Explorations of Tessellated forms for Architectural Application - Dissertation Example

?Explorations of Tessellated forms for Architectural Application This investigation explores the connections in tessellation and architecture focused upon the construction markets in Australia. The history and concept of tessellation is described including historic applications as well as present, modern usages. Trends in the Australian housing market are discussed, with connections drawn to emerging technologies as a means of expediting architecture and the mechanical process of construction. The principles of tessellation and 3-D printing will be discussed with proposed applications for architectural usage, as well as a discussion on the ramifications for the construction market. Table of Contents Introduction …................. p. 4 Literature Review …................. p. 7 Conclusion …................. p. 20 References …................. p. 22 Introduction Architecture is among the principal forms of cultural expression available to any society, and represents a vital synthesis between technology, aesthetics, and the way in which a society perceives itself. Part of this perception is dependent upon the ability to adapt to modern technological realities and adapting to them accordingly. This investigation includes two new technological dimensions applied to theories of architecture to provide a new predictive element for the future growth of the design and construction principles inherent in architecture. The design principles upon which the urban landscape depends touch every aspect of modern life, from the aesthetics involved in fashion or jewelry, to the functional tools developed for mechanical or surgical work, similar design elements and technological solutions are applied which can inform urban design and architectural theories that shake our living societies. Design, architecture, and art in general form and intersect that will be informed by new advances described during the course of this analysis to create potential for a new dimension of design techniques applicable not only in architecture, but throughout a wide range of disciplines. This study will explore a possible future of architecture made available through the adaptation of two relatively novel scientific concepts: 3-D printing and tessellation. In a broad sense, the theories of tessellation will be explored at length, combined with the new potentials inherent in 3-D printing, which will be explored as a functional mechanism. The theoretical principles of tessellation are applicable to the technology of 3-D printing, and this investigation will explore possible developments of this design system for use in architecture. In terms of current trends in Australian architecture, it is necessary to remain cognizant of the aftereffects of the recent global financial crisis, however these international financial issues have affected Australia somewhat less than other countries. Regardless, it is inevitable that an interconnected global economy will still impact every advanced economy. Markets affecting new buildings and home design are nonetheless influenced by international issues, and as a result demand exists for structural flexibility as well as quality (Zenere, 2013). A trend towards increased flexibility underscores the need for a revision of old assumptions, and an adaptation of new technologies to meet the diversity of human needs with a greater range of options than in the past. Renewed interest in a flexibility of options will create a demand for novelty in terms of architectural design options, as old standards are revised to make way for the new economy. In addition to flexibility there is a demand for uniqueness within the reasonable boundaries of domestic comfort. Zenere (2013) describes a modern Australian home market where customers are interested in airy spaces with ample illumination and adaptability in terms of its overall design, in order to accommodate flexible living conditions. The modern economic circumstances prevailing throughout much of the industrialized world creates a social dynamic which reflects the new demands placed upon contractors and by extension architects. In troubling economic times, it is not unusual for growing children to be compelled to return home and as a result of souring financial and employment prospects abroad. Upon returning home, it would be beneficial if the house itself exhibited some degree of universality or customization in the way the rooms are established. This would suggest multipurpose rooms as the most immediate demand, but in Australia – as well as much of the industrialized world the goal is no longer to own one of a number of mass-produced houses all arranged in a neat row for each dwelling is identical to the one before and after it. There is a necessity for the family or organization in the case of a business, to have more latitude to structure the building the way they want, keeping in mind their individual needs, and the most probable permutations of their own individual circumstances. Families able to own homes must contend with the possibility that growing children may need to return. Even if this is not a factor, many homeowners will have elderly parents and they may wish to have the option of converting a room to accommodate these relations. With each family circumstance different, there is a need for more autonomy in terms of the house purchased, and how that house or building is initially designed, or how expansions upon the property might be constructed. A flexible floor plan is needed where building materials are used creatively, and efficiently as well. The end product representing a practical reality where aesthetics are combined with utility. With increased financial constraints, it is important to get the maximum value out of a home or commercial building purchase, this requires more input from the buyer at the planning stage of the construction process, and these needs can be more adequately met by novel technologies employed in the process of constructing the building. On a basic level, this analysis will connect the needs of the new economy in terms of the construction market with two new innovations that demonstrate potential for greater flexibility in terms of the construction process. Initially, it will be necessary to describe the theoretical concept of tessellations, and relate them to the needs of modern construction and architecture. But it is essential to explore the basic theoretical underpinnings before discussing practical applications relating them to the process of design and construction. To that end we will explore tessellations. Literature review In an etymological sense, the word tessellation arises from an original Latin word 'tessera' which refers to a small, regular, stone cube. These were used for artistic purposes in Roman, and later Byzantine mural designs where mosaic patterns were used to decorate walls, floors, and ceilings. Important in both mathematics and architecture for the ways in which they can be manipulated, as indicated by the work of Dutch artist M.C. Escher who prolifically utilized tessellations in his own artworks. A tessellation therefore describes any repeating pattern of symmetrical or interlocking shapes, in fact it is essential that they be both to preserve the perceptual integrity of the image (Dryden et al. 2006), or the architectural integrity as the case may be. They may also be referred to as tilings, but while this term typically refers to a pattern of polygons, it is possible for tessellations to exist which do not conform to straight sided geometric ideals (Avram & Bertsimas, 1993; Barany & Reitzner, 2010; Chen & Cowan, 1999). What matters is whether the shapes interlock. Historically, tessellated forms have been found dating back to ancient Sumerian civilizations circa 4000 BC (Khaira, 2009). The mosaic pictures most individuals are familiar with comprising thousands of differently colored stone cubes or tessera are described collectively as 'tessellata', the complete mosaic image used as a decorative theme. A mechanized outgrowth of this principle may be adapted for modern use. These uses include decorations and floors and tiles, where it is not strictly speaking necessary for tessellations to interlock physically, a geometric pattern of interlocking shapes could certainly appear as a painted facade on a linoleum floor, but whether implied or material these shapes must be non-overlapping with the ability to interlock. However, in modern parlance the terminology of tessellation has been adapted to describe recently developed design principles and objects. In the original Latin, the meeting was limited to a small stone cube, but this narrow view is no longer strictly necessary; a tessellation could refer to a repetitive object arranged in a pattern of virtually any geometric shape or two-dimensional form (totallytessellated.org. 2013). A tessellation could take the shape of a recognizable silhouette, such as the shape of other devices or popular machines, or even animals or recognizable human profiles (Tesellations.org, 2013). What matters is an easily recognizable silhouette form that can be arranged with a regularity along a flat surface. This implies certain design constraints in the original usage of the term, which will be expanded upon later in this analysis. To form a flat image, whatever shape the tessellation takes it must be standardized enough that numerous pieces can interlock together in a predictable manner. Squares obviously work well for this purpose, but are not the only possible design. An intricate shape with numerous irregular projections would not function for this purpose, unless the irregularities are contained within a more basic structural element. Later in this investigation, these principles will be expanded to include shapes not limited to two dimensions. But the same overall constraints must apply: a repetitive shape must exist of the structure that can be interlocked with others of the same shape. Too much irregularity, and the higher meaning or function will be lost. For the construction of an ancient mosaic, if the pieces did not interlocked together with sufficient regularity, it would be more difficult for the human eye to perceive a higher order of structure emerging from the design. The intent of course, is not to focus on the stone cubes themselves – the human mind is able to perceive an image as an emergent property arising from the arrangement of the stone cubes themselves. Regularity becomes more essential if the principle of tessellation is taken off of the two-dimensional floor, and employed as a structural element. In this case, the challenge becomes an issue of satisfying the laws of physics, rather than the perceptions of the human mind. While there are fundamental principles that a tessellation must satisfy in order to qualify as such, there are numerous sources and examples from which to derive inspiration: tessellations frequently occur in nature. The firm markings of certain animals may interlock in some circumstances, in addition to reptilian scales, the scutes on a turtle shell, in addition to the obvious examples of honeycombs. Nor was the Greco-Roman civilization the only one to employ tessellations and architecture. --- Figure 1: The Blue Mosque in Istanbul is an example of the artistic motifs typified in religious settings with a wide range of interlocking images, some of which form radial patterns. With radial patterns it is not strictly speaking necessary to eliminate gaps between interlocking units, as the dome of the Mosque illustrates. Image from Khaira, 2009. Figure 2: An Australian example of the principles of tessellation an architecture would be Federation Square in Melbourne. This structure is comprised of an interlocking pattern of right angled triangles arrayed in a series of interlocking sectors. Khaira, 2009. The structure comprised of the individual units must truly become a picture in order for the design to succeed. This creates a necessity for interlocking forms with the minimum of openings or gaps. These would be distracting in a mosaic image under most cases, and potentially dangerous if used as an architectural element. Individual tessellations also must not overlap one another or the two-dimensional structure will not be truly flat, and over locking sections will prevent the pieces from interlocking with each other. As architectural elements manufactured individually, this becomes a logical impossibility – but it is a characteristic of a tessellation that the pieces should not overlap, but must interlock with each other to form a regular structure (tessellations.org, 2013). There are multiple applications and sources from which tessellations can be derived, as well as a variety of means by which patterns of images can be realized through tessellations, and while interlocking forms are essential – gaps are not entirely undesirable in all cases, as the radial examples from the interior of the blue mosque demonstrate. On the interior of an individual unit such gaps may be part of the information the pattern conveys. But structural regularity where the tessellations meet each other must be maintained. While tessellations represent patterns of predictable regularity, this broad principle is still useful even in an era where many Australians are less interested in cookie-cutter duplicates for homes and other necessary building designs. While it may seem counterintuitive, the desire for flexibility and adaptability can potentially be served very well by the replicant regularity of tessellation techniques. This is a consequence of the potential for greater efficiency of production by incorporating novel techniques for the fabrication and specification of a fundamental components of the building. Later sections will discuss the phenomenon of 3-D printing where these potentials will be more fully realized. For the purposes of this study, it becomes necessary to explore replication or copying between different tessellations as unique cells. If used in an architectural context, some form of mass production will be necessary, and an exploration will be conducted concerning theories of new tessellation shapes, reproductive/iterative process. Research conducted by Cowan, (2009) describes novel theories of random tessellations and ways to generate regular patterns through a form of successive division of the individual cellular components. The theories behind new tessellations through a replication process has important implications for the purposes of large-scale manufacturing, and by extension architecture which will be explored in greater detail in a later section of this analysis. From a series of broad concepts, it is theoretically possible to generate new tessellation models which can be subsequently replicated in large numbers. Further research will expand upon the potential in many dimensions of art and design. The work of Cowan in a broad sense requires unbounded convex cell or window radiating from a centralized origin point. From this starting point, research into mathematical formula provides a series of division functions, also called chords which will trigger a dilation effect in the bounded unit that should lead to a new tessellation. A division process is proposed where a planar dilation occurs once for each generation, or epoch. The division function during each epoch is selected in order to maintain a mean/average unit area equal to one. This theory posits a division of each presumed tessellation in accordance with a linear factor which can be described by an equation as the square root of T +1 divided by the square root of T, where T equals time. This replication system has the potential to produce a predictable tessellation of the unit area with mathematical regularity. Additional functions can be researched to adapt this technique for higher dimensions. To produce regular tessellations with the interlocking properties essential to both structure and aesthetics, fractals should be avoided (Cowan, 1997; Cowan and Chen, 1999; Cowan 2009). This work builds on earlier findings in stochastic geometry by Nagel and Weiss (2005), which explores additional mathematical models for the iteration of tessellations, again seeking to devise a mathematical process whereby the geometric unit can be subdivided into interlocking, nonoverlapping tessellations the repeated replications. Nagel's process requires a frame tessellation from which new versions can be generated through a series of subdivisions. Research from various investigators implies that the iteration process can be replicated repeatedly (Mannion, 1990, Cowan, 2004; Calka, 2010), and under a range of conditions and materials (Gerstein et al. 1995; Goldman & Calka, 2003; Heinrich et al. 2006) and more importantly for the purposes of architecture – the function is scalable. In a purely mathematical perspective, there are uncertainties concerning whether the iterations can proceed to infinity, but this is not entirely relevant for the purposes of architecture. Only a limited degree of scaling would be required. There is another way to describe in the pattern is through the observation of homogeneity. The mathematics by which a tessellation can be predicted depends upon whether or not it is homogeneous, which can be thought of as the degree by which the cell inhibits spatial stationarity (Mecke et al. 2007). This is analogous to the regularity and nonoverlapping requirements described above. These geometric properties must be stable under iteration, which is likened to a process of cellular division – as would occur in biology. Mecke and colleagues described the individual shapes of a tessellation as components realized through a series of point processes which – depending upon the way they are arranged form all the standard, familiar polygons that are readily recognizable (Mecke et al. 2007). The complex theories described by Mecke (2007) and Cowan (2009) form the underpinnings for a process by which greater potential for customization becomes evident through a projected potential to design the design process itself. The ability to devise machines that will replicate a diverse array of modular components has considerable potential to make architecture more cost-effective. This is a high priority according to housing market investigations by Zenere (2013). While it is a truism that homebuyers want high-value at lower costs, the prevailing conditions whereby a large portion of the market for construction work feels the need to tighten their budgets will force builders to comply. Technologies that allow more autonomy in terms of how the building is designed using what components will fit in with the new demand for efficiency. With diverse family circumstances, and the increased drive for efficiency the demand is effectively gone for buildings including large empty spaces that create a drain on the heating and cooling budget for the family or organization. The building design should be tighter, but buyers want more control over the eventual structure of the house or commercial building, which new construction techniques may permit. Some ways to achieve this could be to remove unnecessary walls, allowing additional living space that can be adapted as needed – but within tighter confines than in decades past. Registered architects must look for options to optimize living space based upon the limitations of the construction site, combined with a consideration of the specific needs of the family or organization in question. More customization and communication are therefore essential in the new construction market. One of many ways to achieve this new desired customization is through the new technology of 3-D printing. Once available only to the extremely wealthy, prices are lowering and affordability is increasing for a new series of printing systems with numerous applications in robotics, electronics, as well as engineering. For these fields it is easy to extrapolate the expansion of capabilities to include components useful in the design of buildings and/or houses. In Australia, educational initiatives are making 3-D printing available for universities as well as high schools as the cost of the equipment drops. It is beneficial for educational purposes, which will help to familiarize future generations of potential architects with these techniques for the benefit of society as a whole. 3-D modeling can function through already existing software such as Google sketch up or similar methods by which an individual's own designs can be manufactured automatically using equipment of increasingly smaller sizes and cleaner operation. It is becoming popular in Australia, and is already widespread throughout the UK (Ward, 2011). Some schools attempt to incorporate them as an official component of the curriculum. Various models are available - some of which are complete right out-of-the-box, while others can be assembled by students as part of their education. The relatively inexpensive 3-D printer represents an output device to translate a virtual model into a fully assembled, physical object composed of a malleable plastic or resin that is mechanically assembled within the device to reach the desired shape and dimensions. The plastic is heated to the melting point and piped through channels where it can be released in a coordinated pattern by robotic manipulator/depositors. The engineering applications are self-evident. The devices are useful as sources of experimental test models, as well as finished components (Ward, 2011). In addition to the apparent uses as a source of building materials, 3-D printing also has a attentional in the planning phase of a project in ways that are not immediately obvious. It is also possible to program these devices to replicate not only objects, but also topographical features. It will be possible for an architect to quickly and expediently generate real physical mock ups of the terrain surrounding a construction site in the situation exists where a physical model would be advantageous or supplementary. This has obvious potential in the interest of providing simulations for future projects, and research in similar areas (Fleischer et al. 2008) wherein which it would be possible to create a precise replica of the proposed building and surrounding terrain for planning or demonstration purposes, in addition to the process of actual manufacturing. On a small scale, the purported feasibility of a particular object could be analyzed to better support calculations and estimates regarding dimensions and constraints of a particular building proposal. While most 3-D printing takes the form of monochrome objects using a single layer of resin or plastic, depending upon the model far greater complexity is possible. Devices exist capable of simulating highly complex devices such as medical equipment for training purposes (3D-Print.com.au.2013). At this point, it is not difficult to project extending this technology further to someday create real functional components if a variety of substances were to be utilized during the same printing operation. At advanced levels, 3-D printing represents a means to translate an idea into an object or a system including laser beams, plastics and powdered resin. High-powered lasers can be used to sculpt objects from the basal resin. Another use for laser beams during the process is to employ stop motion photography whereby the path of the low yield laser is scanned and recorded in a computer which later replicates the shape. This technology represents enormous avenues of potential beyond the most obvious in terms of cost effective construction. Many of the expenses entailed in the construction process, as well as many aspects of heavy industry involve storage and warehousing after trade goods or building components have been manufactured. Applications of 3-D printing theoretically allow an elimination of many of these inefficiencies. Even if an individual does not possess access to a 3-D printer themselves, software modeling or stop motion photography could be used to create a data file detailing a three-dimensional object. This data file could then be sent to a retailer or manufacturer that does have access to a large-scale printer, and they would simply be able to produce the device on demand in response to a specific customer request – in quantities determined by the customer. Technique is very, some 3-D printers use very similar tools as a common inkjet printer suitable for flat documents, but employing thin layers of powder glued together in an elaborate pattern leading to the assembly of an object (3D-Print.com.au.2013). These techniques are relevant also due to severe limitations that constrain the marketplace for both construction and industry under present conditions. Manufacturers and retailers must make a series of difficult marketing decisions on a regular basis: what materials to produce and stock in what quantities? Present production techniques require choices concerning what products may or may not be in demand at what time. The most accurate estimations may still overestimate what customers are willing to buy; leading to waste and excess inventory that must be resold at a lower rate, or disposed with. 3d printing technology - if developed sufficiently can partially alleviate this issue. Multiple industries would have the freedom of efficiently fabricating devices and products as needed to fill customer orders placed in advance. In addition to efficiency in terms of expediting manufacture through automation, the capabilities inherent in 3-D printing combined with the knowledge of tessellations could be further researched to lead to advanced materials sciences where structural qualities are optimized from the beginning. Anyone who has observed the reinforced arches supporting a highway or railway bridge above a ravine or river has had a chance to observe the structural advantages inherent with an understanding of weight and force distribution. This principle is readily expressed in a variety of simple products, especially those forged entirely of metal. Many objects composed entirely of metal seek to optimize the mass to weight ratio. Support structures or metal tables might include panels with deliberate gaps and open sections as part of the support. A metal handle for a knife or similar tool may have a hollow structure incorporating numerous gaps. Yet these designs are efficient for their intended purpose based on an investigation of the force distribution. A tool handle composed entirely of solid metal would be heavier, and any benefit in durability would not compensate for the greater weight compared to a hollow design. Therefore, many designers who work with such materials find it to be more cost-effective to include only as much metal as is necessary to fulfill the most probable stresses the object will be placed under, and using gaps in the structure will allow the overall device to weigh much less. Modern engineers working with an understanding of the distribution of force can therefore create a structure in practical terms just as durable as a solid piece of metal, but at a reduced weight. An arch supporting a bridge for practical purposes becomes just as strong as an entirely solid mass, but in a dramatically reduced cost in terms of time and material needed to build it – while allowing traffic to pass underneath the hypothetical bridge. These same principles of force distribution commonly used in metal working could be extrapolated onto a smaller scale. Using the right physical principles, it would be possible to design tessellated micro units that can be assembled efficiently with 3-D printing incorporating interior hollow structures while still bearing for practical purposes the same amount of weight as an entirely solid mass. If implemented on a large scale, it is theoretically possible to develop concrete and steel equivalent materials able to bear the same load if ordered correctly, but without the same weight. While researchers are constantly investigating possible new materials and new strategies for materials engineering now and in the past (Chanson & James, 1998), combining 3-D printing would expedite the synthesis of micro units usable in the actual construction of advanced materials. The mathematics of arches might be employed at a far smaller scale to duplicate the same advantages while reducing even further the need for solid mass. The mathematics of constructing an archway for example, represents the resting dimensions of a hanging a chain supported at either end settling into its most stable configuration possible (Osserman, 2010). Further research might invite investigation into the physics of paired arch like structures perhaps arranged parallel to one another, but inverted in order to support exerted pressure from either direction. Additional research would be necessary to determine what size of tessellation will take best advantage of the physical properties of metal (or a different material under discussion) in order to reduce the effective strength per weight. Thus future research could focus upon the possible physics of an arch made of arches all interlocked together in a non-overlapping matrix. It will be necessary to determine how small such tessellations might become before the difficulty in shaping them outweighs the advantages of reduced weight based upon costs of the material. Reducing the total weight of the building material has considerable advantages in terms of structural integrity and ease of transportation, so long as durability is not compromised. Challenges to overcome would be finding the correct size where the practical strength of the material is not significantly diminished while lowering the weight and mass. It is probable that such tessellations must still be larger than nanoscale structures, the complexity of the instrumentation required at that scale would likely add prohibitive additional costs. Future research should focus on the discovery of a happy medium. In any event, something very like 3-D printing would be necessary to efficiently replicate near microscopic structures such as those proposed. Conclusions At this point additional speculations concerning viable applications for 3-D printing in conjunction with other principles described in previous sections is appropriate. If industrial models were developed with the ability to manipulate heavier materials it will be possible to easily fabricate repetitious objects such as bricks rapidly and on demand. Depending upon the chemistry involved, it is possible to envision an industrial printer able to compile tessellated units composed of calcareous substrates deposited under malleable conditions which could be hardened later into concrete. Moreover, it should also be possible to blend the efficiency of mass production with the flexibility desired by the modern construction market as it exists in Australia and other countries. It will be possible to mass produce structural components very quickly, but with the modification of a simple data file this mass production could be readily and easily modified based on present need rather than extrapolated future market projections. With further development, additional applications become feasible – such as the rapid synthesis of fiber-optic wires combined with their assembly into an ordered cable. Further extrapolations invite speculation on high-powered, industrial-strength 3-D printers with enough power to melt certain malleable metals. At this level of power it is conceivable to create a system capable of automated synthesis of copper wiring with insulation already included. These applications are well worth future research. The changing housing market creates a persistent and pervasive demand for multipurpose functionality in homes as well as commercial buildings. Versatile floor plans are highly desirable, yet without wasting space or creating areas where appropriate air-conditioning based on the season becomes ineffective. Any technologies with the potential to lower costs while permitting greater flexibility in the construction process itself should be favored by market conditions in a technological infrastructure. This versatility is partially demanded by external advancements resulting in improved lifestyles and longer lifespans for most citizens of industrialized countries in any case (Zenere, 2013). The need for elder support, in addition to the aforementioned possibility of adult children returning home creates a need for efficient, yet adaptable dwellings which will benefit from more versatile construction and architectural options. In terms of immediate implementation, Cowan's formula (2009) described above in addition to others like it could readily be incorporated into the software behind 3-D printing as part of an adaptive system to generate tessellated patterns for use in the synthesis of architectural components. If these principles could be Incorporated with the hanging chain guide line of arch construction new possibilities in architecture and heavy industry could be realized. References 3d-print.com.au., 2013. 3D printing Pty Ltd. 3d Printing Systems. http://www.3d-print.com.au/#!. Accessed: 10/19/2013. Avram, F., Bertsimas, D. 1993. On central limit theorems in geometrical probability. Ann. Appl. Probab. 3, 1033–1046 (1993) Barany, I., Reitzner, M. 2010. 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Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration. Adv. Appl. Prob., 37, 859–883. Osserman, R. 2010. Mathematics of the Gateway Arch. Notices of the AMS. Volume 57, Number 2. tessellations.org. 2013. What are Tessellations? http://www.tessellations.org/index.shtml. Accessed: 10/18/2013. totallytessellated.org. 2013. Totally Tessellated. Regular Tessellations. History, Essentials, Mosaic/Tiling, Escher, Beyond. http://library.thinkquest.org/16661/simple.of.regular.polygons/regular.2.html Accessed:10/20/2013. Ward, R. 2011. 3D Printers In Education. 3D Printers Australia. 3d printers for a 3d world. Copyright © 2010-2011 3D Printers Australia. http://www.3d-printers.com.au/2013/10/20/3d-printers-in-education-2/#more-3122. Accessed: 10/19/2013. Zenere, A. 2013. Architecture Trends 2013. Archizen Architects. Buildhome, Architecture Trend Report. http://www.archizen.com.au/file/Architecture_Trends_2013_-_by_Adrian_Zenere.pdf. 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