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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/_ff77d0f0508c7e0bad470e7d6b7f28fa/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_ff77d0f0508c7e0bad470e7d6b7f28fa__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=PA89\u0026q=http://www.ams.org/bookstore\u0026clientid=ca-print-ams\u0026linkid=1\u0026usg=AOvVaw3G_KtOifhl5GdEEKZKBLOv\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%3Dfaces%26source%3Dgbs_word_cloud_r%26hl%3Den\u0026hl=en","isEntityPageViewport":false,"showViewportOnboarding":false,"showViewportPlainTextOnboarding":false},{"page":[{"pid":"PA89","flags":8,"order":99,"vq":"faces"}]},null,{"number_of_results":55,"search_results":[{"page_id":"PR9","page_number":"ix","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e . Plateau\u0026#39;s Problem ..268 CHAPTER V KINEMATICS § 40. Linkages .... § 41. 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The egg is an approximation to\u0026nbsp;..."},{"page_id":"PA11","page_number":"11","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e of revolution . According to whether we rotate a hyperbola about the line connecting the foci or about the ... \u003cb\u003eface\u003c/b\u003e can also be generated by rotating a straight line about a skew axis . ( So far we have have only considered\u0026nbsp;..."},{"page_id":"PA14","page_number":"14","snippet_text":"... \u003cb\u003eface\u003c/b\u003e of revolution , the general hyperboloid of one sheet con- tains two families of straight lines , since a dilatation always transforms straight lines into straight lines . Again the lines are arranged in such a way that every line\u0026nbsp;..."},{"page_id":"PA22","page_number":"22","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e leads to \u0026quot; elliptic coordinates , \u0026quot; which have proved very effec- tive in the treatment of numerous problems , among them problems in astronomy . We can get an over- all picture of the way in which a system of confocal quadrics is\u0026nbsp;..."},{"page_id":"PA46","page_number":"46","snippet_text":"... . The first case is realized in the hexagonal crystals of the type of magnesium , the second in the \u003cb\u003eface\u003c/b\u003e - centered cubic crystals . Cf § 8 . is regular throughout space , for in passing from one. 46 II . REGULar Systems OF POINTS."},{"page_id":"PA48","page_number":"48","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e of the cubes . The resulting set of points also forms a lattice ( the \u003cb\u003eface\u003c/b\u003e - centered cubic lattice ) , for it can be generated by parallel translation from the parallelepiped ABCDEFGH of Figs . 48 and 49. ( These figures\u0026nbsp;..."},{"page_id":"PA50","page_number":"50","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e . To construct this packing , we begin with a sphere K of the tetra- hedral packing and inscribe four smaller spheres k1 to k1 , of equal * The locus of the centers in this packing is not a lattice ; it does not , for example\u0026nbsp;..."},{"page_id":"PA56","page_number":"56","snippet_text":"... \u003cb\u003eface\u003c/b\u003e - centered cubic \u0026quot; lattice , that corresponds to that closest packing of spheres in which the relative positions of two consecutive layers are always the same ( see Figs . 47b and 47c , p . 46 ) . The other form of closest packing\u0026nbsp;..."},{"page_id":"PA85","page_number":"85","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e . Thus the tetra- hedron gives rise to the class 10. T ( Fig . 93 ) . FIG . 93 The line connecting the center of the sphere with a vertex of the tetrahedron also passes through the center of the opposite \u003cb\u003eface\u003c/b\u003e . Since the opposite \u003cb\u003eface\u003c/b\u003e\u0026nbsp;..."},{"page_id":"PA86","page_number":"86","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e meet at every vertex , so that we have three 4 - fold axes . Similarly the eight \u003cb\u003efaces\u003c/b\u003e of the octahedron are opposite in pairs , and are equilateral triangles , so that they give rise to four 3 - fold axes . Finally , the\u0026nbsp;..."},{"page_id":"PA89","page_number":"89","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e . Furthermore , we require that all the \u003cb\u003efaces\u003c/b\u003e be regular polygons . First of all , a polyhedron satisfying these conditions cannot have any re - entrant vertices or edges . For , it is clear that the vertices cannot all be re\u0026nbsp;..."},{"page_id":"PA90","page_number":"90","snippet_text":"... \u003cb\u003eFaces\u003c/b\u003e Vertices Edges \u003cb\u003eFaces Faces\u003c/b\u003e Meeting at a Vertex Tetrahedron ( Fig . 95 ) . Octahedron ( Fig . 96 ) Icosahedron ( Fig . 97 ) · · · Triangles 4 66 6 12 66 12 30 20 Cube ( Hexahedron ) ( Fig . 98 ) Dodecahedron ( Fig . 99 ) · Squares\u0026nbsp;..."},{"page_id":"PA92","page_number":"92","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e of the cube . inscribed in the cube in this way ( see Fig . 102 ) . Similarly , it turns out that the octahedral group is a subgroup of the icosahedral group . This is the reason why a cube can be inscribed in a dodecahedron in\u0026nbsp;..."},{"page_id":"PA93","page_number":"93","snippet_text":"... \u003cb\u003eface\u003c/b\u003e of the dodecahedron there is one edge of each of the cubes , and two cubes meet at each vertex of the dodecahedron . CHAPTER III PROJECTIVE CONFIGURATIONS In this chapter we shall learn. §14 . THE REGULAR POLYHEDRA 93."},{"page_id":"PA121","page_number":"121","snippet_text":"... the triangles determine a common - ordinary or ideal- straight line of intersection ( by Axiom 1 for space ) . Of every FIG . 136 pairs of \u003cb\u003efaces\u003c/b\u003e , the \u0026quot; Desargues. § 19. IDEAL ELEMENTS AND the Principle of DUALITY IN SPACE 121."},{"page_id":"PA124","page_number":"124","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e , the \u0026quot; Desargues plane \u0026quot; ( aa \u0026#39; , ßß \u0026#39; , yy \u0026#39; in Fig . 136 ) , and the \u0026quot; Desargues line \u0026quot; ( VW in the figure ) . The intersection of this three - dimensional figure with any plane that does not contain any of the points V , W , X\u0026nbsp;..."},{"page_id":"PA134","page_number":"134","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e and each of the six diagonal planes passing through a pair of opposite edges . In the figure defined in this way , there are six points lying on each plane : four vertices and two ideal points on each of the planes containing a \u003cb\u003eface\u003c/b\u003e\u0026nbsp;..."},{"page_id":"PA139","page_number":"139","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e of the tetrahedron are labelled I , II , III , and IV , where I is the \u003cb\u003eface\u003c/b\u003e oppo- site the point 1. The plane of the external centers of similitude is called ea , the four planes containing three external and three in- ternal\u0026nbsp;..."},{"page_id":"PA141","page_number":"141","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e of the cube in Fig . 145 . Let the radii be equal and of such length that each sphere goes through the four corners of the \u003cb\u003eface\u003c/b\u003e on which its center lies . Let 1 and 2 be any two planes that are respectively perpendicular to the\u0026nbsp;..."},{"page_id":"PA142","page_number":"142","snippet_text":"... \u003cb\u003eface\u003c/b\u003e - planes of the octahedron belong to the configuration , being the \u003cb\u003eface\u003c/b\u003e - planes I , II , III , and IV ... \u003cb\u003efaces\u003c/b\u003e and vertices respectively of the octahedron , the center of the cube and the six planes through it correspond\u0026nbsp;..."},{"page_id":"PA143","page_number":"143","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e ( e.g. the straight line connecting any vertex of a cube with the center meets the cube at a second vertex ) . The tetrahedron , however , is not symmetrical with respect to a point , ( does not have \u0026quot; central symmetry \u0026quot; ) ; the\u0026nbsp;..."},{"page_id":"PA144","page_number":"144","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e . Just as we stipulated for regular polyhedra that the \u003cb\u003efaces\u003c/b\u003e be regular polygons , so we must stipulate for the regular polytopes in four dimensions that the boundary cells be regular polyhedra . The polytope is called an n - cell\u0026nbsp;..."},{"page_id":"PA145","page_number":"145","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e overlap . The disadvantage can be eliminated by moving the center of projection to a point very close to one of the \u003cb\u003efaces\u003c/b\u003e . For the sake of symmetry we move it to a point at a small distance from the center of one of the \u003cb\u003efaces\u003c/b\u003e and\u0026nbsp;..."},{"page_id":"PA147","page_number":"147","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e of the polyhedron . FIG . 158 FIG . 159 TETRAHEDRON Another series of simple figures is obtained from the symmetrical polyhedra by using a \u003cb\u003eface\u003c/b\u003e plane as image plane , as shown in Fig . 164 for the cube . ( For the tetrahedron this\u0026nbsp;..."},{"page_id":"PA152","page_number":"152","snippet_text":"... \u003cb\u003eface\u003c/b\u003e ( e.g. 1 , 3 , 4 ) that is wholly confined to the finite part of space and from which they extend across ... \u003cb\u003efaces\u003c/b\u003e that do not extend across infinitely distant elements . In Fig . 173b , the ideal plane itself is a boundary\u0026nbsp;..."},{"page_id":"PA156","page_number":"156","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e of the octahedra are triangles , while the planes of sym- metry intersect the octa- hedra in quadrangles , it is seen that each plane of the configuration serves Fig . 176 is simpler than Fig . 175 in. FIG . 177 as symmetry plane\u0026nbsp;..."},{"page_id":"PA166","page_number":"166","snippet_text":"... \u003cb\u003eface\u003c/b\u003e of a cube . The arrange- ment should be apparent from Fig . 181 ( cf. also Fig . 102 , p . 93 ) . We must now prove the assertion made above that there is a straight line l \u0026#39; distinct from 6 \u0026#39; which meets 2 , 3 , 4 , and 5 , and\u0026nbsp;..."},{"page_id":"PA198","page_number":"198","snippet_text":"... \u003cb\u003eface\u003c/b\u003e was based on certain mathematical relations , he had all the parabolic curves marked out on the Apollo Belvidere , a statue renowned for the high degree of classical beauty portrayed in its features . But the curves did not possess\u0026nbsp;..."},{"page_id":"PA208","page_number":"208","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e and the space curves , the latter being one - parameter mani- folds of points . Thus , for example , it is possible to introduce a concept for the ruled surfaces that corresponds to the concept of curvature ; this is the so\u0026nbsp;..."},{"page_id":"PA219","page_number":"219","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e satisfying our condition is the class of canal surfaces . A canal surface is defined as the envelope of a ... \u003cb\u003eface\u003c/b\u003e . In the special case where this curve is a straight line , the result is a surface of revolution ; thus the\u0026nbsp;..."},{"page_id":"PA222","page_number":"222","snippet_text":"... \u003cb\u003eface\u003c/b\u003e . This is particularly simple in the case of the surfaces of revolution . On these surfaces all the meridians are geodesics , because their planes contain the axis and therefore cut the surface at right angles . ( We proved earlier\u0026nbsp;..."},{"page_id":"PA227","page_number":"227","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e of constant mean curvature other than the sphere that have neither a boundary nor any other singular points . It is found that the answer is in the negative , so that the sphere is defined uniquely by our additional condition\u0026nbsp;..."},{"page_id":"PA230","page_number":"230","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e . Bending is impossible in the case of all closed convex surfaces , such as , for example , the ellipsoids . It is likewise impossible to bend any convex surface having boundary curves , provided each boundary curve has the\u0026nbsp;..."},{"page_id":"PA233","page_number":"233","snippet_text":"... \u003cb\u003eface\u003c/b\u003e it is therefore sufficient to describe it as a function of arc length along one of the geodesic curves of the second family . But it is easy to construct surfaces of revolution for which the Gaussian curvature is a prescribed\u0026nbsp;..."},{"page_id":"PA237","page_number":"237","snippet_text":"... \u003cb\u003eface\u003c/b\u003e of constant positive curvature other than the sphere can be taken as an analogue of the plane - if only because all the other surfaces of constant positive curvature have boundaries or singular points ( cf. pp . 227-228 ) . However\u0026nbsp;..."},{"page_id":"PA286","page_number":"286","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e always have a line in common along which they are tangential . This contact motion in space is analogous to ... \u003cb\u003eface\u003c/b\u003e . We have seen ( pp . 208- 209 ) that two ruled surfaces can be tangent along a ruling if and only if they\u0026nbsp;..."},{"page_id":"PA290","page_number":"290","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e , so that it can be set on a flat table top with any \u003cb\u003eface\u003c/b\u003e down . Convexity is not a topological property , for we can change a convex polyhedron into one that is not convex by a topologically neutral transformation . But the\u0026nbsp;..."},{"page_id":"PA292","page_number":"292","snippet_text":"... \u003cb\u003eface\u003c/b\u003e of our polyhedron be bounded by r edges . Then there are altogether rF edges , counted as boundary segments of the \u003cb\u003efaces\u003c/b\u003e . But here we have again counted the edges twice , since every edge forms a boundary segment of two \u003cb\u003efaces\u003c/b\u003e\u0026nbsp;..."},{"page_id":"PA295","page_number":"295","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e , and we get the equation 1632 + 163-3 = 0 . It follows that the connectivity of the prismatic block is exactly 3 . Euler\u0026#39;s formula can be used in the same way in the general case as a convenient means of determining the\u0026nbsp;..."},{"page_id":"PA296","page_number":"296","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e of higher connectivity can be constructed by flattening a sphere made of a plastic material and cutting holes into it ( see Fig . 275 ) . We shall call such surfaces pretzels . It can be proved that a pretzel with p holes must\u0026nbsp;..."},{"page_id":"PA297","page_number":"297","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e of this class can be obtained from the closed surfaces of до h = 1 h = 2 h = 3 h = 4 FIG . 277 a FIG . 278 ... \u003cb\u003eface\u003c/b\u003e will depend on whether we imagine it situated in metric or in 1In contrast to the surfaces shown in Fig . 277\u0026nbsp;..."},{"page_id":"PA302","page_number":"302","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e of a regular octahedron and add the three squares in the planes spanned by the diagonals ( e.g. ABCD in Fig . 288 ) . The eleven \u003cb\u003efaces\u003c/b\u003e obtained in this way do not constitute a polyhedron FIG . 289b as defined earlier , for the\u0026nbsp;..."},{"page_id":"PA303","page_number":"303","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e meeting at each edge is three instead of two . We proceed to remove four triangles : from the front half of ... \u003cb\u003efaces\u003c/b\u003e meet at every edge and that we can travel from any \u003cb\u003eface\u003c/b\u003e to any other by crossing edges . Thus the figure is\u0026nbsp;..."},{"page_id":"PA304","page_number":"304","snippet_text":"... \u003cb\u003eface\u003c/b\u003e starting at a fixed point P. Directly opposite P , on the other side of the thin membrane , there is a point P \u0026#39; that coincides with P if the membrane is replaced by the original surface . Now it might easily be thought that the\u0026nbsp;..."},{"page_id":"PA307","page_number":"307","snippet_text":"... \u003cb\u003eface\u003c/b\u003e in space , the latter to the position of a curve on the surface . Unlike the Möbius strip , the heptahedron has lines along which it intersects itself . It would seem to be a reasonable conjecture that every one - sided closed\u0026nbsp;..."},{"page_id":"PA319","page_number":"319","snippet_text":"... \u003cb\u003eface\u003c/b\u003e is equivalent to the projective plane . Fig . 321 shows a model made of wire netting . The curve in which the surface intersects itself consists of three loops which pass through the point N and which , like the whole surface , are\u0026nbsp;..."},{"page_id":"PA323","page_number":"323","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e can be represented in a form free of singularities or self- intersections . We shall use the notation R , for the four - dimensional space and R , for the three - dimensional space . We can think of the R3 as being imbedded in R\u0026nbsp;..."},{"page_id":"PA324","page_number":"324","snippet_text":"... \u003cb\u003eface\u003c/b\u003e has not yet been investigated . The projective plane can be realized in a surface in R , that is given by very simple equations and has no self - intersections or singularities . The derivation is given in an appendix to this\u0026nbsp;..."},{"page_id":"PA333","page_number":"333","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e of the polyhedron , we must first partition the curved surface into regions . The problem of contiguous regions is the problem of finding the greatest number of regions on a given surface which are such that each of them borders\u0026nbsp;..."},{"page_id":"PA336","page_number":"336","snippet_text":"... way that it becomes a poly- hedron and that the individual regions become the \u003cb\u003efaces\u003c/b\u003e of the 2 The problems for the sphere and the plane are essentially the same . 3 polyhedron . Then it is clear that it is. 336 VI . TOPOLOGY."},{"page_id":"PA337","page_number":"337","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e . We shall prove our result by induction on F. For all F≤ n , the result is trivially true , for then we can simply give each \u003cb\u003eface\u003c/b\u003e of the polyhedron a dif- ferent color . Now let us assume that the theorem is already proved for\u0026nbsp;..."},{"page_id":"PA341","page_number":"341","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e . In E4 , on the other hand , there are surfaces that are iso- metric with the Euclidean plane in the small but are not ruled . We shall present such a surface F .. It is confined to a finite part automorphisms of , 107-109 , 111\u0026nbsp;..."},{"page_id":"PA354","page_number":"354","snippet_text":"... \u003cb\u003efaces\u003c/b\u003e , 17-18 normal , of surface , 183-186 , 315 Self - dual configurations , 118 , 123-124 , 128 , 142-143 , 169 Self - intersection , lines of , 303 , 307-308 , 314-315 , 317 , 319 , 320 , 323 Series , Leibniz \u0026#39; , 37-39 Shortest line\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=7WY5AAAAQBAJ\u0026pg=PA354\u0026vq=faces"}],"search_query_escaped":"faces"},{});</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>