Since I hope that some readers of “the Poincare Conjecture” might be interested in Euclidean & non-Euclidean geometry, I have taken a look through my geometry bookshelf and constructed the following entries.

Until several years ago I had never gone back to look at high school geometry; there was this huge gap in my education, between high school geometry and Riemannian geometry (the geometry of general relativity).

When I perceived that gap, I acted on a couple of recommendations, acquiring Martin’s “Foundations” and Greenberg’s “Non-Euclidean”. The Hartshorne was a later acquisition: he’s known for a challenging book on algebraic geometry, and I just had to see what he did with Euclid.

I also picked up a few ancient used high school geometry books, but I’m not going to include them! And for now, I’ll omit Hilbert’s “Foundations” because we’re getting his work second-hand in the three non-Euclidean geometry books.

I wish I’d known how readable Heath’s “Euclid” was; i’d have bought it early on.

One of the things that fascinates me most in plane geometry is the frieze and wallpaper groups. (Sorry, go look them up! For me, the marvel was that we could say there are only 7 frieze and 17 wallpaper groups.) Martin’s “transformation” and Beyer’s “tessellations” book are devoted to them.

Pedoe I recognized as a co-author of a classic: Hodge & Pedoe “methods of algebraic geometry”; Hodge is he of the “Hodge decomposition” and the “Hodge star operator”. I bought Pedoe’s book as much for his name as for its title.

Beyer, Jinny; Designing Tesselations.

Contemporary Books 1999; ISBN 0 8092 2866 1.

[frieze & wallpaper groups; 3 feb 2008]

this is a rare find. it is written by a woman who designs quilts; she made it her business to learn about symmetry, and to get a couple of mathematicians to help her out. this is a visually magnificent and mathematically accurate presentation of all the wallpaper groups (and the frieze groups).

Greenberg, Marvin Jay; Euclidean and Non-Euclidean Geometries.

W.H.Freeman & Co., (2nd Ed) 1980. ISBN 0 7167 1103 6.

[euclidean & non-euclidean geometry; 3 feb 2008]

this would be my first choice if decide to work through the subject. it has a comfortable writing style despite being a math book. this one i have only browsed, but it reads easier than martin’s “foundations”. oh, it’s intended as lower-division mathmatics. hey, i’m not too proud to read it – once, but not any more.

Hartshorne, Robin; Geometry: Euclid and Beyond.

Springer, 2000; ISBN 0 387 98650 2.

[euclidean & noneuclidean geometry; 3 feb 2008]

and more. a fairly wide-ranging text, upper-division mathematics. i wouldn’t pick it up first, but i’m glad to have it. i would probably go through this after greenberg and after martin, for all the auxiliary material (e.g. field extensions).

Heath, Sir Thomas L.; Euclid: the Thirteen Books of the Elements.

Dover, 1956 (2nd Ed.); ISBN 0 486 60088 2, 60089 0, and 60090 4.

[geometry; 3 feb 2008]

i daresay this is almost the definition of geometry. what makes these books extaordinarily informative is the commentary by Heath. you should have this handy whenever you read a modern book about euclidean geometry. in fact, if you have this book by your side when you’re taking high school geometry, your geometry teacher will either love you or hate you.

Martin, George .; The Foundations of Geometry and the Non-Euclidean Plane.

Springer 1975 (2nd printing 1986); ISBN 0 387 90636 3.

[euclidean & non-euclidean geometry; 3 feb 2008]

upper-division. what i love about this book is its list of 26 equivalents to euclid’s parallel postulate. nevertheless, a fairly dry book. i would probably go through it after i had done greenberg. (i’ve more than browsed this book, but not by much. i read this in preference to greenberg because martin looked more like a math book.)

Martin, George .; Transformation Geometry.

Springer 1982. ISBN 0 387 90636 3.

[geometry, symmetry, frieze & wallpaper groups; 3 feb 2008]

introductory, requiring no college math. the mathematics of frieze and wallpaper groups. i’ve played with them, but i want to work through this book.

Pedoe, Dan; Geometry and the Visual Arts.

Dover, 1983; ISBN 0 486 24458 X

[geometry; 3 feb 2008]

an introductory book. the first three of its nine chapters are devoted to vitruvius, durer, and da Vinci. not your usual geometry book. not a text but a chance to play with geometry as found in art. i think you better have had high school geometry; at least know what a straightedge-and-compass construction is. from the preface: “This book can be taken by the general reader as a diversion into the by-ways of history, with glimpses of the enormous importance of geometry to such people as….”

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February 5, 2008 at 5:30 pm

“go look them up!” i said. easier said than done. “wallpaper groups” and “frieze groups” are what you should search on. here are two URLs, one for wallpaper groups…

http://www.clarku.edu/~djoyce/wallpaper/

and one for frieze groups…

http://www.joma.org/images/upload_library/4/vol1/architecture/Math/seven.html

rip