Vladimir Sedach / QuasiSculpt
QuasiSculpt: Four-dimensional visualization, quasicrystals, and self-assembling systems Vladimir Sedach
Abstract The first aim of this paper is to describe a potential method for visualizing and interacting with 4-dimensional objects by considering how artists currently use computers to work with objects in three dimensions. The second aim is to apply these insights to examine the generation of quasicrystal tilings from projection of regular lattices in 4-space, and to consider the applicability of using quasicrystal tiles in the construction of 2-dimensional artificial self-assembling systems.
1. Introduction 1.1. The fourth dimension The problem of visualizing four dimensional space goes back to before the turn of the 20th century. Although well studied, the existing solutions could be improved upon. The rising importance of four-dimensional visualization of medical data creates a practical impetus to consider the problem from new perspectives. Artists and graphic designers play an important role in information visualization and presentation - getting them involved in the process of four-dimensional visualization could potentially yield great benefits in the clarity of the presented visualizations. In addition, these techniques can potentially go beyond the visualization of preexisting information and yield a new expressive artistic medium. The obvious approach to designing a four-dimensional system for artists is to ask how they currently work in 3d space. In terms of input/output devices, the answer is typical personal computer interfaces: mice and trackballs, and flat display screens, which are all fundamentally twodimensional input and output devices. On the software side subdivision modeling/edge-loop modeling based on the winged-edge datastructure [Bau75] is a popular technique among artists for producing 3d objects [Rai]. Among other advantages, it enables the creation of 3d objects well suited for animation [Sin03]. In addition, the technique can be extended to preserve certain properties of the 3d objects under manipulation by restricting the types of operations performed, without placing additional arbitrary constraints on the artist.
One of the aims of this paper is to consider how this threedimensional paradigm can be extended to four dimensions.
1.2. Quasicrystals and self-assembling systems Quasicrystals [Sen06] are aperiodic crystalline structures, whose geometrical properties can be abstracted to quasiperiodic tilings of the plane. An important property of quasiperiodic tilings is that they can be constructed by projecting regular point lattices from higher-dimensional space, and in particular, a large class of quasiperiodic tilings can be obtained by projection from four-dimensional space. Self-assembly is a mechanism encompassing a broad range of phenomena in nature that can generally be described as the assembly of preexisting components into larger structures. Artificial self-assembling systems are a recent development to attempt to apply the same principles of organization to engineer systems with specific desirable properties from components greater than molecules in size [WB02]. Two dimensional systems represent a promising area of study in self-assembling systems because the space of possibilities they encompass is much richer and more interesting than those of one-dimensional systems but is tractable compared to those of three-dimensional ones, and unlike either one- or three-dimensional self-assembling systems, two-dimensional ones can be realized “in hardware” fairly easily [BB06].
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1.3. Results The paper aims to examine ideas for designing a software framework for letting artists visualize four dimensional space. Using some of these ideas, we show how a quasicrystal tiling generator can be created. We then examine the applicability of quasicrystal tilings to the creation of artificial two-dimensional self-assembling systems, develop some hypotheses on the subject, and describe a way of testing those hypotheses using genetic algorithms. 2. Background 2.1. Geometrical concepts Before proceeding further, we need to nail down the exact definition of three dimensional objects. This section will discuss some elementary topological notions; the reader is referred to [Arm83] for an introduction to topology. First, we need to develop the idea of a polytope. Consider a point, a line segment, and a polygon. A polygon is made up of line segment (from now on referred to as edges), and each of the line segments are delimited by points. Each of a point, line segment, and polygon represent polytopes in 0-, 1-, and 2-dimensions, respectively. Polytopes are formally defined as figures being bounded by a finite number of hyperplanes, with their intersection defining the structure of the polytope. For example, for a 3d object, the intersection of two 3d hyperplanes results in an edge, and the intersection of three or more hyperplanes produces a vertex (point). Generally, in n-dimensional space, intersections of n or more hyperplanes gives a vertex (0d polytope), intersections of n-1 hyperplanes gives an edge (1d polytope), intersections of n-2 hyperplanes gives a face (2d polytope/polygon), etc. A useful notion arises from this definition: that of an ndimensional polygon, which we can define as an n-1 dimensional polytope. While polytopes suffice to deal with any solid objects, we would of course like to examine objects with holes in them. (A connected, orientable surface with k holes in it is said to have genus k). A possibly fitting definition for threedimensional objects could be compact n-dimensional surfaces, however this would admit objects with no volume in n-space (essentially n-1 dimensional objects). n-manifolds turn out to provide us with the necessary definition. An n-manifold is a second-countable (countable base) Hausdorff (all points separated by neighborhoods) space where every point has a neighborhood homeomorphic (topologically isomorphic) to an open ball in Rn . This means we get a nice, smooth, closed object. However, a Mobius strip is also a nice smooth closed object, so we want to impose an additional constraint: the n-dimensional objects we will be working with have to be orientable. Note that for n-dimensional meshes in computer graphics, the planarity condition of polygons has to be relaxed. Not all
of the vertices that make up a polygon may be in a single hyperplane, but they can be made to be, in a bounded number of vertex moves. 2.2. Tiling theory This section will introduce some concepts helpful in understanding quasicrystal tilings and how tilings in generally apply to artificial two-dimensional self-assembling systems. A tile is a set homeomorphic to a disc (ie - it is bounded and is of genus 0). A tiling of the plane is a countable set of tiles which covers the plane entirely and without overlaps (the interiors of the tiles are disjoint). A monohedral tiling is one where every tile is congruent (can be mapped into via a combination of translations, rotations and reflections) to a tile in some finite set. That finite set is called the prototile (the term can also refer to one of its members) of the tiling [GS87]. A necessary concept in understanding how tiles fit together to fill space is the corona. The first corona of a tile is defined as the set of all tiles that meet that tile; the second corona is the set of all tiles that meet that tile and those tiles that meet them, etc. [Sen95]. A tiling in n-space is periodic if we can find translations in n linearly independent directions under which it is invariant. A nonperiodic tiling is one that admits no translations. A tiling in is subperiodic if it admits some translations but they are in less than n linearly independent directions. A set of prototiles is aperiodic if every tiling admitted by them is nonperiodic. An important property of aperiodic tilings is that their prototiles admit infinitely many unique tilings of the plane [GS87]. 2.3. Quasicrystals Quasicrystals were first discovered in nature as a material whose X-ray diffraction patterns exhibited fivefold rotational symmetry [Sen06]. This implies that the material has some regular structure, but it cannot actually be regular since fivefold rotational symmetry in two- or three-space is impossible: the rotational symmetries of structures in two- and threespace can only be 2-, 3-, 4- and 6-fold. This is known as the crystallographic restriction [Baa98]. 2.4. Self-assembling systems Self-assembly as a process encompasses many varied phenomena, but can generally be categorized as “processes that involve pre-existing components..., are reversible, and can be controlled by proper design of the components” [WG02]. Although ideal two-dimensional self-assembly cannot of course occur in the real world, something very close does happen with thin polygonal components on flat (or even curved, such as films) surfaces. Generally, self-assembly of
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physical components is guided by the amount of energy in the system, and such forces as chemical bonds, magnetic and capillary forces, etc. 3. Previous work There exists a wide body of work on four-dimensional visualization using computer graphics, going as far back as 1967 [Nol67]. The reader can find more recent examples in [Rob92] [Mit]. There have been several computer programs written to generate tilings based on multigrid and hypercubic lattice cut-and-project methods. The reader is referred to the author’s ongoing collection of links to quasicrystal tiling software [Sed]. In [BB06], Bhalla considered two-dimensional self assembling systems with polygonal components and an assembly protocol dictated by magnetic fields. Bhalla used evolutionary computation to produce the shape of the components, however the magnetic self-assembly protocol was designed manually. 4. Representing n-dimensional objects on a computer 4.1. Representation of objects in 3-space Polygon meshes are the predominant representation for three dimensional objects in modern computer graphics. Meshes can generally be said to be a tuple of a list of vertices, a list of edges between those vertices (this paper will refer to this as the graph structure of the mesh), and an orientation for each polygon formed by the edges and vertices of the mesh. General polygon meshes can contain holes in the form of missing polygons, unorientable surfaces (two adjacent polygons with different orientations/flipped normals) and have other artifacts, so it makes sense to restrict the implementation of a mesh (both in the datastructures and operations used) to something with more desirable properties. A widely used 3d mesh datastructure that provides some useful restrictions is the winged-edge datastructure [Bau75]. The winged-edge datastructure represents a 3d object mesh as a list of descriptions of an edge, each description consisting of the two vertices which belong to an edge, the two faces that share the edge, and the predecessor and successor edges of the edge when traversing the edges of the two faces that share that edge (in the same orientation). 4.2. Interactive manipulation of objects in 3-space One obvious class of interactivity provided in 3-space is changing the virtual camera parameters to view objects from different perspectives. Although there are many different ways to view three dimensional objects on a two dimensional screen, the most common method employed in modern 3d graphics is one-point perspective.
The next class of manipulations to consider are the ones that manipulate the entire object: change the object’s position, scale, and rotate it. These operations can be thought of (and implemented as) manipulations of the object’s local coordinate system as embedded in the global/scene. Clearly then, they will not change the object’s topology. Since the object is made up in turn of lower-dimensional polytopes, we should then consider the manipulation of these polytopes. While most current 3d modeling software allows for operations on multiple selected components (of the same type - to the author’s knowledge, no interactive 3d modeling software exists that allows the user to select multiple components of different types/dimensions (ex: a vertex and a face) for manipulation; in the case that something like that would be implemented, the operations available would have to be restricted to the set of ones conceptually similar between the different components), all of the operations that will be considered that can be performed on a set of components are done so on each component individually (ie - moving two selected polygons at once is the same as moving each polygon individually). In a recursively related way, we can consider the operations on components that can be decomposed into (temporally, not in the geometric sense) parallel operations on the sub-components of that component (ex: a face move is equivalent to a parallel move of all constituent vertices, resizing a face is equivalent to a set of moves of its constituent vertices, rotating an edge is the same as moving its two vertices, etc.) Further, we can class operations on an object or one or more of its components by the properties of that object they preserve. Some meaningful categories include graphstructure-preserving (ex: move vertex), genus-preserving (ex: extrude), relative-component-distance-preserving (ex: scale/resize), and absolute-component-distance-preserving (ex: rotation) operations. However, the most important category to consider are operations that preserve 2manifoldness. There exist several operations commonly found in today’s 3d modeling software that in certain cases may not preserve the 2-manifold property of object meshes [Flo]. We wish to restrict the set of operations strictly to those that preserve 2-manifoldness of objects since this will ensure that no adverse and difficult to detect artifacts will creep into objects in the course of normal manipulations when the current techniques are extended into four dimensions. An interesting dual to restricting the types of operations that can be performed is to use a more complicated object representation designed specifically to preserve the 2manifold property of meshes [AC99].
Vladimir Sedach / QuasiSculpt
4.3. Representation of objects in 4-space Representing four dimensional objects presents a unique set of challenges. One of the chief ones is the problem of enabling selection of components of the object. The representation has to enable efficient visual selection of component polytopes in all dimensions: vertices (0d), edges (1d), faces (2d), and most importantly this must extend to fourdimensional “polygons” (3d). This is one of the reasons why the above restrictions on three-dimensional objects become important: “degenerate” three-dimensional objects introduce ambiguity into the problem of selection in 4-space. The issue then becomes that of finding an analogue of the winged-edge or other datastructure with “nice” topological properties for four-dimensional objects and by extension its three-dimensional components. The dual of the data structures problem is dealing with operations on them. Some are immediately applicable to 4d objects (vertex moves), others can be extended in a somewhat intuitive way (rotations, which can be thought of as “about a plane” in four-space vs. “about a line” in three-space), while others, such as extrusions, have highly counter-intuitive effects in four dimensions. It is also conceivable that new operations with useful effects which have no analogues in three dimensions will be discovered. 5. Viewing 4-space 5.1. Projection The critical problem in working with 4-space is the difficulty in visualizing it. Three-dimensional objects can be viewed on a flat screen by projection to two dimensions; the same can be done for four-dimensional objects. First, the notion of projection needs to be defined precisely. Several different definitions with differing properties can be found in the literature, and some are too restrictive for current purposes. For the present work, a projection can be defined as an injective map between vertices (zerodimensional polytopes). Every vertex has a unique label that is preserved by the projection (this is how vertices in different dimensions are identified; this also implies that the projection will have an inverse). The relationship between vertices (graph structure) is not affected by the projection. So a vertex is a tuple of ((coordinates), label, dimension). All vertices with the same label have the constraint that their coordinates are related by the specified projections between their corresponding dimensions. It is important to note that here projections of points are defined to be from multiset to multiset - in general, the images of distinct vertices in lowerdimensional space may have coincident coordinates. The above definition leaves it quite open as to what projections are, so now the next step is to identify and classify some projections that are useful in visualizing higher dimensions:
• Projections that preserve parallel lines (ex: axonometric projections) • Projections that don’t preserve (all) parallel lines (ex: perspective projections) • Projections that don’t preserve any parallel lines (ex: fivepoint (curvilinear/"fish-eye") perspective) The most common class of projections used in modern 3d graphics are one-point perspective projections: the “rate of convergence” (difference in slope) of parallel lines is a function of the angle of those lines to the viewing plane (and goes to zero as the lines become parallel to the viewing plane). The projection thus obtained depends on the viewing plane and focal length (distance of the viewing plane to the ideal “pinhole camera” point). Perspective projections can be generalized to be view-plane and focal length independent by extending the space with actual points at infinity, which algebraically can be represented by augmenting the coordinates with one more dimension, called homogeneous coordinates, although this means that now points have multiple representations (much like fractions). This construction is called projective geometry [Cox74] [PP86]. To consider projections from four dimensions to two, note that in general, projections from m to n-space, where |m n| > 1, are really compositions of projections between each intermediate dimension. So there is no reason to restrict all the projections in the composition to a single type - the user of the software should be able to choose the projection that maps 4-space to 3-space, and then again choose the one that maps 3-space to 2-space. The next several sections consider specialized types of projections that have been found to be particularly useful in visualizing 4-space. 5.2. Schlegel diagrams A Schlegel diagram is a projection of an object from n to n − 1 space done by picking one of the faces and projecting onto the plane of that face [Ban90] [Som58]. 5.3. Nets Another way of visualizing the structure of objects in nspace is to “unwrap” them into n-1 space by their polygons. Each polygon is individually projected into the same n-1 space, and then the polygon’s polygons (in the case of 4space, this would be the faces of the three-dimensional components) are identified either in the “normal way” (by connecting the polygons) or by coloring (in places where the n-dimensional object can be thought of as being “cut”). An interesting observation is that the unwrapping of a triangulation of a 2-manifold in 3-space to 2-space is topologically equivalent to (a subdivision of) the fundamental polygon of that 2-manifold.
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6. Quasicrystals In order to experiment with quasicrystal patterns, we need a way to produce them. Fortunately, a well-studied body of work exists on methods for generating quasicrystal tilings, some of which work in a nicely similar way for tilings of any dimensional Euclidean space (for the purposes of the current work, we will of course restrict our attention just to two dimensions). The most common methods can be thought of as a hierarchy [HF]: • Multigrid • Cut-and-project (Euclidean windowed tilings) • – polytopal-windowed tilings (from lattices with polytopal fundamental domains) – ◦ canonical substitution tilings (from hypercubic lattices) The cut-and-project method works by embedding a plane into a higher-dimensional space containing a regular (integral) lattice, such that the plane is at an irrational angle to the translation symmetry vectors of the lattice (otherwise the resulting tiling will obviously be regular), selecting certain such points (those contained in a pre-determined compact set called the window), and projecting them onto the plane [Sen95]. The plane then contains a quasiregular point set, with each point corresponding to a tile. In general, points in the window will be those whose Voronoi cells in the Voronoi diagram of the lattice will be intersected by the plane being projected on. From this point set, a quasicrystalline tiling can be obtained in several ways depending on the characteristics of the set. For some lattices and point sets, the tiling can be obtained by projecting the edges between the projected points that are strictly within the window. In general, the projected points form a Delone point set, the Voronoi diagram of which will have a finite number of uniquely shaped cells - so each unique cell type corresponds to a prototile and therefore the Voronoi diagram must be a quasiperiodic tiling [Lag96]. A method using what is called the Klotz construction can generate tilings directly from higher-dimensional lattices, however it is considerably more complicated than the cutand-project [KS89] [BKSZ90]. For a hypercubic lattice then, the tiling obtained is one made up of parallelotopes (the most famous example of such a tiling is the Penrose tiling). Such tilings obtained from hypercubic lattices are called the canonical substitution tilings. If the higher-dimensional lattice is determined by a more general fundamental domain than hypercubes, quasiperiodic tilings with prototiles made up of any kind of convex polytopes can be obtained [VKR88]. These are the polytopalwindowed tilings, of which the canonical substitution tilings make up a subset. Together they are referred to as the Euclidean windowed tilings. The multigrid method generates tilings by considering a two-dimensional construction only (it can also be extended
to three dimensions and beyond for higher-dimensional quasicrystals). The method produces tilings with five-fold diffraction symmetry by taking five identical sets of line pencils, each rotated by 72 (=360/5) degrees from the other, and using local rules based on the intersections of particular lines from the different pencils to locate the vertices of the tiles, and then another set of local rules to figure out which vertices are supposed to be connected by edges to form tiles. It turns out that the sets of tilings that can be generated by the multigrid and canonical substitution tiling methods are equivalent [Baa98]. So far, we have not considered exactly which dimension the projection will take place from. To generate a two-dimensional quasiperiodic pattern with n-fold diffraction symmetry, the cut-and-project method needs to be performed from a lattice in at least φ(n) dimensions, where φ is Euler’s totient function (number of integers relatively prime to the given integer) [Baa98]. So tilings with 5-fold symmetry need to be projected from at least a 4-dimensional lattice. In practice, 4-dimensional lattices suffice for a wide class (indeed, an infinity) of interesting tilings with 5-, 8- and 10fold diffraction symmetries.
7. Self-assembling systems Self-assembly exists in a wide variety of forms in many systems. In this paper, we consider artificial, static, templated two-dimensional self-assembling systems [WG02]. A self-assembling system is said to be templated when there is some (usually physical) fixed object(s) in the environment that affect the assembly of the system. For example, in crystal growth on a mineral surface, the shape of the surface would be considered a template. Static self-assembling systems are those that reach some equilibrium state when they become stable and finally assembled. This is in contrast to dynamic self-assembling system, like schools of fish or atmospheric storm cells, which never reach a stable energy state [WG02]. Components of the system are essentially tiles, and are defined first and foremost by their shape, at a fixed scale. The simplest kind of assembly protocol for these sorts of systems is probably what the author terms the tiling protocol: the components assemble only when they can fit together without gaps. In terms of complexity, the next addition (or, as in the case of Wang tiles, a replacement) to the protocol can be the restriction that the component edges must be colored correctly (this for example can be used to model magnetic fields). Coloring can be considered a restriction of the generalized self-assembly protocol described in [BJ07].
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7.1. Self-assembling systems and problems in tiling theory An interesting question to consider is that given a set of prototiles, can we say whether they tile the plane or not? This is known as the Domino problem. For a set of prototiles with cardinality greater than one, the problem turns out to be undecidable (in fact, the proof depends on showing that a variation of Wang tiles is equivalent to a Turing machine - today, Wang tiles are being investigated as a possible model for self-assembling DNA computers [PW02]). For a single prototile, the question remains open [GS]. The Domino problem is important since it means that there are shapes which can fill a bounded region but not tile the plane, which is a property that can potentially be useful for artificial self-assembling systems, but given a set of prototiles we can’t mechanically decide whether they will do this or not, which could present an obstacle to applying the theory (for example, this problem can potentially come up in the development films to be assembled at arbitrarily large sizes). The concept of a corona immediately brings up a question: for a given single prototile, what is its maximum corona? This number is known as the Heesch number of the tile [Sen95]. The concept has immediate applications to two-dimensional self-assembling systems as it means we can potentially construct a particular figure out of just a single tile shape. Unfortunately, the largest finite Heesch number for which a set of prototiles is known to exist is currently 5 [Man]. 7.2. Self-assembling systems and quasicrystal tilings Quasicrystal tilings exhibit aperiodicity but are composed of only a small number of prototiles with fairly simple geometrical shapes. The author hypothesizes that given a twodimensional shape (a template), at some fixed scale, a quasicrystal tiling can be found that contains an arrangement of tiles that would fit that template “fairly well.” A further hypothesis (that will not be investigated in this work) is that due to the aperiodicity of quasicrystals, given those exact tiles that fit the template in any initial configuration, they will assemble (without environmental constraints) into the templated configuration (and not any other) with a probability inversely proportional to the number of tiles, which can be increased to unity with the use of relatively simple assembly protocols and environmental constraints. The exciting thing about the above hypotheses is that if there is a small number of prototiles with simple assembly protocols, then there is the possibility that the tiles can either be mass produced, or perhaps found to be present in already existing materials, and employed to self-assemble into more complicated structures which would be unfeasible to manufacture by other methods. For example, currently there is a great deal of promising research into efficient manufacture
of basic nanostructures (nanotubes, nanowires), but assembly of these components into larger configurations remains problematic [WG02]. The question remains of how the first hypothesis can be tested. A possible approach is to examine many aperiodic prototiles and see how well they can possibly tile the desired shape, both the tiles and the shape at some fixed scale in relation to each other. A viable method for doing this is by employing genetic algorithms, where the parameters for generating a particular tiling by the cut-and-project method would be the genotype, and the prototiles produced would be the phenotype. The genotype itself would consist of the angle of the plane (which must always irrational with respect to the lattice), the window, and, since we can only consider finite regions of the tiling of the plane, a point in 4-space through which the plane will pass. The fitness function would examine how well the tiles could be assembled to approximate the given shape, perhaps by choosing a single prototile, and examining some fixed corona of the occurrences of that tile in the quasiperiodic tiling of the plane generated by the cut-and-project method. The evaluation parameter could be something along the lines of “excess area” (either where the tiles don’t fill the shape fully or when they jut outside of the shape). The choice of modification operators (mutation, recombination, selection, etc.) is an open question. 8. Conclusions Artificial self-assembling systems and four-dimensional visualization are interesting and relevant topics of study. In addition to showing the link between the two subjects, this paper has described some possible novel directions for implementing four-dimensional visualization systems, and has provided background on tiling theory and quasicrystal tilings and how they can be applied to artificial two-dimensional self-assembling systems. 9. Acknowledgments The author wishes to thank Marina Gavrilova for supervising the current work and Brian Wyvill for heading the class. The ideas in the present work germinated as the result of a series of ongoing discussions with Jerry Hushlak. Navneet Bhalla provided invaluable guidance and advice on the topic of self-assembling systems. Additionally, the author wishes to acknowledge the support of Rob Kremer. References [AC99] A KLEMAN E., C HEN J.: Guaranteeing 2manifold property for meshes. In Proceedings of the International Conference on Shape Modeling International (March 1999). [Arm83] 1983.
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