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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_viewport_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/v2_a33f2a89320471e58c940b9287b9d4eb__en.js"></script><script>_OC_Hooks = ["_OC_Page", "_OC_SearchReload", "_OC_TocReload", "_OC_EmptyFunc", "_OC_SearchPage", "_OC_QuotePage" ];for (var _OC_i = 0; _OC_i < _OC_Hooks.length; _OC_i++) {eval("var " + _OC_Hooks[_OC_i] + ";");}function _OC_InitHooks () {for (var i = 0; i < _OC_Hooks.length; i++) {var func = arguments[i];eval( _OC_Hooks[i] + " = func;");}}</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. 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permission of \u003ca class=link_aux href=\"https://books.google.com.sg/url?id=7WY5AAAAQBAJ\u0026pg=PA82\u0026q=http://www.ams.org/bookstore\u0026clientid=ca-print-ams\u0026linkid=1\u0026usg=AOvVaw2LfLlqm7r2Dte65RB-fyO2\u0026source=gbs_pub_info_r\"\u003eAmerican Mathematical Soc.\u003c/a\u003e","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},"publisher":"American Mathematical Soc.","publication_date":"1999","subject":"Mathematics","num_pages":357,"sample_url":"https://play.google.com/books/reader?id=7WY5AAAAQBAJ\u0026source=gbs_vpt_hover","synposis":"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.","my_library_url":"https://www.google.com/accounts/Login?service=print\u0026continue=https://books.google.com.sg/books%3Fop%3Dlibrary\u0026hl=en","is_magazine":false,"is_public_domain":false,"last_page":{"pid":"PA359","order":369,"title":"359"}},{"enableUserFeedbackUI":true,"pseudocontinuous":true,"is_cobrand":false,"sign_in_url":"https://www.google.com/accounts/Login?service=print\u0026continue=https://books.google.com.sg/books%3Fid%3D7WY5AAAAQBAJ%26q%3Dtranslation%26source%3Dgbs_word_cloud_r%26hl%3Den\u0026hl=en","isEntityPageViewport":false,"showViewportOnboarding":false,"showViewportPlainTextOnboarding":false},{"page":[{"pid":"PA82","flags":8,"order":92,"vq":"translation"}]},null,{"number_of_results":44,"search_results":[{"page_id":"PR1","page_number":"i","snippet_text":"... Vossen. GEOMETRY AND THE IMAGINATION D. HILBERT AND S. COHN - VOSSEN \u003cb\u003eTRANSLATED\u003c/b\u003e BY P. 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In this case the re- peated application of the same \u003cb\u003etranslation\u003c/b\u003e in both directions gives rise to\u0026nbsp;..."},{"page_id":"PA48","page_number":"48","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e from the parallelepiped ABCDEFGH of Figs . 48 and 49. ( These figures illustrate the fact mentioned before , that the same point lattice can be con- structed from various , and quite different , unit cells . ) From Fig . 49\u0026nbsp;..."},{"page_id":"PA49","page_number":"49","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e in the direction of the diagonal AH of the cube through a distance equal to one fourth of its length ( Fig . 50 ) . We FIG . 51 shall see that the points of L together with those of K make up the centers of the desired\u0026nbsp;..."},{"page_id":"PA50","page_number":"50","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e that transforms K into L , the points E , F , G , H ( in Fig . 49 ) and the spheres with centers at these points are moved completely out of the cube , while the spheres with centers at A , B , C , D are moved into its\u0026nbsp;..."},{"page_id":"PA60","page_number":"60","snippet_text":"... \u003cb\u003etranslations\u003c/b\u003e , in which every point of the plane moves through the same distance in the same direction and ... \u003cb\u003etranslation\u003c/b\u003e or one single rotation , a fact that very considerably simplifies the study of plane motions . In order\u0026nbsp;..."},{"page_id":"PA61","page_number":"61","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e also moves B to the same image point B \u0026#39; . This leaves us with only the last case to consider , in which ... \u003cb\u003etranslations\u003c/b\u003e as rota- tions through the angle zero about an infinitely distant point. § 10. PLANE Motions and\u0026nbsp;..."},{"page_id":"PA62","page_number":"62","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e . If we adopt this point of view , we may regard any rigid motion of the plane as a rotation through some definite angle , which is zero in the case of \u003cb\u003etranslations\u003c/b\u003e . Thus it must always be possible to represent the\u0026nbsp;..."},{"page_id":"PA63","page_number":"63","snippet_text":"... \u003cb\u003etranslations\u003c/b\u003e it has , we can also characterize it by a simple geometrical figure known as the unit cell . By ... \u003cb\u003etranslation\u003c/b\u003e t \u0026#39; whose magnitude is the same as that of t and whose direction forms an angle a with the direction of\u0026nbsp;..."},{"page_id":"PA64","page_number":"64","snippet_text":"... \u003cb\u003etranslations\u003c/b\u003e at all can not contain rotations through any angle except л . For , if there were other rotations in the group , the presence of any \u003cb\u003etranslation\u003c/b\u003e would entail the presence of another \u003cb\u003etranslation\u003c/b\u003e in a direction not parallel\u0026nbsp;..."},{"page_id":"PA65","page_number":"65","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e in the group that moves A , to A ; this would move A \u0026#39; to a point A \u0026quot; whose distance from A is less than ... \u003cb\u003etranslations\u003c/b\u003e a and a - 1 applied once or several times . Thus all the groups of type I , 1 are essentially\u0026nbsp;..."},{"page_id":"PA66","page_number":"66","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e a is applied to the whole unit cell , the result is a congruent adjacent strip of the plane . In this way the whole plane can be covered simply by unit cells of the group . The same property is shared by the unit cells of\u0026nbsp;..."},{"page_id":"PA67","page_number":"67","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e , contrary to the assumption that there are no \u003cb\u003etranslations\u003c/b\u003e in the group . 27 A∞ 2 FIG . 67 Let A be the ( one and only ) center of rota- tion of the group and let Q be any other point . Then all the points equivalent to Q\u0026nbsp;..."},{"page_id":"PA68","page_number":"68","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e t in the group and where all additional transla- tions are parallel to t . By the second lemma of page 63 , the angle of d must be л , i.e. , in the notation introduced above , there can only be 2 - fold centers of rotation\u0026nbsp;..."},{"page_id":"PA69","page_number":"69","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e of the group transforms each of the two rows into itself . Fig . 69 also shows some mutually equivalent points other than the centers of rotation . They are arranged α in zig - zag order . Since they are con- tained in a\u0026nbsp;..."},{"page_id":"PA70","page_number":"70","snippet_text":"... \u003cb\u003etranslations\u003c/b\u003e but no rotations . We shall see that in this case the points equivalent to any given point always form a plane lattice . To prove this , we start with any point P and choose a \u003cb\u003etranslation\u003c/b\u003e t of the group that moves P to an\u0026nbsp;..."},{"page_id":"PA71","page_number":"71","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e P \u0026#39; → U , because this is the resultant of the \u003cb\u003etranslation\u003c/b\u003e P \u0026#39; → P , which is contained in the group , and u . But this leads to a contradiction , as follows : We have seen that PR PQ ; consequently the vertex R \u0026#39; is the\u0026nbsp;..."},{"page_id":"PA72","page_number":"72","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e t \u0026#39; that moves A to B \u0026#39; . The motion t - 1t \u0026#39; evidently transforms B into B \u0026#39; . By the additivity theorem for angles of rotation , t - 1t \u0026#39; is a \u003cb\u003etranslation\u003c/b\u003e , and since the magnitude of the \u003cb\u003etranslation\u003c/b\u003e t was chosen to be as\u0026nbsp;..."},{"page_id":"PA73","page_number":"73","snippet_text":"... \u003cb\u003etranslations\u003c/b\u003e contained in the group trans- forms A into the points of a latice ; all these points , then , are equiva- lent to A and are 2 - fold centers of rotation . Let ABCD be one of the generating parallelograms of the lattice\u0026nbsp;..."},{"page_id":"PA74","page_number":"74","snippet_text":"... \u003cb\u003etranslations\u003c/b\u003e must be of. only other angle that might be admissible ; but by the additivity theorem , a rotation ... \u003cb\u003etranslation\u003c/b\u003e A → C also belongs to the group . The lattice corresponding to the subgroup of \u003cb\u003etranslations\u003c/b\u003e can be\u0026nbsp;..."},{"page_id":"PA75","page_number":"75","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e t defined by AB , transforms AB into BD ( Fig . 78 ) . Thus dt must be a rotation d \u0026#39; through the angle ... \u003cb\u003etranslation\u003c/b\u003e , and the ratio of the magnitude of the \u003cb\u003etranslation\u003c/b\u003e to the distance EF equals the ratio of the\u0026nbsp;..."},{"page_id":"PA76","page_number":"76","snippet_text":"... \u003cb\u003etranslations\u003c/b\u003e that generate the lattice of \u003cb\u003etranslations\u003c/b\u003e . A system of equivalent pointers which are not attached to centers FIG . 80 FIG . 81 of rotation ... \u003cb\u003etranslation\u003c/b\u003e of the group having as small a. 76 II . REGULAr Systems of Points."},{"page_id":"PA77","page_number":"77","snippet_text":"... \u003cb\u003etranslations\u003c/b\u003e of the group can be generated from the square ABCD , since the four vertices , but no other points of this square , are lattice points . As in the previous case , we note that the \u003cb\u003etranslation\u003c/b\u003e lattice can not be chosen\u0026nbsp;..."},{"page_id":"PA78","page_number":"78","snippet_text":"... \u003cb\u003etranslations\u003c/b\u003e and rotations through 2л / 3 . The structure of this subgroup is known to us from II , 2 , ß ; we know also that A is a 3 - fold center of rotation for this subgroup . The lattice of the \u003cb\u003etranslations\u003c/b\u003e of this sub- group is\u0026nbsp;..."},{"page_id":"PA79","page_number":"79","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e in the group , the distance between two 6 - fold centers of rotation cannot be less than AC , so that ... \u003cb\u003etranslations\u003c/b\u003e and the rotations through л . From the dis- FIG . 86 cussion of the case II , 2 , a we know that\u0026nbsp;..."},{"page_id":"PA82","page_number":"82","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e . Now it can be proved that any given rigid motion in space is the resultant of a uniquely defined rotation and a uniquely defined \u003cb\u003etranslation\u003c/b\u003e along the axis of the rotation ; the rotations and trans- lations themselves may\u0026nbsp;..."},{"page_id":"PA83","page_number":"83","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e . We have set up a correspondence in which every motion of the group G that is not a \u003cb\u003etranslation\u003c/b\u003e is associated with a motion that leaves M fixed . We shall make the correspondence complete by letting the identity correspond\u0026nbsp;..."},{"page_id":"PA84","page_number":"84","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e lattices . We shall omit the proof here . According to the above result , we can find all the crystallographic classes of groups of space motions by examining only the discon- tinuous groups of motions on the sphere . There\u0026nbsp;..."},{"page_id":"PA88","page_number":"88","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e . A summary of all the classes and groups obtained in this way is given in the following table . Proper ... \u003cb\u003etranslations\u003c/b\u003e . § 14. The Regular Polyhedra The construction of the crystallographic. 88 II . REGULar Systems OF\u0026nbsp;..."},{"page_id":"PA132","page_number":"132","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e in prep . ( Chelsea Publish- ing Company ) . ) Here ON || AC as we have just proved , and MAOB by assumption . It follows by Pappus \u0026#39; theorem that NM | CB . Finally , we consider the Pascal hexagon ON MLC\u0026#39;B \u0026#39; . In this\u0026nbsp;..."},{"page_id":"PA231","page_number":"231","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e along the axis . The rigid motions of this type form a two - parameter family , and there are no further rigid motions that will bring the circular cylinder into self - coincidence . By a motion of this kind we can map any\u0026nbsp;..."},{"page_id":"PA239","page_number":"239","snippet_text":"... . D. Hilbert , Grundlagen der Geometrie ( 7th ed .: Berlin , 1930 ) . ( English \u003cb\u003etranslation\u003c/b\u003e in prep . , New York , Chelsea Publishing Company . ) IV . AXIOM OF PARALLELS Through any point not lying. § 34. ELLIPTIC GEOMETRY 239."},{"page_id":"PA259","page_number":"259","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e group , the sides come in pairs that are equal in length and equivalent ; in Fig . 249 this divi- sion into pairs is indi- cated for one of the unit cells . The vertices of all the unit cells drawn in the figure have been\u0026nbsp;..."},{"page_id":"PA275","page_number":"275","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e . If the \u003cb\u003etranslations\u003c/b\u003e. B g \u0026quot; FIG . 256 ent shapes , and each of these surfaces is free to rotate about the axis g ; hence every point on the rod g \u0026#39; of our linkage has two degrees of free- dom.1 We have seen above that the\u0026nbsp;..."},{"page_id":"PA276","page_number":"276","snippet_text":"David Hilbert, Stephan Cohn-Vossen. rotation or by a single \u003cb\u003etranslation\u003c/b\u003e . If the \u003cb\u003etranslations\u003c/b\u003e are treated as rotations about an ideal point , we may say that every rigid motion of the plane , without exception , can be replaced by a\u0026nbsp;..."},{"page_id":"PA327","page_number":"327","snippet_text":"... like the unit cells of the crystallographic group of plane \u003cb\u003etranslations\u003c/b\u003e ( see Fig . 72 , p . 70 ) , and each rectangle cor- responds to one layer of the covering . We shall. § 49. TOPOLOGICAL MAPPINGS OF A SURFACE ONTO ITSELF 327."},{"page_id":"PA328","page_number":"328","snippet_text":"... \u003cb\u003etranslations\u003c/b\u003e that move the square lattice into itself . Let g be any other topological mapping of U onto itself that , although it need not map every point into a point equivalent to it , maps equivalent pairs of points into equivalent\u0026nbsp;..."},{"page_id":"PA329","page_number":"329","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e group ( t ) in E that corresponds to ( f ) . Let A\u0026#39;B\u0026#39;C\u0026#39;D \u0026#39; be the images of ABCD under the mapping y ; then the parallelogram A\u0026#39;B\u0026#39;C\u0026#39;D \u0026#39; must be a unit cell of ( t ) . Now the mapping h of the torus is a deformation if and\u0026nbsp;..."},{"page_id":"PA330","page_number":"330","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e group in the Euclidean plane : all the mappings in ( t ) , with the exception of the identity , are free of fixed points , and the unit cell of the group has four sides . 330 VI . TOPOLOGY Conformal Mapping of the Torus."},{"page_id":"PA331","page_number":"331","snippet_text":"... \u003cb\u003etranslations\u003c/b\u003e . Since , in addition , ( t ) is dis- continuous and has a finite unit cell , ( † ) must be a crystallographic \u003cb\u003etranslation\u003c/b\u003e group of the type discussed on pages 70-71 . Now let the same argument be carried through for any\u0026nbsp;..."},{"page_id":"PA333","page_number":"333","snippet_text":"... \u003cb\u003etranslation\u003c/b\u003e groups , and two such surfaces can be mapped con- formally onto each other only if their \u003cb\u003etranslation\u003c/b\u003e groups can be transformed into each other by a hyperbolic rigid motion . It is found in hyperbolic geometry that 6p - 6\u0026nbsp;..."},{"page_id":"PA349","page_number":"349","snippet_text":"... \u003cb\u003etranslations\u003c/b\u003e , 64-65 , 70-72 , 84 , 258-259 , 327-328 , 330-333 HARMONIC points , 97 , 101 , 129 Heesch , 50 ... \u003cb\u003etranslation\u003c/b\u003e in , 258-259 , 329 , 330-333 motion of , 245 , 257-259 , 265 , 267- 268 , 330-331 Hyperbolic space , 247\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=7WY5AAAAQBAJ\u0026pg=PA349\u0026vq=translation"},{"page_id":"PA357","page_number":"357","snippet_text":"... \u003cb\u003eTranslation\u003c/b\u003e , in hyperbolic plane , 258- 259 , 329 , 330-333 \u003cb\u003eTranslations\u003c/b\u003e , 60-62 , 82 groups of , 64-65 , 70-72 , 84 , 258-259 , 327-328 , 330-333 Triangle , exterior angle of , 246 in elliptic plane , 241 isosceles , base angles of\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=7WY5AAAAQBAJ\u0026pg=PA357\u0026vq=translation"}],"search_query_escaped":"translation"},{});</script></div></div></div><script>(function() {var href = window.location.href;if (href.indexOf('?') !== -1) {var parameters = href.split('?')[1].split('&');for (var i = 0; i < parameters.length; i++) {var param = parameters[i].split('=');if (param[0] == 'focus') {var elem = document.getElementById(param[1]);if (elem) {elem.focus();}}}}})();</script>