# in Euclidean, Spherical, and Hyperbolic Spaces

## Second Edition

The following are all the errata that I know of. The changes are indicated in red. If you see other errata please send me a message at dwh2@cornell.edu. I thank those (indicated below) who pointed out to me these errors.

Corrected for 2nd printing:

page xvii, line 7- (Preface): "www.math.cornell/~dwh/books/eg00/supplements"
Pointed out to me by Jeff Johannes, University of Nevada, Las Vegas.
page xxi, in the list of people acknowledged add in the proper alphabetic order: "Jeff Johannes (University of Nevada, Las Vegas)", "Konstantin Rybnikov (Cornell University)", "Derek Rhodes", "J. William Helbron, Upland, CA".

page 47, lines 9- - 7- (A Short History of Hyperbolic Geometry): "In 1964, N. V. Efimov [NE: Efimov] extended Hilbert's result by proving that there is no isometric embedding defined by functions whose first and second derivatives are continuous."

Pointed out to me by Konstantin Rybnikov, Cornell University.
page 51, Figure 5.4: Photo should have no lines and labels on it.

page 59, line 1: Equation should be: " "

Pointed out to me by Jeff Johannes, SUNY Geneseo.
page 69, lines 12-15 (Problem 6.3): "It is a part of mathematical folklore that it is impossible to trisect an angle with compass and straightedge, however you will show in Problem 14.4 that, in fact, it is possible. In addition, we will discuss what is a correct statement of the impossibility of trisecting angles." [See errata for page 186 for Problem 14.4 until it is included in later printings.]
I found this mistake while teaching at the University of Latvia.
page 144 bottom - to top of page 145:
" c. On the plane, spheres, and hyperbolic plane, the product of two rotations (in general, about different points) is a single rotation, translation, or horolation. Show how to geometrically determine which specific isometry you obtain, including the center and angle of anythe rotation."
I found this mistake while reading homeworks at the University of Latvia.
page 186, insert Problem 14.4 which you can access by clicking here.

page 195, bottom, and 196, top: The three occurrences of the limit "x®0" should be "t®0".

This was pointed out to me by Gian Mario Besana, Indiana University Northwest.
page 230, "Theorem 17.6b."

page 266, lines 7- - 6- (Problem 20.1): "There's no way within 2-space to move the mitten to fit the other hand (without turning it inside out)."

Pointed out to me by Derek Rhodes.
page 332, line 3: "Heilbron, J.L., ..."
Pointed out to me by J. William Helbron, Upland, CA.
page 336, insert at the top of the page:
"N. V. Efimov, "Generation of singularities on surfaces of negative curvature [Russian]", Mat. Sb. (N.S.) 64 (106), pp. 286-320, 1964.
Efimov proves that it is impossible to have a C2 isometric embedding of the hyperbolic plane onto a closed subset of Euclidean 3-space. These results are clarified for English-reading audiences in [NE: Milnor]."
Pointed out to me by Konstantin Rybnikov, Cornell University.
page 336, 13- - 9-:
"Milnor, Tilla, Efimov's theorem about complete immersed surfaces of negative curvature, Advances in Math., vol. 8, pp. 474-543, 1972.
Milnor clarifies for English readers the result in [NE: Efimov]."
Pointed out to me by Konstantin Rybnikov, Cornell University.
page 339, line 8:
"Hilbert, David, and S. Cohn-Vossen, Geometry and the Imagination, New York: Chelsea Publishing Co., 1983.
They state "it is our purpose to give a presentation of geometry, as it stands today [1932], in its visual, intuitive aspects." It includes an introduction to differential geometry, symmetry, and patterns (they call it "crystallographic groups"), and the geometry of spheres and other surfaces. Hilbert is the most famous mathematician of the first part of the 20th century."
page 347, under "angle":
"trisection, 186"
Corrected for 3nd printing:
page xxi, in the list of people acknowledged add in the proper alphabetic order: "Eric Bray (Cornell University)", "Barbara Edwards (Oregon State University)", "Douglas C. Mitarotonda (Cornell University)", "Laura Ann Ruganis (Cornell University)"

page 10, line 6-: "Problem 14.3b. See [SE: Hilbert, pp. 272-73] for another discussion of"

page 35, lines 14- & 10-: "Problem 17.1"

Pointed out to me by Colm Mulcahy, Spelman College.
page 84, lines 8-9: "and, if you set b = 2a, then the same equations show that ½Ar(b) = Ar(½b)."
Pointed out to me by Barbara Edwards, Oregon State University.
page 138, lines 5- & 4-: "symmetry group {Id,R}. We do not say Note that the letters S and A each have the same number of patterns symmetries, but we do not call them isomorphic"

page 181, line 9: "site: http://pi.math.cornell.edu/~dwh/books/eg00/."

Pointed out to me by Eric Bray, Cornell student.
page 192, line 6-10: "Look in the plane determined by N, A, and P and show that ÐPAA¢ is congruent to ÐAA¢P¢. You probably have already proved this in Part a; if not, look at the intersections of the plane determined by N, A, and P and the plane tangent to the North Pole N and the image plane P. In this plane draw the line PA¢¢ parallel to P¢A¢. Then use similar triangles and 6.2c to show that "
The error was pointed out to me by Nir Yehoshua Etzion, Cornell student.
page 232, line 9-: "these stars must be on the 2-manifold ..."
Pointed out to me by Laura Ann Ruganis, Cornell student
page 236, line 14: "Proposition II 14, which we discussed in Problem 13.1."
Pointed out to me by Laura Ann Ruganis, Cornell student
page 237, below Figure 18.3: "This leads to . ..."
Pointed out to me by Laura Ann Ruganis, Cornell student
Page 291, line 11-: "... and extended in Problem 20.6.
Pointed out to me by Douglas C. Mitarotonda, Cornell student.