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Quasi-crystalline geometry for architectrual structures

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barbara_weinzierl

This document discusses using quasi-crystal geometry for architectural structures. Some key points: 1) Quasi-crystals have an aperiodic structure that breaks from traditional crystal symmetries, allowing new geometric patterns. The document explores using golden cubes and Penrose tilings as the basis for 3D quasi-crystal geometries. 2) For architectural structures, quasi-crystals would need to be stabilized through the addition of connecting bars or plates. The number and placement of stabilizing elements is analyzed to achieve structural stability. 3) Potential architectural applications are explored at multiple scales, from individual building components to urban planning. Quasi-crystals could provide more design flexibility compared to traditional symmetries while still maintaining

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QUASI-CRYSTALLINE GEOMETRY FOR ARCHITECTURAL STRUCTURES Barbara Weinzierl1) and Ture Wester2) 1) Stud. Arch. 2) Associate Professor Royal Danish Academy of Fine Arts, School of Architecture, Copenhagen, Denmark. ABSTRACT The purpose of the paper is to investigate some possibilities for the application of Quasi- Crystal (QC) geometry for structures in architecture. The basis for the investigations is A: to use the Golden Cubes (the two different hexahedra consisting of rhombic facets where the length of the diagonals has the Golden ratio) as basic elements for aperiodic 3D geometries (Ref.1) and B: to raise aperiodic Penrose tilings (Ref.1) and its binary substitutions from their 2D basis into 3D QC geometries and describe the structural behaviour for these spatial configurations. The structural qualities are based on stability considerations for 3D structures consisting of nodes connected with hinged bars (lattice action), and hinges plates connected by shear connections (plate action) (Ref.2, 3). The QC structure was considered basically as an ideal lattice structure stabilised with the necessary number of ideal plates (Ref.4). As the Average Valency (VA) of the nodes (i.e. the number of bars connected to the node) is essential for the kinematic stability of large clusters of QC units and as there seems not to exist any proof for this number, a simple counting showed that the VA - for increasing number of nodes - converges to be identical to a square 2D periodic lattice (Valency equal to 4) and to a cubic 3D periodic lattice (Valency equal to 6). An interesting quality for building purposes is the fractal quality of the binary substitution as described in the paper. 1. HISTORY The QC type of crystallisation was discovered and published in 1984 by Dan Schechtman (Ref.5). The diffraction images showed 3D icosahedral symmetry, which among crystallo- graphers were believed to be impossible and forbidden. 2D aperiodic tiling with fivefold symmetry using a fat and skinny rhombic tile was found by Roger Penrose (Ref.6) a decade before. These so-called penrose tiling turned out to be a 2D version of the 3D QC. QC steel alloys, which are harder but more brittle than normal steel, are recently commercially produced by e.g. Sandvik AB in Sweden for high quality accessories as e.g. razor blades for electric shavers, surgery instruments etc. Haresh Lalvani (Ref.7) and Koji Miyazaki (Ref.8) have done remarkable research on QC geometry including the quality that aperiodic QC can be regarded as a periodic 5D cubic lattice projected down to 3D. Tony Robbin (Ref.9) has worked with QC architecture and created a large-scale QC sculpture at the DTU in Lyngby, Denmark.
An example of a building in which penrose patterns and QC geometry have been used as a design idea is RMITs Storey Hall in Melbourne by the group ARM (Ashton Ragatt McDougall) (Ref.10). 2. MORPHOLOGY The new QC pattern of crystallisation broke totally with the traditional conception of the morphology of close packing cells and symmetry orders of crystals introducing aperiodic fivefold symmetry in 3D space. The 3D QC geometry can form a close-packing configuration consisting of two different distorted cubes with identical rhombic facets. The relation between the facet's diagonals is the Golden Ratio. All cells have therefore identical faces and identical edges, and all nodes fit to a regular dodecahedron all oriented translationally identical. The valency of a node may vary from 4 to 12. A 2D penrose pattern is the projection of a specific layer in an infinite 3D QC. In a penrose tiling there are eight different combinations of rhombs to create so-called vertex-stars, hence one way of building a QC is to stack such vertex-stars overlapping in 3 D (Ref.1). Another possibility is to build a Rhombic Triacontahedron (see fig.3) from ten skinny and ten fat rhombic cubes and using them as cores for a QC, and fill the gaps with rhombic cubes in an aperiodic order. Another interesting geometry is found by making a binary substitution of a penrose pattern. Each fat rhombus is replaced by a vertex-star of three fat ones and one skinny rhombus, and each skinny rhombus is replaced by a vertex-star of one fat and two skinny rhombi (Ref.1). This means that two binary vertex-stars may be used as building blocks of the binary sub- stitution pattern. A binary substitution tiling is still aperiodic but has no fivefold symmetry. An additional possibility is to make this substitution only partly by making switches. These are made by substituting only one rhombus of the binary vertex-stars with another binary vertex-star in the same scale of the pattern. This method creates stronger undulating structures than the ordinary binary substitution, although periodic parts are unavoidable (Fig.1). There is also a fractal possibility in doing a binary substitution of a binary substitution as you can quot;zoomquot; in and out of the pattern, and also change the scale without changing the pattern. The borderline between the substitutions seems to create unavoidable periodic parts in the structure. (Fig.2) Fig.1 Binary substitution of a Penrose Fig.2 Structure with fractal properties structure with switches The average valency (VA) in an infinite equal to the internal nodes of a huge 2D Penrose pattern has been estimated by counting the VA of an increasing number of adjacent nodes. The
VA appears to converge towards 4. This is not surprising as the valency of a periodic tiling of either the fat or the skinny rhombus is 4, because it is topologically equal to the square grid. The similar procedure was performed on a QC, and the 7 VA seems to converge towards 6 6, which is not surprising as Average Valency 3D Quasi Crystal the valency of the periodic 5 2D Penrose packing of the fat or the skinny rhombic cube is 6 - just 4 like the cubic grid. This 3 procedure is of course not a solid proof, but a reasonable 2 1 7 15 20 21 29 40 42 48 51 52 62 68 73 76 reliable conclusion. VA is ab- Number of Nodes solutely essential for the description of the structural Graph 1. Average Valency (VA) of 2D Penrose converge behaviour of Penrose and QC to 4 and a 3D Quasi Crystal converge to 6 with configurations (Ref.4). increasing number of nodes For an infinite QC where V are Vertices, E are Edges, F are Faces and C are Cells the follow- ing can be deduced: All cells have 6 facets and each facet share 2 cells then F=3xC and as VA is six, then E=3xV. The extended Euler Theorem for an infinite cell-configuration in 3D space is R0-R1+R2-R3=0 where R is the number of elements and the index is the dimension of these elements, which means that V-E+F-C=0. When these equations are combined we will get the following geometric equations for infinite Quasi Crystal configurations consisting of the two golden rhombic cubes: F=3xC E=3xV E=3xC F=3xV V=C E=F 3. STRUCTURAL BEHAVIOUR For a large QC pure lattice structure E=3xV means zero internal geometrical and structural redundancy but global instability because of the quot;missingquot; bars along the surface. Therefore the configuration must be supplied with extra stabilising elements to reach kinematic stability. There are of course many possibilities for placing these extra elements in order to stabilise the structure. For a given QC pure lattice structure the Neutral Method (ref.4) the number of extra stabilising plates or bars can easily be found. The Neutral Method calculates the redundancy directly in a simple way, and is based on Moebius' equation for kinematic stability for pure 3D lattice structures B+S 3xN; where B, S and N is the number of Bars (edges), Supports and Nodes (vertices). An unstable QC pure lattice configuration can be stabilised by extra bars - or maybe more obviously - by inserting (identical) plates in the rhombic meshes. If the stabilisers are applied to the QC-rhombs, a plate will have the same stabilising effect as a diagonal bar. If plates are used they may be arranged in such a way that they articulate, emphasise and stress the architectural idea of the structure. According to the above-mentioned Neutral Method for the Golden Rhombic Dodecahedron as QC, there are minimum 10 extra elements needed, for the Golden Rhombic Icosahedron as QC minimum16 elements are needed. The Rhombic Triacontahedron as QC needs minimum 24 extra elements. As the surface of the Rhombic Triacontahedron, which is a semi-regular polyhedron of second order, has 30 facets means that all 24 extra plates can be put on its
surface to stabilise also the inside of it and still keep 6 meshes open. There are many other solutions for arranging the stabilising plates e.g. to emphasise horizontal or other parallel planes (fig.3) or flows of spirals etc. For an quot;emptyquot; Rhombic Triacontahedron envelope with no internal nodes, all rhombs must be supplied with a stabilising plate, and therefore closed. For a space-truss consisting of one layer of QC cells the stabilising plates can be arranged in rows along the so-called quot;snakesquot; (Ref.1 p.163), creating intersecting tubes. Except for the ends, these tubes have no stabilising elements inside of them, as this would not add stability to the system. In the more open parts of the spaceframe it is possible to either arrange the rest of the needed plates or bars in a vertically standing position to the pattern, for the use of creating holes or windows, or in the layer of the pattern to almost close the surface of the frame as a roof or wall. (Fig.4) Fig.3 A QC lattice Rhombic Triaconta- Fig.4 Stable single layer QC space-truss hedron with stabilising plates as plate stabilised lattice. Fig.5 Single-layer dome with torus-ring as Fig.6 Single-layer quot;Bubblequot; as combin- combination of open plate and lattice. ation of open plate and lattice. A single-layer envelope can be stabilised with a QC torus ring at its base if both are constructed as a structurally closed combined lattice and plate structure (Fig.5), here shown with plates as open frames. For a closed single-layer envelope (or quot;bubblequot;) all facets must constructed as a structurally closed combined lattice and plate structure (Fig.6). 4.ARCHITECTURE The aperiodicity of a QC configuration facilitates a huge number of different arrangements of the rhombohedra. Therefore it may adapt to many different architectural requirements as
quality of space, scale, function, form, adaptability to landscape etc. not only regarding the rhombohedra sizes, the overall shape and framework, but also the configuration and the structural material of the added stabilising elements. 1) However, there are only eight different vertex stars and they repeat themselves within a certain distance to each other, so there is a limited but very large set of space combinations (Ref.1). 2) The fractal possibilities of QCs are especially important for being able to use QC structures in all scales and for all uses, from a skyscraper to a trash-bin, and also for the detailing within a building (Ref.8). 3) The two unit rhombohedra can - as indicated earlier - also be separately arranged in a strictly periodic way forming a distorted cubic grid. As all three above-mentioned options can be combined in one configuration, there are an almost countless number of possible combinations for architectural uses. It is up to the Architect or Urban planner to investigate and select between all these possibilities. Architecture using QC geometry is really not a stiff one-way modular kit - which would rather limit - but is able to give more freedom to create structural and morphological interesting configurations. 4.1 Architecture theory The theory of post-modern deconstruction after Jacques Derrida (Ref.11) fits very well with QC Architecture, as it can be seen as a displacement of existing cubic architecture with twofold and fourfold symmetries with a representation of higher dimensional cubes which in 3D space physically visualises the gap between the binary terms of our western occidental culture. 4.2 Urban scale There are mainly three types of geometry in historical and modern city plans: - the linear type, along a communication and transportation line e.g. a road or a river - the concentric/radial type, around a fortification e.g. on a hill; or an important crossing of roads or rivers - the grid type, which already needs supervised and ordered urban planning from the foundation of the city on e.g. Chinese cities of one planning concept. The first two types are a logic reaction to the environment. The third type is a rational decision of human planning and the need of ordering nature for different reasons. It is indeed very interesting, that the diffraction pattern of a QC with its tenfold symmetry is very similar to radial city plans. One idea is to superimpose a penrose pattern on a radial city plan, so to say to show the virtual QC lying above the city, creating the city-plan with its diffraction pattern. The penrose pattern would in general be a way of planning new city-grids, still ordering spaces to a certain extent, but also regaining the liveliness of a more flexible and less prohibiting grid like twofold or fourfold symmetry. Its snakes could be the streets; a vertex star consisting of five fat rhombs in fivefold symmetry could be a street-circus or a plaza. Chaos equals with Nature, and Order with Culture and Civilisation in our society. Cities grow from village to megalopolis, become huge urban areas which need to be restructured to give a more natural feeling of environment to the people that live in them, as real nature gets pushed back to the worlds most hostile regions as deserts and beyond the polar circle. The fractal possibilities by binary substitution can be used for constituting spaces in different scales like the city centre, or the centre of a suburb and still the overall grid would not be broken. Todays cities are a mix of individuals from different cultures, countries, religions,
colours and continents. Therefore it has become necessary not to symbolise the equality of individuals in a democratic system by a simple order, equal for all its users, but a complex and diverse pattern with possibilities for all the different individuals coexisting in one place. Architecture has always been aiming to control spaces and therefore also actions and users. We need to give freedom and possibilities to develop now, so QC buildings on a penrose grid would combine this freedom without totally loosing control of the city grid, and slipping into anarchy - the essence of quasi crystallisation. 4.3 Wide Span and High-rise Buildings QC buildings in a large scale can be used to room all kinds of uses and functions. The tilting of the walls of QC buildings gives tension to the perception of the street space, as they seem unstable and to come towards the pedestrian, not like the usual psychological stability of vertical lines in buildings reacting to earths gravity. An example for this type of tension, although not a QC, is the Royal Library Extension in Copenhagen by the Danish architects quot;Smidt, Hammer & Lassen, a distorted cube also called the Black Diamond. QC buildings are by this a real statement of human design and its interaction, and a starting point to overcome physical laws. The angles of the tilting floors can be used for large auditoriums, e.g. for an opera, concert- halls or universities, horizontal split-levels can be mounted in a QC space-grid with connected air-spaces, as horizontal plates hovering through space. A large Rhombic Triacontahedron with its five horizontal planes spiralling around the axis would be one possibility for a communication organisation within such a building. The stairs lead from one horizontal floor to the next, the elevator-shafts along the axis through the middle, and the auditoriums leading down and away from the floors opening to the outside views into all directions. (Fig.7) QC domes and space-frames can be mounted over street spaces or plazas to create Galleries looking like reptile skins growing over the citys open spaces. Single layer QC structure can also react and move with or against the landscape, creating amorphous convex and concave spaces between the structure and the ground and in the quot;foldingsquot; of the undulating QC surface. This can be used to cover large areas with temporary structures e.g. for fairs (Fig.8). Fig.7 Spaces within a Rhombic Triaconta- Fig.8 Single-layer structure, stabilised hedron with combined lattice and plate action. The advantage is, that as ordinary twofold and fourfold building structures usually need flat ground, and do not react at all to their environment with their unnatural shape, QC structures are possible literally everywhere, not violating the Nature to fit them.
The fractal possibilities can be used to shift the scale within a building from big halls to normal room size areas to an open space with small-scale space-frames above it. For example this could structure an airport from the check-in hall to offices, waiting rooms, shops, and communication and transportation areas. It can also be used to create large buildings with about brick-size QC rhombohedra, as if the rooms of the building where just carved out of a gigantic QC. The walls and floors of such rooms would then be undulating in a small-scaled pattern, which would create the texture of an organic grown structure. 4.4 Dwellings QC structures for dwellings can be used in all the existing building types, as detached houses, single houses, for rather low suburban housing areas, mixed with garden spaces, or middle- scale housing blocks, with public park areas. If one rhombohedron is one room, there are e.g. the eight vertex-stars that could be a way of organisation. In the penrose pattern the quot;snakesquot; could be a possibility for the detached houses. The quot;snakesquot; could also be the streets, as the borderlines for the housing clusters in various shapes with inner courtyards and gardens. Another way is to cluster the single rhombohedra to a 3D QC into all directions of three- dimensional space. Fig.10 Irregular stacking of rhombic Fig.9 Regular stacking of rhombic triacontahedra triacontahedra Fig.11 QC structure with growth Defect Fig.12 Interior Spaces of a QC A Rhombic Triacontahedron could be the minimal living unit in a 3D cluster, with its rhombohedra as the different rooms, with some split-levels or stepped podiums to create some necessary straight floors. As there is regular and irregular ways of stacking these units and attaching them to each other, houses can be adapted to different family sizes. The regular stacking (Fig.9) creates holes and is not a complete QC aperiodic structure. However these
gaps can be used for courtyards and gardens, or shafts to bring daylight also to the lower levels of higher clusters. The irregular stacking (Fig.10) creates a 3 D QC grid, in which gaps for the above mentioned uses can be made by leaving out some rhombohedra. There is also a mathematical algorithm for producing penrose patterns with incorporated local growth defects, which yet retain the semblance of fivefold symmetry (ref.12, pp.86-96). These non-tileable parts could grow through a real 3D QC like wormholes (Fig.11). QC-colonies have almost no limits in how high or how far they can go, as each Rhombic Triacontahedron can be constructed as a stable unit and therefore able to be a rigid building block in a larger configuration. The QC grid in the cluster is supporting itself and all its parts (Fig.12). 5. CONCLUSION It seems that the special QC geometry can be used in a new and advantageous way as a grid at all levels for architectural structures, town planning, interior design etc. not least because of the fascinating relation to both chaos and order. Edwin A. Abbott's quot;Flatland - a romance of many dimensionsquot;, written in 1884 (ref.13) inspires to consider if there might be a 5D world with 5D beings somewhere in our universe - or rather beyond it - and how it might look like. As the 2D creatures in Flatland only perceive two dimensions as living in a surface, we do not in our world have any sense which can perceive a higher dimension than our usual three. As QCs are regular and periodic in 5D space a QC is just another quot;boringquot; cubic building for a 5D inhabitant It appears extremely inspiring that we can build interesting architecture of their 3D representations. Higher dimensional spaces and structures might be used as perfect tools for the architecture of the third millennium. Only time will show if this potential is going to be utilised in our societies, and not only as futuristic ideas for space stations and moon settlements. It is about time to start something new. 7. REFERENCES 1) Senechal, M. (1995) Quasicrystals and Geometry. Cambridge University Press. 2) Wester, T. (1984) Structural Order In Space, the plate lattice-dualism. School of Architecture, Copenhagen. 3) Wester, T. (1997) The Structural Morphology of Basic Polyhedra in Beyond the Cube (ed. J.F.Gabriel) pp.301-342, John Wiley and Sons, Inc. 4) Wester, T. (1987) Plate Domes, Proceedings of the International Conference on the Design and Construction of Non-Conventional Structures, pp 241-250, Civil-Comp Press, London 5) Schectman, D. et al (1984) Metallic Phase with Long-Range Orientational Order and No Translational Symmetry. Physical Review Letters, Vol. 53, 1951-3 6) Penrose, R. (1974) Pentaplexity, Bulletin of the Institute for Mathematics and Applications, Vol.10. pp.266-271 7) Lalvani, H. (1994) Hyper-Geodesic Structures, IASS Atlanta. American Society of Civil Engineers. 8) Miyazaki, K. (1986) An Adventure in Multidimensional Space, John Wiley and Sons 9) Robbin, T. (1996) Engineering a New Architecture, Yale University Press. 10) Jencks, C. (2000) Architecture 2000 and Beyond - Success in the Art of Prediction, Wiley Academy. 11) DOMUS no. 671 (April 1986) pp 17-24. Interview with Jaques Derrida 12) Peterson, I. (1990) Islands of Truth, A Mathematical Mystery Cruise 13) Abbott, E.A. (1984) Flatland, A Romance of many Dimensions. (first published in 1884) Signet Classic.

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Quasi-crystalline geometry for architectrual structures

  • 1. QUASI-CRYSTALLINE GEOMETRY FOR ARCHITECTURAL STRUCTURES Barbara Weinzierl1) and Ture Wester2) 1) Stud. Arch. 2) Associate Professor Royal Danish Academy of Fine Arts, School of Architecture, Copenhagen, Denmark. ABSTRACT The purpose of the paper is to investigate some possibilities for the application of Quasi- Crystal (QC) geometry for structures in architecture. The basis for the investigations is A: to use the Golden Cubes (the two different hexahedra consisting of rhombic facets where the length of the diagonals has the Golden ratio) as basic elements for aperiodic 3D geometries (Ref.1) and B: to raise aperiodic Penrose tilings (Ref.1) and its binary substitutions from their 2D basis into 3D QC geometries and describe the structural behaviour for these spatial configurations. The structural qualities are based on stability considerations for 3D structures consisting of nodes connected with hinged bars (lattice action), and hinges plates connected by shear connections (plate action) (Ref.2, 3). The QC structure was considered basically as an ideal lattice structure stabilised with the necessary number of ideal plates (Ref.4). As the Average Valency (VA) of the nodes (i.e. the number of bars connected to the node) is essential for the kinematic stability of large clusters of QC units and as there seems not to exist any proof for this number, a simple counting showed that the VA - for increasing number of nodes - converges to be identical to a square 2D periodic lattice (Valency equal to 4) and to a cubic 3D periodic lattice (Valency equal to 6). An interesting quality for building purposes is the fractal quality of the binary substitution as described in the paper. 1. HISTORY The QC type of crystallisation was discovered and published in 1984 by Dan Schechtman (Ref.5). The diffraction images showed 3D icosahedral symmetry, which among crystallo- graphers were believed to be impossible and forbidden. 2D aperiodic tiling with fivefold symmetry using a fat and skinny rhombic tile was found by Roger Penrose (Ref.6) a decade before. These so-called penrose tiling turned out to be a 2D version of the 3D QC. QC steel alloys, which are harder but more brittle than normal steel, are recently commercially produced by e.g. Sandvik AB in Sweden for high quality accessories as e.g. razor blades for electric shavers, surgery instruments etc. Haresh Lalvani (Ref.7) and Koji Miyazaki (Ref.8) have done remarkable research on QC geometry including the quality that aperiodic QC can be regarded as a periodic 5D cubic lattice projected down to 3D. Tony Robbin (Ref.9) has worked with QC architecture and created a large-scale QC sculpture at the DTU in Lyngby, Denmark.
  • 2. An example of a building in which penrose patterns and QC geometry have been used as a design idea is RMITs Storey Hall in Melbourne by the group ARM (Ashton Ragatt McDougall) (Ref.10). 2. MORPHOLOGY The new QC pattern of crystallisation broke totally with the traditional conception of the morphology of close packing cells and symmetry orders of crystals introducing aperiodic fivefold symmetry in 3D space. The 3D QC geometry can form a close-packing configuration consisting of two different distorted cubes with identical rhombic facets. The relation between the facet's diagonals is the Golden Ratio. All cells have therefore identical faces and identical edges, and all nodes fit to a regular dodecahedron all oriented translationally identical. The valency of a node may vary from 4 to 12. A 2D penrose pattern is the projection of a specific layer in an infinite 3D QC. In a penrose tiling there are eight different combinations of rhombs to create so-called vertex-stars, hence one way of building a QC is to stack such vertex-stars overlapping in 3 D (Ref.1). Another possibility is to build a Rhombic Triacontahedron (see fig.3) from ten skinny and ten fat rhombic cubes and using them as cores for a QC, and fill the gaps with rhombic cubes in an aperiodic order. Another interesting geometry is found by making a binary substitution of a penrose pattern. Each fat rhombus is replaced by a vertex-star of three fat ones and one skinny rhombus, and each skinny rhombus is replaced by a vertex-star of one fat and two skinny rhombi (Ref.1). This means that two binary vertex-stars may be used as building blocks of the binary sub- stitution pattern. A binary substitution tiling is still aperiodic but has no fivefold symmetry. An additional possibility is to make this substitution only partly by making switches. These are made by substituting only one rhombus of the binary vertex-stars with another binary vertex-star in the same scale of the pattern. This method creates stronger undulating structures than the ordinary binary substitution, although periodic parts are unavoidable (Fig.1). There is also a fractal possibility in doing a binary substitution of a binary substitution as you can quot;zoomquot; in and out of the pattern, and also change the scale without changing the pattern. The borderline between the substitutions seems to create unavoidable periodic parts in the structure. (Fig.2) Fig.1 Binary substitution of a Penrose Fig.2 Structure with fractal properties structure with switches The average valency (VA) in an infinite equal to the internal nodes of a huge 2D Penrose pattern has been estimated by counting the VA of an increasing number of adjacent nodes. The
  • 3. VA appears to converge towards 4. This is not surprising as the valency of a periodic tiling of either the fat or the skinny rhombus is 4, because it is topologically equal to the square grid. The similar procedure was performed on a QC, and the 7 VA seems to converge towards 6 6, which is not surprising as Average Valency 3D Quasi Crystal the valency of the periodic 5 2D Penrose packing of the fat or the skinny rhombic cube is 6 - just 4 like the cubic grid. This 3 procedure is of course not a solid proof, but a reasonable 2 1 7 15 20 21 29 40 42 48 51 52 62 68 73 76 reliable conclusion. VA is ab- Number of Nodes solutely essential for the description of the structural Graph 1. Average Valency (VA) of 2D Penrose converge behaviour of Penrose and QC to 4 and a 3D Quasi Crystal converge to 6 with configurations (Ref.4). increasing number of nodes For an infinite QC where V are Vertices, E are Edges, F are Faces and C are Cells the follow- ing can be deduced: All cells have 6 facets and each facet share 2 cells then F=3xC and as VA is six, then E=3xV. The extended Euler Theorem for an infinite cell-configuration in 3D space is R0-R1+R2-R3=0 where R is the number of elements and the index is the dimension of these elements, which means that V-E+F-C=0. When these equations are combined we will get the following geometric equations for infinite Quasi Crystal configurations consisting of the two golden rhombic cubes: F=3xC E=3xV E=3xC F=3xV V=C E=F 3. STRUCTURAL BEHAVIOUR For a large QC pure lattice structure E=3xV means zero internal geometrical and structural redundancy but global instability because of the quot;missingquot; bars along the surface. Therefore the configuration must be supplied with extra stabilising elements to reach kinematic stability. There are of course many possibilities for placing these extra elements in order to stabilise the structure. For a given QC pure lattice structure the Neutral Method (ref.4) the number of extra stabilising plates or bars can easily be found. The Neutral Method calculates the redundancy directly in a simple way, and is based on Moebius' equation for kinematic stability for pure 3D lattice structures B+S 3xN; where B, S and N is the number of Bars (edges), Supports and Nodes (vertices). An unstable QC pure lattice configuration can be stabilised by extra bars - or maybe more obviously - by inserting (identical) plates in the rhombic meshes. If the stabilisers are applied to the QC-rhombs, a plate will have the same stabilising effect as a diagonal bar. If plates are used they may be arranged in such a way that they articulate, emphasise and stress the architectural idea of the structure. According to the above-mentioned Neutral Method for the Golden Rhombic Dodecahedron as QC, there are minimum 10 extra elements needed, for the Golden Rhombic Icosahedron as QC minimum16 elements are needed. The Rhombic Triacontahedron as QC needs minimum 24 extra elements. As the surface of the Rhombic Triacontahedron, which is a semi-regular polyhedron of second order, has 30 facets means that all 24 extra plates can be put on its
  • 4. surface to stabilise also the inside of it and still keep 6 meshes open. There are many other solutions for arranging the stabilising plates e.g. to emphasise horizontal or other parallel planes (fig.3) or flows of spirals etc. For an quot;emptyquot; Rhombic Triacontahedron envelope with no internal nodes, all rhombs must be supplied with a stabilising plate, and therefore closed. For a space-truss consisting of one layer of QC cells the stabilising plates can be arranged in rows along the so-called quot;snakesquot; (Ref.1 p.163), creating intersecting tubes. Except for the ends, these tubes have no stabilising elements inside of them, as this would not add stability to the system. In the more open parts of the spaceframe it is possible to either arrange the rest of the needed plates or bars in a vertically standing position to the pattern, for the use of creating holes or windows, or in the layer of the pattern to almost close the surface of the frame as a roof or wall. (Fig.4) Fig.3 A QC lattice Rhombic Triaconta- Fig.4 Stable single layer QC space-truss hedron with stabilising plates as plate stabilised lattice. Fig.5 Single-layer dome with torus-ring as Fig.6 Single-layer quot;Bubblequot; as combin- combination of open plate and lattice. ation of open plate and lattice. A single-layer envelope can be stabilised with a QC torus ring at its base if both are constructed as a structurally closed combined lattice and plate structure (Fig.5), here shown with plates as open frames. For a closed single-layer envelope (or quot;bubblequot;) all facets must constructed as a structurally closed combined lattice and plate structure (Fig.6). 4.ARCHITECTURE The aperiodicity of a QC configuration facilitates a huge number of different arrangements of the rhombohedra. Therefore it may adapt to many different architectural requirements as
  • 5. quality of space, scale, function, form, adaptability to landscape etc. not only regarding the rhombohedra sizes, the overall shape and framework, but also the configuration and the structural material of the added stabilising elements. 1) However, there are only eight different vertex stars and they repeat themselves within a certain distance to each other, so there is a limited but very large set of space combinations (Ref.1). 2) The fractal possibilities of QCs are especially important for being able to use QC structures in all scales and for all uses, from a skyscraper to a trash-bin, and also for the detailing within a building (Ref.8). 3) The two unit rhombohedra can - as indicated earlier - also be separately arranged in a strictly periodic way forming a distorted cubic grid. As all three above-mentioned options can be combined in one configuration, there are an almost countless number of possible combinations for architectural uses. It is up to the Architect or Urban planner to investigate and select between all these possibilities. Architecture using QC geometry is really not a stiff one-way modular kit - which would rather limit - but is able to give more freedom to create structural and morphological interesting configurations. 4.1 Architecture theory The theory of post-modern deconstruction after Jacques Derrida (Ref.11) fits very well with QC Architecture, as it can be seen as a displacement of existing cubic architecture with twofold and fourfold symmetries with a representation of higher dimensional cubes which in 3D space physically visualises the gap between the binary terms of our western occidental culture. 4.2 Urban scale There are mainly three types of geometry in historical and modern city plans: - the linear type, along a communication and transportation line e.g. a road or a river - the concentric/radial type, around a fortification e.g. on a hill; or an important crossing of roads or rivers - the grid type, which already needs supervised and ordered urban planning from the foundation of the city on e.g. Chinese cities of one planning concept. The first two types are a logic reaction to the environment. The third type is a rational decision of human planning and the need of ordering nature for different reasons. It is indeed very interesting, that the diffraction pattern of a QC with its tenfold symmetry is very similar to radial city plans. One idea is to superimpose a penrose pattern on a radial city plan, so to say to show the virtual QC lying above the city, creating the city-plan with its diffraction pattern. The penrose pattern would in general be a way of planning new city-grids, still ordering spaces to a certain extent, but also regaining the liveliness of a more flexible and less prohibiting grid like twofold or fourfold symmetry. Its snakes could be the streets; a vertex star consisting of five fat rhombs in fivefold symmetry could be a street-circus or a plaza. Chaos equals with Nature, and Order with Culture and Civilisation in our society. Cities grow from village to megalopolis, become huge urban areas which need to be restructured to give a more natural feeling of environment to the people that live in them, as real nature gets pushed back to the worlds most hostile regions as deserts and beyond the polar circle. The fractal possibilities by binary substitution can be used for constituting spaces in different scales like the city centre, or the centre of a suburb and still the overall grid would not be broken. Todays cities are a mix of individuals from different cultures, countries, religions,
  • 6. colours and continents. Therefore it has become necessary not to symbolise the equality of individuals in a democratic system by a simple order, equal for all its users, but a complex and diverse pattern with possibilities for all the different individuals coexisting in one place. Architecture has always been aiming to control spaces and therefore also actions and users. We need to give freedom and possibilities to develop now, so QC buildings on a penrose grid would combine this freedom without totally loosing control of the city grid, and slipping into anarchy - the essence of quasi crystallisation. 4.3 Wide Span and High-rise Buildings QC buildings in a large scale can be used to room all kinds of uses and functions. The tilting of the walls of QC buildings gives tension to the perception of the street space, as they seem unstable and to come towards the pedestrian, not like the usual psychological stability of vertical lines in buildings reacting to earths gravity. An example for this type of tension, although not a QC, is the Royal Library Extension in Copenhagen by the Danish architects quot;Smidt, Hammer & Lassen, a distorted cube also called the Black Diamond. QC buildings are by this a real statement of human design and its interaction, and a starting point to overcome physical laws. The angles of the tilting floors can be used for large auditoriums, e.g. for an opera, concert- halls or universities, horizontal split-levels can be mounted in a QC space-grid with connected air-spaces, as horizontal plates hovering through space. A large Rhombic Triacontahedron with its five horizontal planes spiralling around the axis would be one possibility for a communication organisation within such a building. The stairs lead from one horizontal floor to the next, the elevator-shafts along the axis through the middle, and the auditoriums leading down and away from the floors opening to the outside views into all directions. (Fig.7) QC domes and space-frames can be mounted over street spaces or plazas to create Galleries looking like reptile skins growing over the citys open spaces. Single layer QC structure can also react and move with or against the landscape, creating amorphous convex and concave spaces between the structure and the ground and in the quot;foldingsquot; of the undulating QC surface. This can be used to cover large areas with temporary structures e.g. for fairs (Fig.8). Fig.7 Spaces within a Rhombic Triaconta- Fig.8 Single-layer structure, stabilised hedron with combined lattice and plate action. The advantage is, that as ordinary twofold and fourfold building structures usually need flat ground, and do not react at all to their environment with their unnatural shape, QC structures are possible literally everywhere, not violating the Nature to fit them.
  • 7. The fractal possibilities can be used to shift the scale within a building from big halls to normal room size areas to an open space with small-scale space-frames above it. For example this could structure an airport from the check-in hall to offices, waiting rooms, shops, and communication and transportation areas. It can also be used to create large buildings with about brick-size QC rhombohedra, as if the rooms of the building where just carved out of a gigantic QC. The walls and floors of such rooms would then be undulating in a small-scaled pattern, which would create the texture of an organic grown structure. 4.4 Dwellings QC structures for dwellings can be used in all the existing building types, as detached houses, single houses, for rather low suburban housing areas, mixed with garden spaces, or middle- scale housing blocks, with public park areas. If one rhombohedron is one room, there are e.g. the eight vertex-stars that could be a way of organisation. In the penrose pattern the quot;snakesquot; could be a possibility for the detached houses. The quot;snakesquot; could also be the streets, as the borderlines for the housing clusters in various shapes with inner courtyards and gardens. Another way is to cluster the single rhombohedra to a 3D QC into all directions of three- dimensional space. Fig.10 Irregular stacking of rhombic Fig.9 Regular stacking of rhombic triacontahedra triacontahedra Fig.11 QC structure with growth Defect Fig.12 Interior Spaces of a QC A Rhombic Triacontahedron could be the minimal living unit in a 3D cluster, with its rhombohedra as the different rooms, with some split-levels or stepped podiums to create some necessary straight floors. As there is regular and irregular ways of stacking these units and attaching them to each other, houses can be adapted to different family sizes. The regular stacking (Fig.9) creates holes and is not a complete QC aperiodic structure. However these
  • 8. gaps can be used for courtyards and gardens, or shafts to bring daylight also to the lower levels of higher clusters. The irregular stacking (Fig.10) creates a 3 D QC grid, in which gaps for the above mentioned uses can be made by leaving out some rhombohedra. There is also a mathematical algorithm for producing penrose patterns with incorporated local growth defects, which yet retain the semblance of fivefold symmetry (ref.12, pp.86-96). These non-tileable parts could grow through a real 3D QC like wormholes (Fig.11). QC-colonies have almost no limits in how high or how far they can go, as each Rhombic Triacontahedron can be constructed as a stable unit and therefore able to be a rigid building block in a larger configuration. The QC grid in the cluster is supporting itself and all its parts (Fig.12). 5. CONCLUSION It seems that the special QC geometry can be used in a new and advantageous way as a grid at all levels for architectural structures, town planning, interior design etc. not least because of the fascinating relation to both chaos and order. Edwin A. Abbott's quot;Flatland - a romance of many dimensionsquot;, written in 1884 (ref.13) inspires to consider if there might be a 5D world with 5D beings somewhere in our universe - or rather beyond it - and how it might look like. As the 2D creatures in Flatland only perceive two dimensions as living in a surface, we do not in our world have any sense which can perceive a higher dimension than our usual three. As QCs are regular and periodic in 5D space a QC is just another quot;boringquot; cubic building for a 5D inhabitant It appears extremely inspiring that we can build interesting architecture of their 3D representations. Higher dimensional spaces and structures might be used as perfect tools for the architecture of the third millennium. Only time will show if this potential is going to be utilised in our societies, and not only as futuristic ideas for space stations and moon settlements. It is about time to start something new. 7. REFERENCES 1) Senechal, M. (1995) Quasicrystals and Geometry. Cambridge University Press. 2) Wester, T. (1984) Structural Order In Space, the plate lattice-dualism. School of Architecture, Copenhagen. 3) Wester, T. (1997) The Structural Morphology of Basic Polyhedra in Beyond the Cube (ed. J.F.Gabriel) pp.301-342, John Wiley and Sons, Inc. 4) Wester, T. (1987) Plate Domes, Proceedings of the International Conference on the Design and Construction of Non-Conventional Structures, pp 241-250, Civil-Comp Press, London 5) Schectman, D. et al (1984) Metallic Phase with Long-Range Orientational Order and No Translational Symmetry. Physical Review Letters, Vol. 53, 1951-3 6) Penrose, R. (1974) Pentaplexity, Bulletin of the Institute for Mathematics and Applications, Vol.10. pp.266-271 7) Lalvani, H. (1994) Hyper-Geodesic Structures, IASS Atlanta. American Society of Civil Engineers. 8) Miyazaki, K. (1986) An Adventure in Multidimensional Space, John Wiley and Sons 9) Robbin, T. (1996) Engineering a New Architecture, Yale University Press. 10) Jencks, C. (2000) Architecture 2000 and Beyond - Success in the Art of Prediction, Wiley Academy. 11) DOMUS no. 671 (April 1986) pp 17-24. Interview with Jaques Derrida 12) Peterson, I. (1990) Islands of Truth, A Mathematical Mystery Cruise 13) Abbott, E.A. (1984) Flatland, A Romance of many Dimensions. (first published in 1884) Signet Classic.