Discrete Groups and Visualization of Three-Dimensional Manifolds Charlie Gunn The Geometry Center, The University of Minnesota 1.1 The circle and the line x) Abstract When we evaluate the expression sin (2 we are only interested in ( x) = ( (x + x mod 1, since sin is a periodic function: sin 2 sin 2 k )) We describe a software implementation for interactive visualization , where k is an integer. The set of all motions of the real line of a wide class of discrete groups. In addition to familiar Euclidean R by integer amounts forms a group , which leaves invariant the x) R= space, these groups act on the curved geometries of hyperbolic function sin (2 . We can form the quotient , which is the set and spherical space. We construct easily computable models of of equivalence classes with respect to this group. This quotient can 0; 1] our geometric spaces based on projective geometry; and establish be represented by the closed interval [ , with the understanding algorithms for visualization of three-dimensional manifolds based that we identify the two endpoints. But identifying the two endpoints x) upon the close connection between discrete groups and manifolds. yields a circle. Once we know the values of sin (2 on the circle, We describe an object-oriented implementation of these concepts, we can compute it for any other value y , simply by subtracting or [0; 1) and several novel visualization applications. As a visualization adding integers to y until the result lies in the range . tool, this software breaks new ground in two directions: interactive In this example the discrete group is the set of transformations x ! x + k k exploration of curved spaces, and of topological manifolds modeled of R given by all translations , where is an integer. on these spaces. It establishes a generalization of the application of is discrete since no non-trivial sequence in converges to the the 1 S projective geometry to computer graphics, and lays the groundwork identity element. The quotient of R under this action is , the unit 1 = S for visualization of spaces of non-constant curvature. circle. We write R= . CR Categories and Subject Descriptors : I.3.3 [Picture/Image Generation] display algorithms I.3.5 [Computational Geometry and 1 Object Modeling Graphics]: geometric algorithms, hierarchy and S geometric transformations, I.3.7 [Three dimensional Graphics and a 1 Realism] color, shading, shadowing, and texture R Additional Key Words and Phrases: discrete group, tessel- a-1 a a+1 lation, quotient space, projective geometry, hyperbolic geometry, -1 0 1 spherical geometry, curvature, geodesic. Figure 1: The circle is the quotient of R by the integers. 1 Discrete Groups = [0; 1) I is a fundamental domain for this group action. We Symmetry, broadly speaking, implies a redundant supply of infor- can recover R from the fundamental domain and : the union mation. A mirror image contains the same information as the scene that it mirrors. The theory of discrete groups has been developed [ g I 2 over the past 100 years as a formalization of the process of extract- g ing a single copy of the information present in symmetric con®g- urations. The discrete groups which we study here are groups of covers R without overlap. motions which act on a geometric space, such as Euclidean space, We move into two dimensions to bring out other features of the to produce tessellations by congruent non-overlapping cells. Fa- concepts introduced in this example. miliar examples include wallpaper patterns, and the interlocking designs of M. C. Escher. We consider two simple examples before introducing mathematical de®nitions. 1.2 The torus and the plane 2 R Permission to copy without fee all or part of this material is granted Instead of R we now work with . Let be the group of trans- Permission to copy without fee all or part of this material is granted 2 (x; y ) ! (x + ; y ) (x; y ) ! providedprovided that that thethe copiescopies areare notnot mademade oror distributeddistributed forfor directdirect lations of R generated by 1 and commercial advantage, the ACM copyright notice and the title of the (x; y + commercial advantage, the ACM copyright notice and the title of the 1 ), that is, unit translations in the coordinate directions. publication and its date appear, and notice is given that copying is by 2 = publication and its date appear, and notice is given that copying is by What is the quotient R ? Instead of the unit interval with its permissionpermission of of thethe AssociationAssociation forfor ComputingComputing Machinery.Machinery. ToTo copycopy otherwise, or to republish, requires a fee and/or specific permission. endpoints identi®ed, we are led to a unit square that has its edges otherwise, or to republish, requires a fee and/or specific permission. identi®ed in pairs. If we imagine the square is made of rubber and ©199©19933 ACMACM--00--8979189791--601601--8/93/0088/93/008/0015…$1.50…$1.50 that we can perform the identi®cations by bending the square and 2 Current address: SFB 288, MA 8-5, Technische Universitat,Strasse des gluing, we ®nd that the resulting surface is the torus T . See Figure 17 Juni 136, 1 Berlin 12, Germany, [email protected] 2. 1 255 2 2 R P l P P l P of T sees, we will make them in : Light follows geodesics, which appear to be very complicated on the rolled-up torus, but 2 m m in R are just ordinary straight lines. A complicated closed path m m based at P which wraps around the torus several times unrolls in the hP universal cover to be an ordinary straight line connecting P and 2 P l P for some h . See Figure 3. An immediate consequenceof this is that an observer on the torus based at P sees many copies of himself, one for every closed geodesic on the surface passing through P . For example, if he looks to the left he sees his right shoulder; if he looks P straight ahead he sees his back. See [Wee85] for a complete and elementary description of this phenomenon. We say the rolled-up l torus represents the outsider's view; while the unrolled view we m term the insider's view, since it shows what someone living inside the space would see. The importance of the insider's view becomes more telling in three dimensional spaces, since to ªroll upº our Figure 2: Making a torus from a square fundamental domains requires four or more dimensions. In this case the insider's view becomes a practical necessity. 1.3 Algebra and geometry: the fundamental group A key element of this approach is the interplay of algebraic and (-2,2) (-1,2) (0,2) (1,2) geometric viewpoints. To clarify this, we introduce the fundamental P (-2,1) (-1,1) (0,1) (1,1) (2,1) group of a space, formed by taking all the closed paths based at some (0,0) (-2,0) (-1,0) m (1,0) (2,0) point P in the space. We get a group structure on this set: we can l l add paths by following one and then the other, and subtract by going m (-2,-1) (-1,-1) (0,-1) (1,-1) (2,-1) around the second path in the reverse order. The zero-length path is the identity element. If one path can be moved or deformed to another path, the two paths correspond to the same group element. Figure 3: Outside and inside views of a complicated torus It is easy to check that different P 's yield isomorphic groups. We path say a space is simply connected if every closed path can be smoothly shrunk to a point, like a lasso, without leaving the space. [Mun75] When we try to perform the analogous construction for the two- The fundamental group of a simply connected space consists of just 2 holed torus, instead of a square in the Euclidean plane R , we are the identity element. 2 led to a regular octagon in the hyperbolic plane H [FRC92]. We 2 2 T In the above example R is simply connected; while , the describe hyperbolic geometry in more detail below. quotient, isn't. When X is the quotient of a simply connected Y X space Y , we say that is the universal covering space of . The importance of simply connected spaces in the study of discrete 1.5 De®nition of discrete group G groups is due to a basic result of topology that (subject to technical A discrete group is a subgroup of a continuous group such that G U \ = I constraints which we will consider satis®ed) every space has a there is a neighborhood U of the identity in with , the unique universal covering space [Mun75]. So in considering group identity element. 2 R actions, we need only consider actions on simply connected spaces. In the example of the torus above, the group acts on . Such The interplay of algebra and geometry reveals itself in the fact an action on a topological space X is called properly discontinuous X 2 that the fundamental group of the quotient, a purely topological if for every closed and bounded subset K of , the set of \ K 6= object, is isomorphic to the group of symmetries , which arises in such that K is ®nite. In the cases to be discussed here, a purely geometric context. is discrete if and only if the action of is properly discontinuous. If in addition the quotient space X= is compact, we say that 1.4 Inside versus Outside Views is a crystallographic, or crystal, group.