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<!DOCTYPE html><html><head><title>Geometry and the Imagination - David Hilbert, Stephan Cohn-Vossen - Google Books</title><link rel="stylesheet" href="/books/css/_a33f2a89320471e58c940b9287b9d4eb/kl_about_this_book_kennedy_full_bundle.css" type="text/css" /><link rel="stylesheet"href="https://fonts.googleapis.com/css2?family=Product+Sans:wght@400"><script src="/books/javascript/atb_a33f2a89320471e58c940b9287b9d4eb__en.js"></script><link rel="canonical" href="https://books.google.com/books/about/Geometry_and_the_Imagination.html?id=7WY5AAAAQBAJ"/><meta property="og:url" content="https://books.google.com/books/about/Geometry_and_the_Imagination.html?id=7WY5AAAAQBAJ"/><meta name="title" content="Geometry and the Imagination"/><meta name="description" content="This remarkable book endures as a true masterpiece of mathematical exposition. The book is overflowing with mathematical ideas, which are always explained clearly and elegantly, and above all, with penetrating insight. It is a joy to read, both for beginners and experienced mathematicians. Geometry and the Imagination is full of interesting facts, many of which you wish you had known before. The book begins with examples of the simplest curves and surfaces, including thread constructions of certain quadrics and other surfaces. The chapter on regular systems of points leads to the crystallographic groups and the regular polyhedra in $\mathbb{R}^3$. In this chapter, they also discuss plane lattices. By considering unit lattices, and throwing in a small amount of number theory when necessary, they effortlessly derive Leibniz's series: $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + - \ldots$. In the section on lattices in three and more dimensions, the authors consider sphere-packing problems, including the famous Kepler problem. One of the most remarkable chapters is ``Projective Configurations''. In a short introductory section, Hilbert and Cohn-Vossen give perhaps the most concise and lucid description of why a general geometer would care about projective geometry and why such an ostensibly plain setup is truly rich in structure and ideas. The chapter on kinematics includes a nice discussion of linkages and the geometry of configurations of points and rods that are connected and, perhaps, constrained in some way. This topic in geometry has become increasingly important in recent times, especially in applications to robotics. This is another example of a simple situation that leads to a rich geometry. It would be hard to overestimate the continuing influence Hilbert-Cohn-Vossen's book has had on mathematicians of this century. It surely belongs in the pantheon of great mathematics books."/><meta property="og:title" content="Geometry and the Imagination"/><meta property="og:type" content="book"/><meta property="og:site_name" content="Google Books"/><meta property="og:image" content="https://books.google.com.sg/books/publisher/content?id=7WY5AAAAQBAJ&printsec=frontcover&img=1&zoom=1&edge=curl&imgtk=AFLRE72NtNzpLA1qVsV5s5Owtj22fzkwYtpjctNDMyGSEw9v4blbsZp1kWZNIHZKV_YtmUzisWWsyJuNnZXjazRYNEnpcPTeBNfgn_aXnO2IdyKQqMLKM7O_wQ45X0pvB1vtmESkOPWF"/><link rel="image_src" href="https://books.google.com.sg/books/publisher/content?id=7WY5AAAAQBAJ&printsec=frontcover&img=1&zoom=1&edge=curl&imgtk=AFLRE72NtNzpLA1qVsV5s5Owtj22fzkwYtpjctNDMyGSEw9v4blbsZp1kWZNIHZKV_YtmUzisWWsyJuNnZXjazRYNEnpcPTeBNfgn_aXnO2IdyKQqMLKM7O_wQ45X0pvB1vtmESkOPWF"/><script></script><style>#gbar,#guser{font-size:13px;padding-top:1px !important;}#gbar{height:22px}#guser{padding-bottom:7px !important;text-align:right}.gbh,.gbd{border-top:1px solid #c9d7f1;font-size:1px}.gbh{height:0;position:absolute;top:24px;width:100%}@media all{.gb1{height:22px;margin-right:.5em;vertical-align:top}#gbar{float:left}}a.gb1,a.gb4{text-decoration:underline !important}a.gb1,a.gb4{color:#00c !important}.gbi .gb4{color:#dd8e27 !important}.gbf .gb4{color:#900 !important} #gbar { padding:.3em .6em !important;}</style></head><body class=""><div id=gbar><nobr><a target=_blank class=gb1 href="https://www.google.com.sg/search?tab=pw">Search</a> <a target=_blank class=gb1 href="https://www.google.com.sg/imghp?hl=en&tab=pi">Images</a> <a target=_blank class=gb1 href="https://maps.google.com.sg/maps?hl=en&tab=pl">Maps</a> <a target=_blank class=gb1 href="https://play.google.com/?hl=en&tab=p8">Play</a> <a target=_blank class=gb1 href="https://www.youtube.com/?tab=p1">YouTube</a> <a target=_blank class=gb1 href="https://news.google.com/?tab=pn">News</a> <a target=_blank class=gb1 href="https://mail.google.com/mail/?tab=pm">Gmail</a> <a target=_blank class=gb1 href="https://drive.google.com/?tab=po">Drive</a> <a target=_blank class=gb1 style="text-decoration:none" href="https://www.google.com.sg/intl/en/about/products?tab=ph"><u>More</u> »</a></nobr></div><div id=guser width=100%><nobr><span id=gbn class=gbi></span><span id=gbf class=gbf></span><span id=gbe></span><a target=_top id=gb_70 href="https://www.google.com/accounts/Login?service=print&continue=https://books.google.com.sg/books%3Fid%3D7WY5AAAAQBAJ%26redir_esc%3Dy%26hl%3Den&hl=en&ec=GAZACg" class=gb4>Sign in</a></nobr></div><div class=gbh style=left:0></div><div class=gbh style=right:0></div><div role="alert" style="position: absolute; left: 0; right: 0;"><a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&redir_esc=y&hl=en&output=html_text" title="Screen reader users: click this link for accessible mode. 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The book is overflowing with mathematical ideas, which are always explained clearly and elegantly, and above all, with penetrating insight. It is a joy to read, both for beginners and experienced mathematicians. Geometry and the Imagination is full of interesting facts, many of which you wish you had known before. The book begins with examples of the simplest curves and surfaces, including thread constructions of certain quadrics and other surfaces. The chapter on regular systems of points leads to the crystallographic groups and the regular polyhedra in $\mathbb{R}^3$. In this chapter, they also discuss plane lattices. By considering unit lattices, and throwing in a small amount of number theory when necessary, they effortlessly derive Leibniz's series: $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + - \ldots$. In the section on lattices in three and more dimensions, the authors consider sphere-packing problems, including the famous Kepler problem. One of the most remarkable chapters is ``Projective Configurations''. In a short introductory section, Hilbert and Cohn-Vossen give perhaps the most concise and lucid description of why a general geometer would care about projective geometry and why such an ostensibly plain setup is truly rich in structure and ideas. The chapter on kinematics includes a nice discussion of linkages and the geometry of configurations of points and rods that are connected and, perhaps, constrained in some way. This topic in geometry has become increasingly important in recent times, especially in applications to robotics. This is another example of a simple situation that leads to a rich geometry. It would be hard to overestimate the continuing influence Hilbert-Cohn-Vossen's book has had on mathematicians of this century. It surely belongs in the pantheon of great mathematics books.</div></div></div><div class="search_box_wrapper"><form action=/books id=search_form style="margin:0px;padding:0px;" method=get> <input type=hidden name="redir_esc" value="y"><input type=hidden name="id" value="7WY5AAAAQBAJ"><table cellpadding=0 cellspacing=0 class="swv-table"><tr><td class="swv-td-search"><span><input id=search_form_input type=text maxlength=1024 class="text_flat swv-input-search" aria-label="Search in this book" name=q value="" title="Search inside" accesskey=i></span></td><td class="swv-td-space"><div> </div></td><td><input type=submit value="Search inside"></td></tr></table><script type="text/javascript">if (window['_OC_autoDir']) {_OC_autoDir('search_form_input');}</script></form><div id="preview-link"><a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&printsec=frontcover" class="primary"><span dir=ltr>Preview this book</span> »</a></div></div></td> </tr></table><div id="summary-second-column"></div></div></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="selected_pages_anchor"></a>Selected pages</h3><div id=selected_pages class=about_content><div id=selected_pages_v><div class="selectedpagesthumbnail"><a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&pg=PR1&source=gbs_selected_pages&cad=1" ><img src="https://books.google.com.sg/books/publisher/content?id=7WY5AAAAQBAJ&pg=PR1&img=1&zoom=1&sig=ACfU3U3xRl7z4LCHiygqoBk7XPba0xECfQ" alt="Title Page" title="Title Page" height="160" border="1"></a><br/><a class="primary" href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&pg=PR1&source=gbs_selected_pages&cad=1" >Title Page</a></div><div class="selectedpagesthumbnail"><a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&pg=PR7&source=gbs_selected_pages&cad=1" ><img src="https://books.google.com.sg/books/publisher/content?id=7WY5AAAAQBAJ&pg=PR7&img=1&zoom=1&sig=ACfU3U3Eokwop3RCb7_5b9NBNGsPoVdr5w" alt="Table of Contents" title="Table of Contents" height="160" border="1"></a><br/><a class="primary" href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&pg=PR7&source=gbs_selected_pages&cad=1" >Table of Contents</a></div><div class="selectedpagesthumbnail"><a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&pg=PA343&source=gbs_selected_pages&cad=1" ><img src="https://books.google.com.sg/books/publisher/content?id=7WY5AAAAQBAJ&pg=PA343&img=1&zoom=1&sig=ACfU3U2nmwB_saWQHl-IirgMManPq3lDVg" alt="Index" title="Index" height="160" border="1"></a><br/><a class="primary" href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&pg=PA343&source=gbs_selected_pages&cad=1" >Index</a></div><div style="clear:both;"></div></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="toc_anchor"></a>Contents</h3><div id=toc class=about_content><div id=toc_v><div class="first_toc_column"><div class="first_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>CHAPTER </span></span></div></td><td class="toc_number" align=right>1</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap">4 </span></div></td><td class="toc_number" align=right>19</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>APPENDICES TO CHAPTER I </span></span></div></td><td class="toc_number" align=right>25</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>CHAPTER II </span></span></div></td><td class="toc_number" align=right>32</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Crystals as Regular Systems of Points </span></span></div></td><td class="toc_number" align=right>52</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Plane Motions and their Composition Classification of </span></span></div></td><td class="toc_number" align=right>59</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>The Crystallographic Groups of Motions in the Plane </span></span></div></td><td class="toc_number" align=right>70</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Crystallographic Classes and Groups of Motions in Space </span></span></div></td><td class="toc_number" align=right>81</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr></table></div></div><div class="second_toc_column"><div class="second_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Space Curves </span></span></div></td><td class="toc_number" align=right>178</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>The Spherical Image and Gaussian Curvature </span></span></div></td><td class="toc_number" align=right>193</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Developable Surfaces Ruled Surfaces </span></span></div></td><td class="toc_number" align=right>204</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>The Twisting of Space Curves </span></span></div></td><td class="toc_number" align=right>211</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Bendings Leaving a Surface Invariant </span></span></div></td><td class="toc_number" align=right>232</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Hyperbolic Geometry and its Relation to Euclidean and </span></span></div></td><td class="toc_number" align=right>242</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Stereographic Projection and CirclePreserving Trans </span></span></div></td><td class="toc_number" align=right>248</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>CHAPTER V </span></span></div></td><td class="toc_number" align=right>272</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr></table></div></div><br style="clear:both;"/></div><span onclick="_OC_setListSectionVisible('toc_h', 1)" class=morelesslink id=toc_hc0 style="display:none"><br>More</span><div id=toc_hd1><div class="first_toc_column"><div class="first_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>The Regular Polyhedra </span></span></div></td><td class="toc_number" align=right>89</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Perspective Ideal Elements and the Principle of Duality </span></span></div></td><td class="toc_number" align=right>112</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Ideal Elements and the Principle of Duality in Space </span></span></div></td><td class="toc_number" align=right>119</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Comparison of Pascals and Desargues Theorems </span></span></div></td><td class="toc_number" align=right>128</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Reyes Configuration </span></span></div></td><td class="toc_number" align=right>134</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Regular Polyhedra in Three and Four Dimensions </span></span></div></td><td class="toc_number" align=right>143</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Enumerative Methods of Geometry </span></span></div></td><td class="toc_number" align=right>157</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Schläflis DoubleSix </span></span></div></td><td class="toc_number" align=right>164</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>CHAPTER IV </span></span></div></td><td class="toc_number" align=right>171</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr></table></div></div><div class="second_toc_column"><div class="second_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>An Instrument for Constructing the Ellipse and its Roul </span></span></div></td><td class="toc_number" align=right>283</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Polyhedra </span></span></div></td><td class="toc_number" align=right>290</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>OneSided Surfaces </span></span></div></td><td class="toc_number" align=right>302</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>The Projective Plane as a Closed Surface </span></span></div></td><td class="toc_number" align=right>313</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Topological Mappings of a Surface onto Itself Fixed </span></span></div></td><td class="toc_number" align=right>324</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Conformal Mapping of the Torus </span></span></div></td><td class="toc_number" align=right>330</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>APPENDICES TO CHAPTER VI </span></span></div></td><td class="toc_number" align=right>340</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><a class="primary" href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&printsec=copyright" ><span title="Copyright" style="white-space:nowrap"><span dir=ltr>Copyright</span></span></a></div></td><td class="toc_number" align=right></td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr></table></div></div><br style="clear:both;"/><span onclick="_OC_setListSectionVisible('toc_h', 0)" class=morelesslink id=toc_hc1 style="display:none"><br>Less</span></div><script type="text/javascript">if (window['_OC_setListSectionVisible']) {_OC_setListSectionVisible('toc_h', 0);}</script></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="book_other_versions_anchor"></a>Other editions - <a href='https://books.google.com.sg/books?q=editions:ISBN0821819984&id=7WY5AAAAQBAJ'>View all</a></h3><div id=book_other_versions class=about_content><div id=book_other_versions_v><div class="one-third-column"><div class="crsiwrapper"><table class="rsi" cellspacing=0 cellpadding=0 border=0><tr><td class="coverdstd" align="center"><a href="https://books.google.com.sg/books?id=n3oqEAAAQBAJ&source=gbs_book_other_versions_r&cad=3" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/publisher/content?id=n3oqEAAAQBAJ&printsec=frontcover&img=1&zoom=5&edge=curl" border="0" height="80"></a></td><td valign=top><div class=resbdy><a class="primary cresbdy" href="https://books.google.com.sg/books?id=n3oqEAAAQBAJ&printsec=frontcover&source=gbs_book_other_versions_r&cad=3"><span dir=ltr>Geometry and the Imagination</span></a><br><span style="line-height: 1.3em; font-size:-1;"><span><a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22D.+Hilbert%22" class="secondary"><span dir=ltr>D. Hilbert</span></a>,<a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22S.+Cohn-Vossen%22" class="secondary"><span dir=ltr>S. Cohn-Vossen</span></a></span><br/><span><span style="color:#99522e">Limited preview</span> - 2021</span><br/></span></div></td><td align=right></td></tr></table></div></div><div class="one-third-column"><div class="crsiwrapper"><table class="rsi" cellspacing=0 cellpadding=0 border=0><tr><td class="coverdstd" align="center"><a href="https://books.google.com.sg/books?id=SOtLAAAAMAAJ&source=gbs_book_other_versions_r&cad=3" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/publisher/content?id=SOtLAAAAMAAJ&printsec=frontcover&img=1&zoom=5" border="0" height="80"></a></td><td valign=top><div class=resbdy><a class="primary cresbdy" href="https://books.google.com.sg/books?id=SOtLAAAAMAAJ&source=gbs_book_other_versions_r&cad=3"><span dir=ltr>Geometry and the Imagination</span></a><br><span style="line-height: 1.3em; font-size:-1;"><span><a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22David+Hilbert%22" class="secondary"><span dir=ltr>David Hilbert</span></a></span><br/><span><span style="color:#999">Snippet view</span> - 1952</span><br/></span></div></td><td align=right></td></tr></table></div></div><div class="one-third-column"><div class="crsiwrapper"><table class="rsi" cellspacing=0 cellpadding=0 border=0><tr><td class="coverdstd" align="center"><a href="https://books.google.com.sg/books?id=Z8EfAQAAIAAJ&source=gbs_book_other_versions_r&cad=3" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/publisher/content?id=Z8EfAQAAIAAJ&printsec=frontcover&img=1&zoom=5" border="0" height="80"></a></td><td valign=top><div class=resbdy><a class="primary cresbdy" href="https://books.google.com.sg/books?id=Z8EfAQAAIAAJ&source=gbs_book_other_versions_r&cad=3"><span dir=ltr>Geometry and the Imagination</span></a><br><span style="line-height: 1.3em; font-size:-1;"><span><a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22David+Hilbert%22" class="secondary"><span dir=ltr>David Hilbert</span></a>,<a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22Stephan+Cohn-Vossen%22" class="secondary"><span dir=ltr>Stephan Cohn-Vossen</span></a></span><br/><span><span style="color:#999">Snippet view</span> - 1952</span><br/></span></div></td><td align=right></td></tr></table></div></div><div style="clear:both"></div><div class="showall"><a class="secondary" href="https://books.google.com.sg/books?q=editions:ISBN0821819984&id=7WY5AAAAQBAJ&source=gbs_book_other_versions_r&cad=3">View all »</a></div><script>(function () {var fn = window['_OC_WSBookList'] || window['_OC_BookList'];fn && fn('book_other_versions', [{"title":"Geometry and the Imagination","authors":"D. 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{color: #6057CC;font-size: 11px;}.cloud6 {color: #574BCC;font-size: 11.5px;}.cloud5 {color: #4E3DCC;font-size: 12px;}.cloud4 {color: #4632CC;font-size: 14px;}.cloud3 {color: #3D26CC;font-size: 16px;}.cloud2 {color: #341ACC;font-size: 18px;}.cloud1 {color: #2B0DCC;font-size: 20px;}.cloud0 {color: #2200CC;font-size: 22px;}.cloud {margin-top: 4px;line-height: 24px;}.cloud a {margin-right: 6px;text-decoration: none;}.cloud a:hover {text-decoration: underline;}</style><div class=cloud><a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=angles&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>angles</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=axes&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>axes</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=axioms&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>axioms</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=axis&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>axis</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=boundary&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>boundary</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=called&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>called</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=centers+of+rotation&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>centers of rotation</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=centers+of+similitude&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>centers of similitude</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=circle&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>circle</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=circular&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>circular</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=closed+curve&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>closed curve</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=closed+surface&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>closed surface</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=cone&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>cone</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=configuration&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>configuration</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=conformal+mapping&source=gbs_word_cloud_r&cad=4" class="cloud0"><span dir=ltr>conformal mapping</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=congruent&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>congruent</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=conics&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>conics</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=connectivity&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>connectivity</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=consider&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>consider</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=construction&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>construction</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=contains&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>contains</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=corresponding&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>corresponding</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=cube&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>cube</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=cylinder&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>cylinder</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=defined&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>defined</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=Desargues&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>Desargues</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=discontinuous+groups&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>discontinuous groups</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=distance&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>distance</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=edges&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>edges</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=ellipse&source=gbs_word_cloud_r&cad=4" class="cloud0"><span dir=ltr>ellipse</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=ellipsoid&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>ellipsoid</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=elliptic+geometry&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>elliptic geometry</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=equal&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>equal</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=equivalent&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>equivalent</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=Euclidean+plane&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>Euclidean plane</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=example&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>example</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=F%E2%82%81&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>F₁</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=faces&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>faces</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=figure&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>figure</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=finite&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>finite</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=fixed+point&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>fixed point</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=focal&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>focal</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=follows&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>follows</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=four&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>four</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=Gaussian+curvature&source=gbs_word_cloud_r&cad=4" class="cloud0"><span dir=ltr>Gaussian curvature</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=geodesic&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>geodesic</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=geodesic+lines&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>geodesic lines</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=given&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>given</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=groups+of+motions&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>groups of motions</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=Hence&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>Hence</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=heptahedron&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>heptahedron</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=hexagon&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>hexagon</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=hyperbolic+plane&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>hyperbolic plane</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=hyperboloid&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>hyperboloid</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=incidence&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>incidence</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=lattice+points&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>lattice points</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=mean+curvature&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>mean curvature</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=minimal+surface&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>minimal surface</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=M%C3%B6bius+strip&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>Möbius strip</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=move&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>move</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=obtained&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>obtained</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=octahedron&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>octahedron</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=parabolic+points&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>parabolic points</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=parallel&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>parallel</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=pass&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>pass</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=perpendicular&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>perpendicular</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=point+of+intersection&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>point of intersection</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=pointers&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>pointers</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=polygon&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>polygon</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=polyhedron&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>polyhedron</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=position&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>position</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=projective+plane&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>projective plane</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=proved&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>proved</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=quadrics&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>quadrics</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=radius&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>radius</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=region&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>region</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=Reye%27s+configuration&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>Reye's configuration</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=rigid+motions&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>rigid motions</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=ruled+surfaces&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>ruled surfaces</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=segment&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>segment</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=sides&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>sides</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=sphere&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>sphere</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=spherical+image&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>spherical image</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=square&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>square</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=straight+lines&source=gbs_word_cloud_r&cad=4" class="cloud0"><span dir=ltr>straight lines</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=surface+of+revolution&source=gbs_word_cloud_r&cad=4" class="cloud1"><span dir=ltr>surface of revolution</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=symmetry&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>symmetry</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=systems+of+points&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>systems of points</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=tangent+plane&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>tangent plane</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=tetrahedron&source=gbs_word_cloud_r&cad=4" class="cloud9"><span dir=ltr>tetrahedron</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=theorem&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>theorem</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=three-dimensional&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>three-dimensional</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=tion&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>tion</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=topological&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>topological</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=torus&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>torus</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=translation&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>translation</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=triangles&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>triangles</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=umbilical+points&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>umbilical points</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=unit+cell&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>unit cell</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=vertex&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>vertex</span></a> <a href="https://books.google.com.sg/books?id=7WY5AAAAQBAJ&q=vertices&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>vertices</span></a></div></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="about_author_anchor"></a>About the author <span style="color:#666; font-weight:normal">(1999)</span></h3><div id=about_author class=about_content><div id=about_author_v><div class=textmodulecontent>Born in Konigsberg, Germany, David Hilbert was professor of mathematics at Gottingen from 1895 to1930. Hilbert was among the earliest adherents of Cantor's new transfinite set theory. Despite the controversy that arose over the subject, Hilbert maintained that "no one shall drive us from this paradise (of the infinite)" (Hilbert, "Uber das Unendliche," Mathematische Annalen [1926]). It has been said that Hilbert was the last of the great universalist mathematicians and that he was knowledgeable in every area of mathematics, making important contributions to all of them (the same has been said of Poincare). Hilbert's publications include impressive works on algebra and number theory (by applying methods of analysis he was able to solve the famous "Waring's Problem"). Hilbert also made many contributions to analysis, especially the theory of functions and integral equations, as well as mathematical physics, logic, and the foundations of mathematics. His work of 1899, Grundlagen der Geometrie, brought Hilbert's name to international prominence, because it was based on an entirely new understanding of the nature of axioms. Hilbert adopted a formalist view and stressed the significance of determining the consistency and independence of the axioms in question. In 1900 he again captured the imagination of an international audience with his famous "23 unsolved problems" of mathematics, many of which became major areas of intensive research in this century. Some of the problems remain unresolved to this day. At the end of his career, Hilbert became engrossed in the problem of providing a logically satisfactory foundation for all of mathematics. As a result, he developed a comprehensive program to establish the consistency of axiomatized systems in terms of a metamathematical proof theory. In 1925, Hilbert became ill with pernicious anemia---then an incurable disease. However, because Minot had just discovered a treatment, Hilbert lived for another 18 years. </div></div></div></div><div class=vertical_module_list_row><h3 class="about_title">Bibliographic information</h3><div class="about_content" id="metadata_content" style="padding-bottom:.3em"><div class="metadata_sectionwrap"><table id="metadata_content_table"><tr class="metadata_row"><td class="metadata_label">Title</td><td class="metadata_value"><span dir=ltr>Geometry and the Imagination</span><br><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=bibliogroup:%22AMS+Chelsea+Publishing+Series%22&source=gbs_metadata_r&cad=6"><i><span dir=ltr>Issue 87 of AMS Chelsea Publishing Series</span></i></a></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Authors</span></td><td class="metadata_value"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22David+Hilbert%22&source=gbs_metadata_r&cad=6"><span dir=ltr>David Hilbert</span></a>, <a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22Stephan+Cohn-Vossen%22&source=gbs_metadata_r&cad=6"><span dir=ltr>Stephan Cohn-Vossen</span></a></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Edition</span></td><td class="metadata_value"><span dir=ltr>illustrated, reprint</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Publisher</span></td><td class="metadata_value"><span dir=ltr>American Mathematical Soc., 1999</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>ISBN</span></td><td class="metadata_value"><span dir=ltr>0821819984, 9780821819982</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Length</span></td><td class="metadata_value"><span dir=ltr>357 pages</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Subjects</span></td><td class="metadata_value"><div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics%22" itemprop="url" dir=ltr><span itemprop="title">Mathematics</span></a></div> › <div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+Geometry%22" itemprop="url" dir=ltr><span itemprop="title">Geometry</span></a></div> › <div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+Geometry+General%22" itemprop="url" dir=ltr><span itemprop="title">General</span></a></div><br><br><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+/+Geometry+/+General%22&source=gbs_metadata_r&cad=6"><span dir=ltr>Mathematics / Geometry / General</span></a><br/><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+/+Geometry+/+Non-Euclidean%22&source=gbs_metadata_r&cad=6"><span dir=ltr>Mathematics / Geometry / Non-Euclidean</span></a><br/><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+/+History+%26+Philosophy%22&source=gbs_metadata_r&cad=6"><span dir=ltr>Mathematics / History & Philosophy</span></a></td></tr><tr class="metadata_row"><td>&nbsp</td><td>&nbsp</td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Export Citation</span></td><td class="metadata_value"><a class="gb-button " href="https://books.google.com.sg/books/download/Geometry_and_the_Imagination.bibtex?id=7WY5AAAAQBAJ&output=bibtex"><span dir=ltr>BiBTeX</span></a> <a class="gb-button " href="https://books.google.com.sg/books/download/Geometry_and_the_Imagination.enw?id=7WY5AAAAQBAJ&output=enw"><span dir=ltr>EndNote</span></a> <a class="gb-button " href="https://books.google.com.sg/books/download/Geometry_and_the_Imagination.ris?id=7WY5AAAAQBAJ&output=ris"><span dir=ltr>RefMan</span></a></td></tr></table></div><div style="clear:both"></div></div></div><script>_OC_addFlags({Host:"https://books.google.com.sg/", IsBooksUnifiedLeftNavEnabled:1, IsZipitFolderCollectionEnabled:1, IsBrowsingHistoryEnabled:1, IsBooksRentalEnabled:1});_OC_Run({"is_cobrand":false,"sign_in_url":"https://www.google.com/accounts/Login?service=print\u0026continue=https://books.google.com.sg/books%3Fid%3D7WY5AAAAQBAJ%26redir_esc%3Dy%26hl%3Den\u0026hl=en"}, {"volume_id":"7WY5AAAAQBAJ","is_ebook":false,"volumeresult":{"has_flowing_text":false,"has_scanned_text":true,"can_download_pdf":false,"can_download_epub":false,"is_pdf_drm_enabled":false,"is_epub_drm_enabled":false},"sample_url":"https://play.google.com/books/reader?id=7WY5AAAAQBAJ\u0026source=gbs_atb_hover","is_browsable":true,"is_public_domain":false}, {});</script><div id="footer_table" 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