Tessellated, Tiled, and Woven Surfacesin Architecture

Michael J. Ostwald

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Background to Tiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Tiling in Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Abstract

This chapter is focused on the tessellation, tiling, and weaving of architecturalsurfaces. These three processes result in the production of a geometric patternof connected shapes which cover a plane. In architecture, such techniques aretypically employed to create a more durable or weatherproof finish for a floor,wall, or ceiling. But they also provide a means of decorating a surface to achievean aesthetic, poetic, or symbolic outcome, some of which are used to evokeparticular mathematical properties. This chapter provides an overview of thedevelopment of architectural tiling, highlighting key connections to mathematics.The architectural examples range from simple Neolithic weaving and stonecutting practices to late twentieth century aperiodic cladding systems in majorpublic buildings. The chapter also refers to past research into tiling in architectureand the primary themes which have been examined in the past.

KeywordsArchitecture · Tiling · Tessellation · Weaving · Aperiodic tiling

M. J. Ostwald (�)UNSW Built Environment, University of New South Wales, Sydney, NSW, Australiae-mail: m.ostwald@unsw.edu.au

© Springer Nature Switzerland AG 2019B. Sriraman (ed.), Handbook of the Mathematics of the Arts and Sciences,https://doi.org/10.1007/978-3-319-70658-0_86-1

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Introduction

Nikos Salingaros (1999) argues that throughout history, patterns have provided afundamental connection between architecture, mathematics, and society. The humanmind is not only drawn to identify patterns in nature and society, but also to developnew patterns and visually represent them to make sense of their applications orconsequences. Salingaros postulates that at its most basic level, this socioculturalaffinity for patternmaking and identification “explains the ubiquitousness of visualpatterns in the traditional art and architecture of mankind” (1999: 76). He evensuggests that tiled, tessellated, or woven geometric arrangements are the “visibletip” (Salingaros 1999: 76) of mathematics in society, providing an importantlink between abstract ideas and their practical application. In a sense, even themost basic tiled patternmaking in architecture is significant because it representsboth a pragmatic and poetic connection between mathematics and the world. Butfor Salingaros (1999), these basic geometric patterns that were once pivotal tomaintaining this connection have not always been celebrated or respected. Forexample, in the first half of the twentieth century, he proposes that Modernistarchitects devalued the role of the tile, and since then, most architects have eitherignored it or forgotten its potential. The reasons why Modernist architects devaluedtiling provides an interesting insight into the differences between architects andmathematicians attitudes to tessellations.

The Modern movement in architecture, which developed in Europe and Americain the early decades of the twentieth century, had a fascination with large-scalesculptural form-making, functional expression, and mechanized production (Ost-wald and Dawes 2018). As such, Modernist architects tended to valorize materials,like concrete, steel, and glass, which did not need applied finishes, and they alsocalled for the rejection of all ornamentation and decoration. For example, Europeanarchitectural theorists like Adolf Loos (1998) rejected tiling on the basis that it wasunhygienic (having too many unnecessary joints), socially biased (as it requireda particular class of artisan workers), and culturally inferior (because geometricpatterns were often used in primitive or tribal art in Africa and Asia).

The only tessellations or patterns that Modernist architects found acceptablewere those that arose as a by-product of large-scale industrial manufacturing.Thus, the simple rectilinear gridded or “stacked” pattern produced by the use ofprefabricated components was considered “honest” because it was an expression ofthe basic properties of the material (Fig. 1). The standard “stretcher” bond tilingarrangement was considered “functional” because its overlapping pattern enhancedits structural properties (Fig. 2). However, a “herringbone” pattern was regarded as“dishonest,” because the material was being used to achieve a decorative and therebymorally debased outcome (Fig. 3). These three tessellations – stacked, stretcher,and herringbone – are all somewhat trivial in a mathematical sense. Indeed, theycan be constructed using exactly the same components or tiles and they all cover asurface in an efficient manner. But for a Modernist architect or theorist, the choiceof patterns had additional ethical, moral, or symbolic implications which also hadto be taken into account.

Tessellated, Tiled, and Woven Surfaces in Architecture 3

Fig. 1 Stacked pattern

Fig. 2 Stretcher or runningpattern

While a detailed analysis of early Modernist attitudes to patternmaking is beyondthe scope of this chapter, they do provide an insight into the complexity of thetopic. For a mathematician, a tessellation may be expressed using an algebraic oralgorithmic notation, and its essence may be interrogated or classified using severalrelated methods. For an architect, a tiled or patterned surface must be appropriate

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Fig. 3 Herringbone pattern

for the environmental conditions of a site, the proposed function of the building,and its budget. In addition, a pattern might also evoke social or cultural propertiesand symbolize values or communicate particular messages. Paradoxically, it is theselatter, more aesthetic or poetic properties of architectural tiling which have beenmore closely associated with developments in mathematics. Indeed, in the lastdecade of the twentieth century, there was a renewed interest in complex tiling inarchitecture which has continued to the present day.

Despite shifting attitudes to tiling, a recent framework for classifying connec-tions between architecture and mathematics identifies “surface articulation” as aconsistent theme throughout the history of architecture. Surface articulation isdefined as the “use of mathematics to achieve an efficient or controlled coverageof a defined plane” (Ostwald and Williams 2015: 40). This category encompasses“empirically or intuitively derived methods for achieving a waterproof, or wind-proof [ . . . ] surface,” along with the use of geometric tiles “to achieve an intricate,patterned surface covering” (40). These definitions encompass both a sense ofthe pragmatic and aesthetic implications of tiling in architecture. However, in thischapter, the focus is only briefly on the practicalities of tiling, as some of the mostinteresting connections between architectural and mathematical tessellations oftenoccur because designers are inspired by particular developments in mathematics.

This chapter provides a background to the history of tiling in the built environ-ment and an overview of parallel developments that have occurred in mathematics.However, before progressing, there are five caveats that must be stated. First, themajority of the “architectural” examples discussed in this chapter are interior orexterior surfaces, rather than complete buildings, and as such they could also be

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regarded as being drawn from the fields of interior design or façade design. Second,this chapter also discusses woven surfaces, treating them as a special type of tiling.The reason for this addition will become clearer in the following section, but theearliest patterned surfaces created in primitive societies were typically woven, andtheir resultant geometric expression was similar to a tiled or tessellated surface,even if they were not produced in the same way. Third, the majority of this chapteris about surface or “plane-filling” tiling. However, it is also possible to talk aboutthe tiling or tessellation of space, which is associated with “space-filling” properties.Just as a square tile can readily cover a surface creating a gridded pattern, a cubictile can readily fill a volume in a lattice. With a few exceptions, architecture’sinterest in tiling has been dominated by the former category, surface tessellations,and examples of the use of volumetric tessellations are less common. Finally, whenartists, designers, or architects are inspired by mathematics, they tend to adapt,rather than adopt, the language of mathematicians. As such, the ways architectsdescribe tiling are not consistent with the nomenclature used in mathematics.Furthermore, architects’ knowledge of contemporary tessellation theory is limitedand most often tiling patterns are applied without any deep understanding of theirmathematical properties. This last caveat should be kept in mind when reading thischapter and looking at its examples.

Background to Tiling

In architecture and mathematics, the words “tiling” and “tessellation” are often usedinterchangeably to describe the process of covering a surface in polygons. Whilesome mathematical texts use the word “tessellation” when the pattern has a clearunderlying rule and “tiling” when it does not, this use is not consistent. The word“tessellation” is derived from the Latin tessela, which was a small, semi-regularcubic tile used for mosaics in the ancient Greek and Roman worlds. The origins ofthe word “tile” are typically traced to the Latin tegula, meaning to cover something.In essence, both words have their origins in the process of covering large walls,floors, or ceilings with smaller, more durable or decorative elements.

A more mathematical definition of tiling is that it is the process of covering asurface using plane shapes such that two conditions are met. First, there must be nooverlap between any tiles, and second there must be no gaps in the resultant surface.If these two conditions are met, then the tiling is said to be “complete” or “true.”In addition to this, every instance of a particular polygon in a pattern is defined asa “tile,” and the number of different polygon types used to create the tessellation iscalled the “tile-set.” Thus, if a surface is covered in identically sized square tiles, thisis a tile-set of one as there is only one type of tile (Fig. 4). The stacked, stretcher,and herringbone examples in the previous section all have a tile-set of one. If asurface is covered in a mixture of square tiles, and right-angled triangular tiles (madeby dividing a square across its diagonal), it is described as a tile set of two, andso on. In theory, there is no maximum number for a tile-set, as it is possible thatevery tile differs from every other one. In contrast, there are only three “regular”

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Fig. 4 Square

or “congruent” tessellations with a tile-set of one – squares, equilateral triangles,and hexagons (Figs. 4, 5, and 6). Tessellations such as these, which display multiplelines of symmetry or repeat their configuration, are usually called “periodic” sets.Some of the most recognizable periodic patterns are based on squares, rectangles,trapezoids, or parallelograms. Furthermore, several three-dimensional shapes alsoperfectly fill a volume of space, by repeating only a single shape. The most obviousexample of this is the cube, but also the rhombic dodecahedron has this propertyalong with a number of triangular and rhombic prisms.

Whereas periodic tiles must have lines of symmetry or repeat their configuration,there are also two types of “aperiodic” tiles. The first type can fill a plane eitherperiodically and aperiodically, depending on the rule used to assemble them.Gardner’s quadrilaterals are an example of the first type of aperiodic tiles (Ostwald1998). The second type, which are also the most interesting to mathematiciansand architects, are those that can only tile aperiodically. While in recent years thiscategory has been the focus of considerable research, prior to the 1960s it was noteven apparent that it existed at all.

In 1961, Hao Wang set out to determine if, given a particular polygonal tile-set, there was a way of determining its capacity to tile a plane periodically (that is,to determine if they would repeat their configuration). However, in 1965, RobertBerger proved that it was not possible to develop a decision procedure for periodictiling, and thus aperiodic tile-sets and patterns must exist (Rubinsteim 1996). Thisrealization triggered a series of developments in the field, with Berger proposingthe first aperiodic tile-set which had 20,426 different shapes. In 1967, Bergerreduced this number to 104; in 1968, Donald Knuth reduced it to 92; and in 1971,

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Fig. 5 Equilateral triangulartessellation

Fig. 6 Hexagonaltessellation

Raphael Robinson reduced the set to just 6 tiles. Two years later, Roger Penrosedemonstrated that using a parallelogram tiling period, the set could be reduced tojust two tiles, known today as “Penrose tiles” or the “darts and kites” set (Fig. 7).In the early 1990s, John Conway identified a further two-tile aperiodic set known asthe “pinwheel” set.

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Fig. 7 Penrose “dart andkite” aperiodic tiling

Interestingly, the Penrose “darts and kites” set exhibits a type of “quasi-symmetry” reminiscent of the fivefold symmetry found in quasi-crystal geometryin the 1960s and in aluminum-manganese crystals in the 1980s (Stewart andGolubitsky 1993; Ostwald 1998). A peculiar characteristic of Conway tiles is thatlike fractal geometric forms, they can be scaled, such that a smaller version of thetile-set can nest within a larger (Fig. 8). There are multiple periodic sets where thiscan occur, but the Conway pattern is the only well-known aperiodic set with thisproperty. Despite this, such tile sets tend to be what Benoit Mandelbrot calls “trivialfractals,” meaning that the connection between tessellations and fractal geometry isnot necessarily as significant or interesting as this example might suggest (Ostwaldand Vaughan 2016). Some tessellation patterns repeat their small-scale geometryat larger scales, which is part of the definition of a fractal. But there are manyexamples of nonfractal patterns with these same properties, including congruentsquare (Fig. 4) and equitriangular (Fig. 5) tessellations.

In parallel with the development of examples of plane-filling aperiodic tiling,there have also been several proposals for space-filling aperiodic sets. Both KojiMiyazaki (1977) and Robert Ammann identified two tile-sets that will fill spaceaperiodically. Ammann’s tile-set comprises a pair of rhombohedra formed bycreating two solids, each of which have six sides that are all the same as Penrose’sstarting rhombus for the formation of the dart and kite set. Plane-filling and space-filling variations of aperiodic Voronoi tessellations have also been developed. Whilethese Voronoi sets do not possess the mathematical purity of Ammann’s andMiyazaki’s sets, they are interesting because they appear to replicate several naturalphenomena, which have in turn inspired architects, designers, and artists.

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Fig. 8 Conway “pinwheel”aperiodic tiling

Tiling in Architecture

Throughout history, tiling has been closely associated with architecture. Even theearliest primitive shelters were often floored with woven matting, walled withroughly stacked masonry and carved with geometric patterns. One of the oldestfoundation myths of architecture, the “primitive hut of the ancients,” depictsprehistoric humans platting together branches in a tight grid to create the firstshelter (Rykwert 1991). Evidence of this approach to weaving baskets, mats, andscreens can be found in Pre-Mesolithic and Neolithic societies (10,000 BC). Theseearly constructions were produced by platting dried leaves or vines to create largersurfaces. By repeating a regular pattern, these woven objects became more durable,and by combining different materials, they become decorative surfaces which oftengrew to take on cultural significance (Gerdes 1999).

Indra Kagis McEwen (1993) suggests that the origins of architecture – meaningnot only the first constructed works, but also those cited in mythology as the startingpoint for architecture – can be traced to either woven structures or mosaic floors.While, from a mathematical perspective, weaving and tiling are different processes,they share several common linguistic ancestors and their surface conditions havesimilar properties. For example, in Greek mythology, Daedalus is credited as beingthe first architect because he created the Labyrinth at Knossos and Ariadne’s choros,an intricate mosaic dance-floor. However, there remains some confusion aboutwhich of these is regarded as the first work of architecture. In mythology, both areclassified as architecture by virtue of the fact that they are geometric weaves thatanimate human behaviors. In the first instance, Daedalus’ Labyrinth at Knossos is

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a complex spatial pattern that describes the pathways taken by the Minotaur at itscore, and the sacrificial victims released into its confines. Conversely, the choros ofAriadne encapsulates the movement patterns followed by dancers as they engagein a choreographed rite. In both cases, the surface pattern is significant because itshapes or animates the actions of people. Regardless of which of these examplesmight be considered the first work of architecture, it is clear that in Greek andRoman mythology, the origins of architecture are tied to the act of creating patternedsurfaces.

For Grünbaum and Shephard (1987: 1), the practical “art of tiling” commencedwith the first use of stones to cover a floor. Such irregular floors can be found inNeolithic ruins, and from around 7000 BC, evidence is available that more regularstone and timber floors were used in granaries. In Mesopotamia and ancient Egypt,primitive bricks were made by putting clay into square or rectangular molds (oftenwith thickened or hollowed central sections, to facilitate stacking) and then firing theclay in a kiln or drying it in the sun. These bricks were used to create roads, paths,floors, and walls, being tiled together in a range of patterns which were intended toincrease their strength or durability.

Grünbaum and Shephard (1987: 1) also specifically connect the “art of tiling”with architecture because in ever the earliest societies, people realized that tes-sellated patterns add richness, decoration, or ornamentation to the surface of abuilding. Furthermore, at any point in history, “whatever kind of tiling was infavor, its art and technology always attracted skillful artisans, inventive practitioners,and magnanimous patrons” (Grünbaum and Shephard 1987: 1). For example,some of the earliest mosaics can be traced to Mesopotamia, when colored tileswere combined together to create patterns or pictures on walls or floors. Coloredtiles, which celebrated geometry, rather than pictorial representation, were alsoextensively used in Ancient Persian and Indian buildings, and today they are oftenlinked to Islamic architecture. While the use of such tiling patterns was moreubiquitous than this, the Mosques, Madrassas, and Palaces of North Africa andthe Middle East represent some of the most advanced tiling applications ever seen,and remain a source of fascination and debate to the present day (Bonner 2017;Wichmann and Wade 2017).

Tiling was an important part of Islamic architecture because religious beliefsrestricted the use of pictorial or figurative representations. As such, it is notsurprising that elaborately carved and inlaid tilings and arabesques reached possiblytheir highest level of refinement in Islamic architecture. For example, the Moorishpalace of the Alhambra in Granada (Andalusia) features examples of betweenthirteen and sixteen different types of congruences or isometries. These tiles, whichare mostly from the late thirteenth and early fourteenth century expansion ofthe building, are today celebrated for their richness and inventiveness. The latterdimension, inventiveness, is significant because it was only in the seventeenthcentury that the first serious study on tessellation was completed by JohannesKepler, and the nineteenth century that Yevgraf Fyodorov identified that periodictilings all conform to one of seventeen isometries or “wallpaper” groups. Thus,three hundred years before the different types of congruences in periodic tiling were

Tessellated, Tiled, and Woven Surfaces in Architecture 11

classified and understood by mathematicians, the designers of the Alhambra hadalready used many of them.

In a further example, Lu and Steinhardt (2007) note that several tiling patternsin the mid-fifteenth Iranian architecture have similarities to Penrose tiling in thatthey appear to possess quasi-periodicity. Lu originally observed these properties inthe Madrasa in Bukhara (Uzbekistan) and in a more advanced version in the Darb-iImam shrine in Isfahan (Iran). While both the veracity of claims about the tilingpatterns in the Darb-i Imam shrine and the date they were produced have beendisputed (Lauwers 2018), Kukeldash Madrasah in Bukhara also possess severalcomplex tiled surfaces within three dimensional ribbed-ceiling domes or arches(Makovicky 2018).

A completely different set of tiling traditions emerged across Europe, with thefloors of monasteries, public buildings, and commercial buildings often tiled insymmetrical, repetitive colored patterns, while churches and palaces were lined inmosaic representations of biblical scenes, famous battles, or landscapes. As such,a bifurcation occurred in European tiling practices with more functional patternsreserved for interior thoroughfares or axes, and a pictorial tradition, sometimesoccurring in parallel with frescos or tromp l orl, serving as a type of communicationor entertainment.

Examples of the first type, and especially so-called “checkerboard” tilings, canbe found throughout history in classical Greek temples, Roman villas, Medievalwineries, and Victorian hospitals. In general, such geometric tiles tend be usedin entries, hallways, or atria. While the base patterns were often simple, theywere also regularly framed with “border tiles” or punctuated with decorative tiles,sometimes depicting heraldic coats-of-arms or guild signs and emblems. Despitethe use of colored tiles and those with different textures or illustrations, many ofthese tiles could be regarded as being largely functional applications. However, thereare notable exceptions including the tiled floors in Masonic lodges and temples.Such ceremonial floors are typically rectilinear in shape and are tiled (often onthe diagonal) in a black and white checkerboard, with a border of triangular (ordiamond-shaped) tiles enclosing them. Many of these floors feature decorativecentral tiles depicting crosses (“saltires”) and six-pointed stars, or they are inlaidat their corners with images of Masonic tools; the square, compass, plumb-line, andtrowel.

An example of the second type of tiling, where the purpose is more rep-resentational or pictorial, is the famous second century mosaic from Palencia,Spain. This tiled panel depicts the four seasons on a square field that has beendivided diagonally creating four triangular regions, which are further subdividedinto octagonal, trapezoidal, and cruciform tilings. Each of the triangular regionsis centered on a human face (within an octagonal tile), surrounded by depictionsof birds, plants, and mythical beasts, representing the passage of time in a year.In another example, in the sixth century Basilica di San Vitale in Ravenna, Italy,the elaborate mosaics depict Old Testament stories in the rich golden and brownhues of the Helenistic-Roman decorative tradition. The underlying geometry ofthe tiling patterns in these mosaics is not significant, because their purpose is to

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illustrate a combination of biblical themes and natural elements (flora and fauna).The Basilica di San Vitale also has a famous mosaic floor depicting a labyrinth, asdoes the thirteenth century Chartres Cathedral in France. While the actual purposeof labyrinth tilings of this type remains unknown, they are often interpreted as eithersymbols representing the challenging paths taken by pilgrims to reach the placeof worship, or else as the literal path a kneeling penitent follows around the floorof the cathedral while seeking absolution. Such examples, much like the DaedalicLabyrinth at Knossos and Ariadne’s dance-floor, use tiling to illustrate potential ordesired movement.

By the early nineteenth century, pictorial tiling had fallen out of favor acrossmost parts of Europe, being both prohibitively expensive and, in some countries,less socially acceptable than it previously was. In contrast, simple, decorative tilinghad become sufficiently affordable that it could be found in many houses in fireplace surrounds, window and door sills, and even the walls of kitchens. Duringthe Victorian era in England and America, tiles were not only hard-wearing, theywere a means of displaying middle class wealth and they also offered a level ofhygiene which the previous stone, timber, or stucco surfaces could not. However,in the closing years of the nineteenth century a range of conflicting forces began tochange the ways tiles were used. On the one hand, the industrial revolution increasedthe scale of mass production such that tiles were more available than ever before,but on the other, with the rise of the Arts and Crafts movement in England, there wasa new-found enthusiasm for handmade tiles. Modern Architecture, which grew todominate the architectural canon of the early twentieth century, rejected Victorian,Gothic-Revival and Arts and Crafts style architecture, along with almost all of theirtiling traditions. Oscar Niemeyer was possibly the only Modernist architect whoincluded elaborate murals and tiled surfaces in his designs for religious structures inPampulha and Brasilia (Brasil).

By the 1970s and 1980s, a growing number of architects and urban planners hadbegun to criticize Modernism for its often cold, inhuman spaces and forms. Thebacklash against Modernism, which was collectively known as Post-modernism,sought to reintroduce a connection to history and a sense of humanity. Onedesign strategy Post-modern architects used to achieve these goals was to employostentatious tiled decorations on walls and floors, some of which were intended toreplicate or recall historic places or events. For example, architect Charles Moore’sPiazza d’Italia in New Orleans (USA) was completed in 1978. This design fora public space features multiple decorative, oversized, and brightly colored tilingpatterns, which generally evoke ancient Roman (or classical Greek) architecture(Jencks 1991). The ground-plane of Piazza d’Italia is tiled in concentric black andwhite rings, recalling both the labyrinth floors of Basilica di San Vitale and ChartresCathedral and the iconography of Las Vegas Casinos and shopping malls. As thetiles Moore chose were often cheap and mass-produced, the whole affect was likea temporary stage set. Robert Venturi and Denise Scott Brown’s 1982 Gordon WuHall (Princeton, USA) uses grand, decorative tiling patterns above its entry façade,and their 1983 Lewis Thomas Laboratory for Molecular Biology (Princeton, USA)is covered in multiple different brick tessellation patterns. In both cases, the tiling

Tessellated, Tiled, and Woven Surfaces in Architecture 13

patterns tend to be simple, but the way they are positioned so prominently andboldly was intended as an ironic counterpoint to Modernism and a means of evokinghistoric building styles (von Moos 1987).

One of the first direct architectural engagements with the mathematics of tilingoccurred in the final decade of the twentieth century. In Melbourne (Australia),ARM architects were commissioned to refurbish the historic Storey Hall auditorium.Their design, which was completed in 1996, overlays lime green Penrose Tiles onthe historic façade, as well as throughout the new foyer and auditorium. While itmight be possible to classify ARM’s use of tiles as an extension of the Post-moderntradition, their design includes multiple references to the actual mathematicalproperties of aperiodic tiles and their explanations directly reference Penrose’s work(Ostwald 1998). Not only does their version of the dart and kite set partially functionin three dimensions, but it emphasizes the importance of those parts of surfaceswhich are either untiled or which cannot be tiled because of a deliberate flaw in thelogic followed. In another more recent example, Federation Square in Melbourneis a large arts and entertainment complex which was completed in 2002. It wasdesigned by LAB architects and it uses a pinwheel aperiodic tile-set as both exteriorcladding and to shape parts of its planning. The exterior pinwheel tiles are made ofzinc (both solid and perforated), glass (translucent and frosted), and sandstone. Athree-dimensional glass tile set also serves as an atrium roof and the public spacesof Federation Square are lined with cobblestones (often inlaid with fragments oftext), beneath which is a hidden environmental “labyrinth.”

Federation Square and Storey Hall are not alone in their flamboyant adoptionof recent mathematical tile-sets for generating architectural space and form. Therehave also been multiple examples of Voronoi tessellations including Minifie Nixon’sCentre for Ideas (Melbourne) in 2004 and ARM’s 2009 Melbourne Recital Hall(Melbourne). Such is the popularity of aperiodic and quasi-periodic paving patternsthat they have been used in many public buildings and space – for example, theMathematical Institute in Oxford, the Science Centre in Paris and the Zamet Centrein Rijeka – and have even been used in commercially produced bathroom andkitchen tiles.

In most of the examples given in this chapter, the focus has been on theseamlessly covered surfaces, but certain combinations of periodic tile-sets can createareas that are unable to be tiled. Conventionally, this is regarded as a flaw or errorand is rectified by removing a number of surrounding tiles and reworking the patternuntil there are no holes. But what if the holes, their shapes and frequency, are viewedas a different dimension of the tiling pattern? John Conway describes this way ofthinking about tessellations as “hole theory” and it as akin to imagining “a vasttemple with a floor tessellated by Penrose tiles and a circular column exactly inthe center. The tiles seem to go under the column, [but] the column covers a holethat can’t be tessellated” (Gardner 1989: 26–27). ARM’s Storey Hall offers an earlyarchitectural interpretation of the significance of untiled, or un-tile-able parts ofsurfaces.

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Conclusion

In the first century BC, the Roman author and military architect Marcus VitruviusPollio published De Architectura. In that work, he famously defined the threecore properties of architecture as firmness, commodity, and delight. These sameproperties are also behind the enduring appeal of tiled, tessellated, and wovensurfaces in architecture. Tiling occurs in part to increase the strength or stabilityof a surface, it also improves the function or usability of a surface, and it providesan opportunity to develop decorative, symbolic, or poetic properties. Thus, forpractical, social, and cultural reasons, tiling has remained significant in architecture.However, the relationship between architectural and mathematical tiling has beenless consistent.

It could be argued that architectural tiling, and particularly in Islamic archi-tecture, effectively predates the development of formal mathematical knowledgeabout tiling. It must also be noted that more recently, architects and designers havelearnt from and openly adapted advances in aperiodic tiling. Whether or not thisrelationship will continue is currently unknown, as multiple cultural and socialfactors shape the way architects and designers work with mathematical concepts.But regardless of the evolving relationship between architecture and mathematics,tiling remains one of its most obvious and visible points of contact.

Cross-References

� Fractal Geometry in Architecture�Greek and Roman Architecture�Labyrinths

References

Bonner J (2017) Islamic geometric patterns: their historical development and traditional methodsof construction. Springer, Cham

Gardner M (1989) Penrose tiles to trapdoor ciphers. W. H. Freeman, New YorkGerdes P (1999) Geometry from Africa: mathematical and educational explorations. Mathematical

Associate of America, Washington, DCGrünbaum B, Shephard GC (1987) Tilings and patterns. W. H. Freeman, New YorkJencks C (1991) The language of post modern architecture. Wiley, LondonLauwers L (2018) Darb-e imam tessellations: a mistake of 250 years. Nexus Network J 20:321–329Loos A (1998) Ornament and crime: selected essays. Ariadne Press, RiversideLu PJ, Steinhardt PJ (2007) Decagonal and quasi-crystalline tilings in medieval Islamic architec-

ture. Science 315:1106–1110Makovicky E (2018) Vault mosaics of the kukeldash madrasah, Bukhara, Uzbekistan. Nexus

Network J 20:309–320McEwen IK (1993) Socrates ancestor: an essay on architectural beginnings. MIT Press, Cam-

bridge, MAMiyazaki K (1977) On some periodical and non-periodical honeycombs. University Monographs,

Kobe

Tessellated, Tiled, and Woven Surfaces in Architecture 15

Ostwald MJ (1998) Aperiodic tiling, Penrose tiling and the generation of architectural forms. In:Williams K (ed) Nexus II: Architecture and mathematics. Edizioni dell’Erba, Florence, pp 99–111

Ostwald MJ, Dawes MJ (2018) The mathematics of the modernist villa: architectural analysis usingspace syntax and isovists. Birkhäuser, Cham

Ostwald MJ, Vaughan J (2016) The fractal dimension of architecture. Birkhäuser, ChamOstwald MJ, Williams K (2015) Mathematics in, of and for architecture: a framework of types.

In: Williams K, Ostwald MJ (eds) Architecture and mathematics from antiquity to the future:Volume I. Birkhäuser, Cham, pp 31–57

Rubinsteim H (1996) Penrose tiling. Transition 52(53):20–21Rykwert J (1991) On Adam’s house in paradise: the idea of the primitive hut in architectural

history. MIT Press, Cambridge, MASalingaros N (1999) Architecture, patterns and mathematics. Nexus Netw J 1(1):75–85Stewart I, Golubutsky M (1993) Fearful symmetry. Penguin, Londonvon Moos S (1987) Venturi, Rauch and Scott Brown: buildings and projects. Rizzoli, New YorkWichmann B, Wade D (2017) Islamic design: a mathematical approach. Springer, Cham