, Exercise 3.31. Other cases besides m and n even or odd need to be considered here.<\/p>\n The exercise should appear as:<\/p>\n 3.31. Describe the different topological spaces that result (and the conditions on m and n from which they arise) when we identify the endpoints m and n in a general digital interval {m, m + 1,…,n}.<\/p>\n Page 107, Example 3.31. The configuration space is incorrect. Here is a correct description:<\/p>\n The space of configurations corresponding to folding along the horizontal axis first, then (if possible) folding along the vertical axis.<\/p>\n Page 109, Example 3.33. In each case R+<\/sup> should be [0, \u221e).<\/p>\n Page 122, Proof of Theorem 4.13. In the second to last line, it should be x \u2208 f \u22121<\/sup>(U) rather<\/p>\n than x \u2208 U.<\/p>\n Page 134, Exercise 4.33. It is necessary to assume that X is not empty.<\/p>\n Page 161, Exercise 5.29(b). It should be \u201c> c1<\/sub>\u201d and \u201c< c2<\/sub>\u201d rather than \u201c= c1<\/sub>\u201d and \u201c= c2<\/sub>\u201d,<\/p>\n respectively.<\/p>\n Page 176, Exercise 6.7(a). It necessary to assume that X is nonempty.<\/p>\n Page 183, Exercise 6.19. It necessary to assume that n \u2265 2.<\/p>\n Page 197, Exercises 6.43 and 6.44. These exercises go together and should be 6.43(a) and<\/p>\n 6.43(b) rather than 6.43 and 6.44. The following is a hint for 6.43(b):<\/p>\n Hint: Given x, y \u2208 Rn<\/sup> \u2212 C, find a path in Rn<\/sup> \u2212 C going from x to y in a plane in Rn <\/sup>containing x and y.<\/p>\n Page 209, Proof of Theorem 7.6, & Page 212, Exercise 7.2. Although Theorem 7.6 is<\/p>\n presented prior to Theorems 7.7 and 7.8, the intent is to have Theorems 7.7 and 7.8 available in the exercise to prove Theorem 7.6.<\/p>\n Page 229, Example 7.15. It is not necessary to assume that s is an element of A with minimum absolute value. The argument carries through if s is simply any element in the set A.<\/p>\n Pages 286-7. The notions of a point of discontinuity and set of discontinuities are used, but they were not previously defined. One remedy is to introduce the following definition and theorem:<\/p>\n DEFINITION. Given a function f : X \u2192 Y and a point x \u2208 X, then f is continuous at x if for every open set V containing f(x) there exists an open set U containing x such that f(U) \u2282 V . If f is not continuous at x \u2208 X, then x is a point of discontinuity of f, and the set of all points of discontinuity of f is the set of discontinuities of f.<\/p>\n THEOREM. A function f : X \u2192 Y is continuous if and only if its set of discontinuities is empty.<\/p>\n Page 293, Proof of Theorem 9.20. For clarification, the \u201caforementioned open balls\u201d are those in the finite collection of open balls of radius \u03b5\/2 that cover f(S1<\/sup>).<\/p>\n Page 327, Equation 10.11. It should be \u201cpi = 0\u201d rather than \u201cpi \u2265 0\u201d.<\/p>\n Page 356. The following should appear at the top of the page:<\/p>\n (ii) In order to store the image, construct the cartoon determined by the partition.<\/p>\n (We can also store the color information by indicating which side of each digital simple<\/p>\n closed curve in the cartoon has which color.)<\/p>\n Page 407. The first sentence should read: \u201cAlthough the theorem is intuitively clear,….\u201d<\/p>\n Page 443, Example 14.4. Another approach to identifying a compact surface such as the one in this example is to use the relationships K = P #P and T #P = P #P #P to express the connected sum as a connected sum of projective planes only. In the case of K#K#P #P #T #T , each K contributes a P #P and each P #T contributes a P #P #P, so that the result is then 10P.<\/p>\n <\/p>\n IV. The Knot Book<\/strong>, AMS Publishing, 2004. See the last page of this printing for corrections.<\/p>\n <\/p>\n Articles:<\/u><\/strong><\/p>\n <\/p>\n Check that the list of 3-spiral knots of nine or fewer crossings is complete.<\/p>\n 2. \u201cNoncompact Fuchsian and Quasi-Fuchsian Surfaces in Hyperbolic 3-Manifolds\u201d, C. Adams, Algebraic and Geometric Topology 7 (2007) 565-582.<\/p>\n Morwen Thistlethwaite and Anastasiia Tsvietkova pointed out that the proof of the following theorem is incomplete.<\/p>\n Theorem 1.9<\/u>: Let K be an alternating knot in the 3\u2013sphere with hyperbolic complement. Let S be a checkerboard surface obtained from a reduced alternating projection of K. Then S is quasi-Fuchsian.<\/p>\n This theorem has been proved in \u00a0Quasifuchsian State Surfaces, David Futer, Efstratia Kalfagianni,\u00a0 Jessica S. Purcell, Transactions of the AMS ,Volume 366, Number 8, August 2014, Pages 4323\u20134343.<\/p>\n <\/p>\n Thanks to Erica Flapan and Hugh Howards and for catching these errors. See their preprint \u201cSplittings of Tangles and Spatial Graphs\u201d.<\/p>\n 2.The correct statement of Theorem 3.1 should be:<\/p>\n Let G be an n-composite alternating graph with reduced alternating projection \u03c0(G ), with n < 4. Then there is a disc D in S2<\/sup> such that the boundary of D meets edges of \u03c0(G ) transversely in n non-double points and \u03c0(G) is non-trivial to either side of the boundary of D.<\/p>\n <\/p>\n Note that the original only spoke of \u03c0(G) nontrivial to the inside of D. This same misstatement occurs in W. Menasco\u2019s original version of this theorem for alternating knots and links as well.<\/p>\n <\/p>\n 3. \u201cNoncompact Hyperbolic 3-Orbifolds of Small Volume\u201d, C. Adams, Topology 90, Proceedings of Topology Year at Ohio State University, ed. by B. Apanasov, W. Neumann, Reid and L. Siebenmann, de Gruyter (1992)1-16.<\/p>\n Thanks to Simon Drewitz and Ruth Kellerhals for catching these errors in that paper. See Appendix A of \u00a0\u201cThe non-arithmetic cusped hyperbolic 3-orbifold of minimal volume\u201d, S. Drewitz and R. Kellerhals, arXiv 2106.12279 for the details.<\/p>\n p. 10: The volume given for a certain orbifold with a {6,3,2}-cusp of volume \u221a21\/24 was incorrect. The stated volume was .47 but the actual volume is \u00a0approximately 0.3383. This does not affect Theorem 3.4 as the volume is still larger than the\u00a0 requisite v0<\/sub>\/4.<\/p>\n Inequality (11) appears to be incorrect. In fact, in contradiction to the claim in the paper, there is an orbifold corresponding to d=\u221a(1+\u221a(3)). See the material starting at the second to last paragraph of page 24 of \u201cThe non-arithmetic cusped hyperbolic 3-orbifold of minimal volume\u201d, and the depiction in Figure 13. However, again the volume is greater than v0<\/sub>\/4, so none of the theorems of \u00a0\u201cNoncompact Hyperbolic 3-Orbifolds of Small Volume\u201d are impacted.<\/p>\n","protected":false},"excerpt":{"rendered":" Errata Page This page contains all of the essential errata that I am currently aware of in books and papers that I have either authored or co-authored, listed in reverse chronological order. It does not include all inconsequential typos. … Continue reading \n
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