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. It also includes additional references and results. I appreciate the various mathematicians and readers who have sent me corrections/comments. Please forward to me any other corrections that you might find.
Books:
- The Tiling Book (AMS Publishing, 2022)
(Thanks especially to Craig Kaplan and Dirk Frettloeh for sending corrections.)
- 28: Definition 1.10. “For any subset S” should be “For any connected subset S”
p. 29: An additional exercise for Section 1.1 is: “Find 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.”
p.30, Problem 11: Replace Figure 1.6 with Figure 1.3. And replace “blue”, “yellow” and “purple” by “red”, “orange” and “yellow”.
p. 31: Bottom: Replace two occurrences of q by x.
p.39: Problem 4: Replace “Use the classification of isometries into five types” by “Use the fact every isometry is a product of 0,1,2, or 3 reflections”
Problem 7 (b) Add “angles between -pi and pi”
p.43: Figure 1.26 (b): In the lowermost right, theta should be -theta.
p.44: Line -8 “there would be two rotation centers (see problem 8)” should be “ by Exercise 1.2/9, we would have a glide reflection.”
p.47: Figure 1.29: One black triangle is missing.
p.51: Definition 1.19: Remove stray )
p.54: Caption for Figure 1.39: “eight isohedral tilings” should be “seven isohedral tilings”
p.56: Problem 6: Replace “Figure 1.37” by “Figure 1.34”.
p.69: Add triangle to figure for p31m
p.70: At the top, “International Union of IUCr” should be “International Union of Crystallography (IUCr)”.
p.73: Problem 10: Switch order of 1.40 and 1.37
p.82: Third line: “In the top row we first see the regular hexagonal tiling and then two non-periodic tilings” should be “In 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.”
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.
Problem 10: Drop symmetry group pm.
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?”
That there is one that is periodic has been proved in the case the tiles are polyominoes. See:
“Arbitrary versus periodic storage schemes and tessellations of the plane using one type of polyomino”, H.A.G Wijshoff, J. van Leeuwen, Information and Control, Vol. 62, issue 1, July 1984, 1-25.
p.104: Figure 2.19(d): The second row of squares is offset incorrectly.
p.107: Figure 2.24: the number 34 in the lower left corner should be 33.
p.110: Line 12: “any two nonadjacent sides” should be “any two nonadjacent sides or a vertex and a side if there are no nonadjacent sides”.
p.112: Line 8: “And once again, all vertices on the boundary of the new patch are incident to three triangles or a square and a triangle.” Should be replaced by “ 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.”
p.119: Problem 1: This is the same as Problem 2 from Section 2.1.
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.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.148: 3 lines above exercises, fix types of T’s.
p.148: Problem 2. Replace “and has a translational symmetry” by “and has only one direction of translational symmetries”
p.149: Problem 4: Remove “Use 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.” This is false. (Could replace with “Why 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.”
Problem 12: “a a” should be “a”.
p.154 bottom: “Martin Garner” should be “Martin Gardner”.
p.159: Problem 13. “unit area hexagon” should be “unit area regular hexagon”.
p.165: Problem 1: (T) and (T) should be overline{v}(T) and overline{e}(T).
p.178: Problem 7: “polygonal tiles. Then show” should be “polygonal tiles, such”
p.180: Figure 3.15: (b) should be yellow, (c) should be green, (d) should be purple and € should be red.
p.180: Line -10: “three” should be “limited” and “ Either they appear as every…in Figure 3.16.” should be replaced by “ 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).”
p.181: Figure 3.16 is slightly misleading, because these are more than three configurations of cornered and cornerless tiles. In particular, in (a) each row can be shifted horizontally by a whole tile, and likewise for the columns in (b).
p.184: Figure 3.20. Add (a) , (b), (c). Figure (c) is slightly wrong.
p.185: Line 1: “of on eof” should be “of one of”
p.185: Regarding the Project, this is possible if you allow the use of any number of modified hexagonal prototiles. In particular, any set of Wang tiles be converted into an equivalent set of modified hexagons. Pick two opposite edges of a regular hexagon and decorate them with unique markings that force them to mate with each other. Then transcribe the markings from each Wang tile to the other four edges.
p.185: Problem 2 Replace “Robinson protoset?” by “Robinson protoset appearing in Figure 3.15?”
Problem 8: Replace “what fraction of the tiles are crosses?” with “what fraction of the tiles are cornered crosses?
p.186: In Exercise 4, replace “Starting with the uniform tiling (3.6^2) as appears in Figure 2.3” by “Starting with the uniform tiling (3.6.3.6) as appears in Figure 2.5”
p.193: It appears this open problem was solved in N. Dolbilin, “The Countability of a Tiling Family and the Periodicity of a Tiling”, N. Dolbilin, Discrete and Computational Geometry, 12: 403-414 (1995).
p.194: Problem 6: “patch touch a “ should be “ patch share a”
p.199: Line (: After the limit, it should be F_{2n-1}/F_{2n}
p.200: line 5: Replace “by adding heights to the corners of the resulting 2-dimensional tiling to obtain a tiling in 3-space.” should be “by projection to 3-space.”
p.200: – 8 “from 4-dimensional up to 12-dimensional spaces”. Should be “from 4-dimensional to higher dimensional spaces”
p.217, line -2 “snub cuboctahedron,k” should be “snub cuboctahedron,”
p.220: Problem 5: Replace “Show that the only positive integer solutions” by “Show that for p , q > 2, the only positive integer solutions”.
p,222: Line -2: The lower limit should be 0, not a.
p.226: line -1 “Although the octahedra” should be “Although the octagons”
p.230: Exercise 5(b). Note that the (p-2)(q-2) > 4 is equivalent to 1/p+1/q < ½, which occurs at various other points in the book.
p.233: Figure 4.21: The yz-plane is missing its lower half.
p.236: Figure 4.26: The caption is missing a closing parenthesis
p.258: Line 12 “we can define the unit 3-sphere to be…” The exponent above S should be a 3.
p.264: “FigureA.3” should be “Figure A.3”
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: “Penrose tiling”, Andrew Glassner’s Notebook, IEEE Computer Graphics and Applications, July/August 1998, p. 78-86.
p.270: Line 1: “In Section 3.6, we will see” should be “In Section 3.6, we saw”
p.272: Line 18: “side-length” should be “side length”
p.273: Notes for Section 2.7: “if a finite polygonal protoset admits a tiling” should be “if a finite polygonal protoset admits a tiling with a translational symmetry”
p.274: Notes for Section 4.3: “the article “should be “The article”.
p.276: 11. “Edmund Harris should be “Edmund Harriss”
p.290: [43] It should be “Aufbau der Ebene aus kongruenten Bereichen”
p.291 [64] f should be F
[66] should be followed by Delone
[69] Remove ,
[78] “tetrhedra” should be “tetrahedra”
p.292: [91] kelvin should be capitalized.
II. Calculus by Jon Rogowski, Colin Adams and Robert Franzosa, Macmillan Publishing 4th edition, 2017. This book will continue to be updated in its forthcoming electronic versions.
III. Introduction to Topology: Pure and Applied, by Colin Adams and Robert Franzosa, Prentice Hall/Pearson, June, 2007.
Page 21, Example 0.14. It should read “h(0, 0) = h(0, 1), but (0, 0) = (0, 1).”
Page 72, Theorem 2.18. The assumption that B is regularly closed is unnecessary; B can be any subset of X.
Page 87, Exercise 3.21. A modified, clarified version:
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 [−1, 1) × (−1, 1), and C = {(x, y) | − 1 ≤ x + y < 1 and − 1 < x − 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 × R, Rl × R, and Rl × Rl, where Rl is the real line in the lower limit topology.
Page 93, Example 3.18. In the last sentence “only” is incorrect. Replace the last sentence with,
“Digital circles arise when both m and n are odd. Do they arise in any other cases?”
Page 97, Exercise 3.31. Other cases besides m and n even or odd need to be considered here.
The exercise should appear as:
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}.
Page 107, Example 3.31. The configuration space is incorrect. Here is a correct description:
The space of configurations corresponding to folding along the horizontal axis first, then (if possible) folding along the vertical axis.
Page 109, Example 3.33. In each case R+ should be [0, ∞).
Page 122, Proof of Theorem 4.13. In the second to last line, it should be x ∈ f −1(U) rather
than x ∈ U.
Page 134, Exercise 4.33. It is necessary to assume that X is not empty.
Page 161, Exercise 5.29(b). It should be “> c1” and “< c2” rather than “= c1” and “= c2”,
respectively.
Page 176, Exercise 6.7(a). It necessary to assume that X is nonempty.
Page 183, Exercise 6.19. It necessary to assume that n ≥ 2.
Page 197, Exercises 6.43 and 6.44. These exercises go together and should be 6.43(a) and
6.43(b) rather than 6.43 and 6.44. The following is a hint for 6.43(b):
Hint: Given x, y ∈ Rn − C, find a path in Rn − C going from x to y in a plane in Rn containing x and y.
Page 209, Proof of Theorem 7.6, & Page 212, Exercise 7.2. Although Theorem 7.6 is
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.
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.
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:
DEFINITION. Given a function f : X → Y and a point x ∈ 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) ⊂ V . If f is not continuous at x ∈ 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.
THEOREM. A function f : X → Y is continuous if and only if its set of discontinuities is empty.
Page 293, Proof of Theorem 9.20. For clarification, the “aforementioned open balls” are those in the finite collection of open balls of radius ε/2 that cover f(S1).
Page 327, Equation 10.11. It should be “pi = 0” rather than “pi ≥ 0”.
Page 356. The following should appear at the top of the page:
(ii) In order to store the image, construct the cartoon determined by the partition.
(We can also store the color information by indicating which side of each digital simple
closed curve in the cartoon has which color.)
Page 407. The first sentence should read: “Although the theorem is intuitively clear,….”
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.
IV. The Knot Book, AMS Publishing, 2004. See the last page of this printing for corrections.
Articles:
- “The Spiral Index of Knots”, C. Adams, W. George*, R. Hudson*, R. Morrison*, L. Starkston*, S. Taylor*, O. Turanova*, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 149, Issue 2, (2010) 297-315.
Check that the list of 3-spiral knots of nine or fewer crossings is complete.
2. “Noncompact Fuchsian and Quasi-Fuchsian Surfaces in Hyperbolic 3-Manifolds”, C. Adams, Algebraic and Geometric Topology 7 (2007) 565-582.
Morwen Thistlethwaite and Anastasiia Tsvietkova pointed out that the proof of the following theorem is incomplete.
Theorem 1.9: Let K be an alternating knot in the 3–sphere with hyperbolic complement. Let S be a checkerboard surface obtained from a reduced alternating projection of K. Then S is quasi-Fuchsian.
This theorem has been proved in Quasifuchsian State Surfaces, David Futer, Efstratia Kalfagianni, Jessica S. Purcell, Transactions of the AMS ,Volume 366, Number 8, August 2014, Pages 4323–4343.
- “Alternating Graphs”, C. Adams, K. Foley*, J. Kravis*, R. Dorman,* S. Payne*, Journal of Combinatorial Theory, Series B, v. 77, no. 1 (1999) 96-120.
Thanks to Erica Flapan and Hugh Howards and for catching these errors. See their preprint “Splittings of Tangles and Spatial Graphs”.
- In the definition of what it means to be an alternating region in a projection of a spatial graph, each region should be considered as a complementary region to a regular neighborhood of the projection, not just to the projection itself.
2.The correct statement of Theorem 3.1 should be:
Let G be an n-composite alternating graph with reduced alternating projection π(G ), with n < 4. Then there is a disc D in S2 such that the boundary of D meets edges of π(G ) transversely in n non-double points and π(G) is non-trivial to either side of the boundary of D.
Note that the original only spoke of π(G) nontrivial to the inside of D. This same misstatement occurs in W. Menasco’s original version of this theorem for alternating knots and links as well.
3. “Noncompact Hyperbolic 3-Orbifolds of Small Volume”, 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.
Thanks to Simon Drewitz and Ruth Kellerhals for catching these errors in that paper. See Appendix A of “The non-arithmetic cusped hyperbolic 3-orbifold of minimal volume”, S. Drewitz and R. Kellerhals, arXiv 2106.12279 for the details.
p. 10: The volume given for a certain orbifold with a {6,3,2}-cusp of volume √21/24 was incorrect. The stated volume was .47 but the actual volume is approximately 0.3383. This does not affect Theorem 3.4 as the volume is still larger than the requisite v0/4.
Inequality (11) appears to be incorrect. In fact, in contradiction to the claim in the paper, there is an orbifold corresponding to d=√(1+√(3)). See the material starting at the second to last paragraph of page 24 of “The non-arithmetic cusped hyperbolic 3-orbifold of minimal volume”, and the depiction in Figure 13. However, again the volume is greater than v0/4, so none of the theorems of “Noncompact Hyperbolic 3-Orbifolds of Small Volume” are impacted.