Labyrinth

Tessa Morrison

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Topology of Labyrinths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Mnemonic Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Abstract

The labyrinth is a fusion of architecture and symbol, and it has permeated culturesince ancient times. It is also a synthesis of reality, religion, and myth that hasmerged through the ages. It has been a prison for the Minotaur in ancient Creteand an ancient Egyptian palace as described by Herodotus and Pliny. In ancientRome, it appears as graffiti in Pompeii. For pseudo-Dionysius (late fifth centuryto early sixth century), it was the dance of the angels which has formed the basisfor Christian processions and rituals and for shaping the architectural boundariesof early churches. In mediaeval times, it was the pathway to Jerusalem thatdecorated the floors of cathedrals, particularly in France. Although labyrinthscome in various forms, a very precise structure of the symbol of the labyrinthemerges in ancient times that is repeated in various cultures, sometimes roundand sometimes square but exactly the same structure. The word “labyrinth”seems to have so many meanings, yet many of these ancient labyrinths haveone precise geometrical structure that has been retained for millennia and is

T. Morrison (�)The School of Architecture and Built Environment, The University of Newcastle, Newcastle,NSW, Australiae-mail: Tessa.morrison@newcastle.edu.au

© Springer International Publishing AG, part of Springer Nature 2018B. Sriraman (ed.), Handbook of the Mathematics of the Arts and Sciences,https://doi.org/10.1007/978-3-319-70658-0_4-1

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commonly called the Cretan labyrinth. The labyrinth, architecture, and geometryhave been entwined through time. This chapter considers this relationship byexamining a paradigm that makes it possible to assess these simple structuresand how they have changed in appearance, while the geometrical structure hasstayed the same.

KeywordsCretan labyrinths · Roman labyrinths · Church labyrinths

Introduction

In modern speech, the words “maze” and “labyrinth” are often interchanged. Yetthey both have very different meanings. The word maze originated from the oldEnglish mazen, which means to bewilder, and it was first recorded in c.1385 byThomas Chaucer in The Legend of the Good Woman (Bradley et al. 1989, 507).It is mostly associated with English maze gardens where there is more than onepath to the center – these are multicursal patterns. The most famous of these is atHampton Court in England that dates to the Elizabethan era of the sixteenth century.Labyrinths, on the other hand, are unicursal patterns, meaning they only have onepath to the center, and once on the path, no deviation is possible except to returnto the beginning. These labyrinths have been dated back to the early Bronze Age,2500–2000 BC; however, they are notoriously difficult to date. They were carvedinto stone and are found in Italy, Spain, Iran, England, Ireland, and Sardinia andare all circular. The first precise dating of this geometric structure was in Pylos,in Greece. They are found as a square preserved on a clay tablet; on one side is alabyrinth, and on the other side of the clay tablet was inventory. An early exampleof the labyrinth appears as a doodle that was preserved by fire in the destruction ofthe city of Pylos in the thirteenth century BC (see Fig. 1a). Exactly the same symbolwas found over 1400 years later, and it was also preserved by disaster. It camein the form of graffiti on the peristyle of a villa in Pompeii, with the inscription“labyrinths. Here lies the Minotaur” (Fig. 1b). The same pattern occurs on coinsfrom Crete from the second and third century BC (Fig. 1c), on a seventh-centuryBC Etruscan wine pitcher (see Fig. 1d), and a ninth-century AD Biblical manuscriptwith the inscription “the walls of Jericho” (Fig. 1e). Sometimes this symbol is roundand sometimes square, but it is the same structure. The symbol has been called theCretan labyrinth after the myth of the Minotaur. In this myth, every 7 years, sevenmales and seven females from Athens were to be sacrificed to the Minotaur, halfman, half bull, who was imprisoned in the labyrinth in Crete. The labyrinth had beenbuilt by the famous architect Daedalus, and it was designed with many confusingpaths, making it difficult to find the way out. However, Ariadne, daughter of theKing of Crete, fell in love with one of those to be sacrificed, Theseus. He smuggleda sword into the labyrinth to kill the Minotaur, and Ariadne gave him a ball of stringso that he would be able to retrace his steps and escape the labyrinth (Apollodorus1997).

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Fig. 1 (a) The Pylos labyrinththirteenth century BC, (b) graffiti from Pompeii 79 AD, (c) Cretancoins c.190–100 BC and 267–200 BC, (d) seventh-century BC Etruscan wine pitcher, and (e)nineteenth-century AD Biblical manuscript

Classical writers such as Herodotus, c.485–425 BC (Herodotus 1939), and Plinythe Elder 23–79 AD (Pliny 1949) wrote of an Egyptian labyrinth. This was foundin the Temple of the Twelfth Dynasty Pharaoh Ammenemes III, c.1842–1797 BC –which was built beside his pyramid at Harrow in the Fayum district. Herodotus wasoverwhelmed by the magnificence of the Egyptian labyrinth. He stated that “the

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pyramids, too, are astonishing structures, each one of them equal to many of ourmost ambitious works of Greece; but the labyrinth surpasses them. It has 12 coveredcourts – six in a row facing north, six South – the gates of the one range exactlyfronting the gates of the other, with a continuous wall around the outside of thewhole. Inside, the building is of two storeys and contains 3000 rooms, of which halfare underground, and the other half directly above them” (Herodotus 1939, II.149).By the nineteenth century, none of this magnificent Egyptian labyrinth survivedwhen Flinders Petrie found evidence of its existence; it literally was a hole in theground (Hall 1904). Herodotus’ description of the Egyptian labyrinth sounded morelike that of a multicursal maze rather than a unicursal labyrinth. However, Plinyclaimed “there is no doubt that Daedalus adapted it [the Egyptian labyrinth] as themodel for the labyrinth built by him in Crete, but he reproduced only 100th partof it containing passages that wind, advance and retreat in a bewilderingly intricatemanner” (Pliny 1949, XXXVI. 85). Yet, by this time the unicursal labyrinth structurethat became known as the Cretan labyrinth was well established, as is demonstratedin the graffiti at Pompeii. However, it is often stated that the unicursal labyrinthinestructure was the pattern left by Ariadne’s thread that Theseus took into the labyrinthto find his way out, rather than the labyrinth itself.

Topology of Labyrinths

The main purpose of this chapter is to analyze the ancient labyrinths associatedwith architecture; thus there are restrictions on the type of labyrinths examined. Toexamine the structure of the unicursal labyrinth, it is first necessary to define theanatomy of the unicursal labyrinth.

1. There are an equal number of paths or levels of the labyrinths equidistance fromthe center, so that the levels are concentric levels. These can be concentric circles,squares, or any geometrical shape, as long as they are parallel and an equidistanceto the center. Each level has maximum coverage, meaning the only gaps on thelevel are for turns.

2. The unicursal labyrinths have a restricted anatomy, as described in Fig. 2.3. There are a finite number of levels.

These ancient labyrinths obey the law of alternation, meaning that the directionof the path changes whenever the level of the labyrinth changes. In short, theselabyrinths are simple, alternative, transitive labyrinths or SAT labyrinths. In strictmathematical terms, the unicursal labyrinth’s topology is that of a straight line,which gives very little information on the symbol’s structure. However, in thischapter a paradigm will be outlined that emphasizes its features, such as turns.Figure 2a shows an example of a SAT labyrinth that has alternating rows, spirals, andonly one semiaxis. The semiaxis from the entrance to the center is called the throatof the labyrinth. Figure 2b only obeys the laws of alternation and has four semiaxes.To analyze these symbols, it is necessary to first unroll the labyrinths. This reduces

Labyrinth 5

Fig. 2 Anatomy of the SATlabyrinths

Fig. 3 The process of unrolling a SAT labyrinth

the symbol to its fundamental form (FF), which consists purely of the geometryof its turns, thus making it simpler to analyze. Figure 3 demonstrates the processof unrolling the labyrinth with only one semiaxis, to reveal its FF (Fig. 3e). Oftenthe FF consists of fundamental elements (FE) which are stacked vertically to createthe FF. However, many SAT labyrinths have more than one semiaxis, particularlychurch and Roman mosaic labyrinths.

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Definitions

A fundamental form or FF is a rectangular graft obtained when a labyrinth isunrolled. A FF is constructed from fundamental elements or FE. They are minimalbuilding blocks, or elements, of the FF. One or more FEs is stacked vertically tocreate FFs.

The SAT labyrinth is the most enduring and simplest labyrinth. There are twoother families of labyrinths: the church labyrinths and the ancient Roman labyrinthsthat were found in mosaic floors. Both have generally four semiaxes, although thisnumber can be extended. This gives the labyrinth the appearance of perfect fourfoldrotational symmetry. By using the same process of unrolling the labyrinths, the endresult is a series of FFs. Figure 4 shows a Roman mosaic labyrinth unrolled, andthis reduces the labyrinth to its four sections that make up the FF of the Romanlabyrinth. The labyrinth has ten rows and the four sectors that can be clearly seenwhen reduced to a FF. Each section consists of a single or a collection of stackedvertical FEs, and each section is called a fundamental element sector (FES).

Definition

In a FF of a labyrinth with semiaxis is other than the throat, there is a divisioncoursed by the semiaxis, which divides the FF into sectors. These sectors consist ofa single, or a collection of, stacked vertical FEs, and they are called the fundamentalelement sector or FES.

The Cretan labyrinth has eight levels, and its level sequence from the outside,designated as 0, to the center is 032147658. An example of FE is 3214, which canbe symbolized by γ4; the other part of the sequence, 7658, is isomorphic to 3214, butfour levels further from the entrance are also symbolized by γ4; this is demonstratedin Fig. 3e. Therefore, FF of the Cretan labyrinth is denoted as SAT[γ4

2]. In generalthe notation is SAT[ξn

m . . . ψnm] where ξψ = FE and n,m = 1,2,3, . . . ,s. The level

sequence of the FEs essentially determines the topology of the SAT labyrinths. Thetwo FEs of the Cretan labyrinths are clearly visible in the unrolling of the SATlabyrinths demonstrated in Fig. 3. The subscript gives the number of levels in thisparticular FE sequence, and this superscript gives the number of these particular FEsthat are stacked vertically in succession. The level sequence of the FEs is completelydetermined by the topology of the SAT labyrinths. The level sequences of the FEsare read from the beginning of the right-hand side, and the levels run from the top tothe bottom. The notation of the FE’s stacking is also read from the top to the bottom.Figure 4a demonstrates how the levels of the graph are read, while Fig. 4b illustratesexamples of FEs’ dual or inverse sequence patterns and their level sequences. Theduals of the FEs denote the walls of the labyrinths and the FE and paths of thelabyrinths. However, what is interesting is that there are only three main FEs that areneeded to define that topology of the SAT labyrinth. These are γn which denotes theFE (n − 1) . . . 321n, examples of this are γ 2 = 12, γ 4 = 3214, and γ 6 = 543,216,

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Fig. 4 Demonstrating how the levels of the graph are read

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Fig. 5 Unrolling Romanmosaic labyrinths

ϕn which denotes 1234 . . . . (n − 3) (n − 2) (n − 1)n, and furthermore ηn whichdenotes the FE (n − 1)2 (n − 3)4 (n − 5)6 . . . 5 (n − 4)3 (n − 2)1n. An example ofthis is η8 = 72,543,618. Another two FEs are added: αn and α2[αn] or [αn] α2. Thebrackets indicate nested turns and n is always even. These nested turns are neededfor the FFs of the church and Roman labyrinths. With these five classifications ofFEs, it is possible to classify the overall topology of all unicursal labyrinths.

Excluding spirals, SAT labyrinths are the simplest form of labyrinthine struc-tures. Roman labyrinths are the next significant development in labyrinths. Thetypical arrangement of the Roman labyrinth is in four sections. Only rarely is thisnumber extended, and traditionally Roman labyrinths have a fourfold symmetry.These sections become clear when the labyrinth is unrolled. At first appearance,the Roman labyrinths look far more complex and a completely different structureto the Cretan labyrinth. Figure 5 demonstrates this, showing that by unrolling theRoman labyrinth its structure is simplified. The Roman labyrinth’s topology ofeach of the four FESs becomes a SAT labyrinth. Figure 5 clearly shows a typicalRoman labyrinth which has been unrolled, and the first, second, and third sectionsare isomorphic to SAT[γ4

2] with each sector’s path being linked. But the fourth FESintroduces a new form of FE.

In Fig. 4c is an FE that is labelled α2[αn], this means on the right hand of thegraph of a FE αn is nested within a α2 on the left-hand side. This type of FE cannotconstitute FF alone. In Fig. 5, the fourth FES is labelled (α2[αn])2; the notation that

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will be used for the FF of the Roman labyrinth in Fig. 5 is R[γ42 × 3, α2[αn]2]. This

classifies each FES and is read from left to right. In general, the notation is R[ξ xI, . . . .ψ x j] where ξ.ψ = FES and i,j = 1,2,3, . . . n. This will enable the variationsof each Roman labyrinth to be detected no matter how many FESs there are.

Through looking at the topology of the FES, each FES, with the occasionalexception of the last sector, runs to the center and becomes a SAT labyrinth.

Many of the early Roman labyrinths are associated with cities and the myth ofTheseus and the Minotaur (see Fig. 6). There are thought to be at least 43 recordedRoman labyrinth mosaic floors, but the survey of these labyrinth mosaic floors isincomplete since many of the labyrinths are in war zones, and many have beenreburied to protect them (Kern 2000). The most common topology of these Romanfloor labyrinths is γ4

2, the FF of the Cretan labyrinth strongly suggesting a culturalor ritual connection. The function of these labyrinths is difficult to ascertain, as thereis no ancient written reference to their purpose – mythical, or possibly a ritual. Butthey do indicate some movement through into the center. The earliest labyrinth in achurch is a Roman labyrinth in the Cathedral of Algiers and is dated in the fourthcentury. It is the standard for sector Roman labyrinths and in the center is a matrix ofletters where the word “Sancta” (holy) is spelt out in all four directions in the shapeof a swastika, replacing the more standard image of the Minotaur. The labyrinthis in the entrance of the basilica, seemingly unrelated to the shape of the building.Nevertheless, it was a dominant feature of the original basilica.

Church labyrinths are the third family of unicursal labyrinths. The most famousof these is the floor labyrinth at Chartres Cathedral, France (see Fig. 7). The FFof the Chartres Cathedral labyrinth is isomorphic to Fig. 7d. The standard churchlabyrinth is 12 concentric levels and is divided into four sections. However, Fig.8 shows examples of these labyrinths for levels 8, 10, and 12 with their FFs. Thesymmetry of the labyrinth is highlighted and is unrolled in a similar fashion to theRoman labyrinth. It is possible to detect the variations of the structure and symmetryof the labyrinth. An example of the notation used for the FFs of the church labyrinthsin Fig. 7 is C[α2[α2]2, α3

2, α32, α2[α2]2]. This equation reads from left to right and

classifies each FES. In general the notation is C[ξn, . . . .ψ m] where ξ,ψ = FES andn,m = 1,2,3, . . . s. The dotted and solid line under the FESs indicates an extra lineon the first and last level FES. If the turns on the semiaxes are removed, then theSAT structure underlying the church labyrinth is revealed as shown in Fig. 8.

When the church labyrinth has eight concentric levels, the level sequence of theunderlying SAT labyrinths is 032147658 which are isometric to the Cretan labyrinth(see Fig. 9). This is the only variation of a level 8 standard church labyrinth. Whilelevel 10 church labyrinths have two variations (see Fig. 8b, c), they are symmetricalto each other. The Chartres Cathedral labyrinth has an underlying SAT labyrinthwith a level sequence of 0,5,4,32,1,6,11,10,9,8,7,12, to repeated sequences and aFE γ6. All the church labyrinths underlying SAT labyrinths are all a combinationof the FEs and γ6 and γ4. The 12-level style labyrinth is found on the floors ofchurches, roof bosses, on stone steles, and in many manuscripts. These structures

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Fig. 6 Roman labyrinths

indicate a strong architectural and mathematical continuity with the past (Morrison2009).

By using this process of unrolling and assessing the overall structures of thelabyrinths, it becomes possible to see the cultural transference of the structuresand how they are embedded into other symbols from other cultural symbolicformats. By classifying the FFs of the three types of labyrinths (SAT, Church,and Roman labyrinths), it is possible to examine their structural connections,

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Fig. 7 Chartres Cathedrallabyrinth

variations, and transformations from one to the other. The level 8 church labyrinthis interesting, since it gives the appearance of having no resemblance to the Cretanlabyrinth. The initial thought is that the church labyrinth evolved from the Romanlabyrinth because of the appearance of the four quadrants and the close culturallinks. However, the structure of the SAT labyrinths is embedded into the Romanlabyrinths, which in turn is the foundational structure of the Church labyrinth. Thisbegs the question, how could such a complex structure be preserved, embodied intoother cultural symbols, and transferred through time?

The Cretan labyrinth is an enduring and complex symbol; it is also very difficult(if not impossible) to hand draw with any accuracy, yet this precise structure hasbeen repeated for centuries. However, on close examination of these labyrinths,particularly the ones which have been drawn into clay tablets or as graffiti, thereis clearly an underlying structure to assist in this drawing. The continuous repetitionof this complex symbol appears to support the suggestion that it was drawn througha mnemonics, a small symbol that acted as a guide to construct these more complexsymbols. Using mnemonic devices in training the memory was common in classicaltimes. In a world devoid of printing and notepaper, a highly trained memory wasof paramount importance to recover information, and rhetoric was an importantpart of classical education. The earliest surviving treatise on training the memoryis known as Ad Herennium and is dated c.86–5 BC. However, within the text of AdHerennium are described earlier Greek writings on the art of memory which do notsurvive (Yates 1966).

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Fig. 8 Level 8, 10, and 12 church labyrinths and their FF and underlying SAT FF

Mnemonic Devices

A mnemonic is a device used to remember something that is otherwise too hard torecall in detail. Several mnemonic systems have been suggested. One symbol andmethod of construction is continually pointed to as being the easiest way to drawa Cretan labyrinth (Attali 1999; Kern 2000; Morrison 2009). The mnemonic, or

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Fig. 9 Removing the semiaxis from the level 8 church labyrinth

nucleus, is shown in Fig. 10a and is expanded by beginning at the top vertical of thecross and then inserting a right angle or arc between the vertical of the cross andvertical of the L-shape on the right-hand side (see Fig. 10b). Second is beginningon the vertical on the left-hand side and following the path made in the last stepsand terminating at the dot in the right-hand quadrant (see Fig. 10c). The precedingsteps, following Fig. 9, continue to build up the labyrinths by beginning at the dot orthe line on the left-hand side, leaving the lines that have terminated at the dots, andthen traversing the symbol in the same direction and terminating at the first dot orline on the right-hand side, again leaving the lines that have terminated at the dots.

Causing this nucleus (Fig. 10a) to expand into the complete labyrinth has beenreported to have been a game called “walls of Troy” that was well-known at thebeginning of the twentieth century (Heller 1946). This same nucleus is a symbolthat is found in many ancient pottery shards and may have been used as seen in thethirteenth-century BC clay tablet from Pylos, where a pattern of dots can be seenpressed into the clay (Fig. 1a).

Although there is no doubt that there would be a need for some mnemonicsystem for such a complex structure, and there is evidence of dots, and sometimespinholes in parchment, there is no evidence, however, of the actual algorithm, andthis can only be speculation. Other suggested nuclei have been in the form ofreligious symbols such as the double axes from the Minoan society and the ancientsymbol of the swastika, where the labyrinth is constructed in a similar manner toFig. 9 (Morrison 2009). Nevertheless, these labyrinthine structures endured throughtime and appeared to have some mythical and ritualistic function, and this functionwas adapted and absorbed through changes in culture and religion. By the end ofthe sixth century, the church labyrinth had developed from the Roman labyrinth.The earliest surviving example is from the sixth century at San Vitales, Ravenna.However, it was not until the twelfth century that the church labyrinths becamea significant part of cathedrals, particularly in France, the most famous of allbeing the floor labyrinth of Chartres Cathedral. Originally there was an image ofTheseus and the Minotaur on a bronze plaque at the center of the Chartres Cathedral

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Fig. 10 One of the suggested systems of mnemonic for the Cretan labyrinth

labyrinth. Unfortunately this was removed and melted down for the NapoleonicWars. The original purpose of the church labyrinths is debatable and the connectionwith the Minotaur obscure. Many modern churches are having a church labyrinthincorporated into the church as a mosaic floor, and walking the labyrinth is equatedto the concept of walking a pilgrimage. However, there is no evidence that thiswas their original purpose, but walking or dancing the pattern of all the labyrinths,whether it be Cretan, Roman, or church labyrinths, does appear to be the obvious,although speculative, purpose.

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Conclusion

By examining the labyrinths by eye, there is little to hold them together apartfrom the fact they are unicursal. However, by examining these three types oflabyrinths through the paradigms described above does indicate there is somestructural connection which may have evolved through ritual, and perhaps cultural,connections. Apart from the geometrical structure, the architectural elements ofthe walls of the labyrinths or a city, whether it be Troy or Jericho, are commondecorative features, or sometimes a literary reference or connection is stronglyassociated with these labyrinths. The connection of the labyrinths with architecturehas a mythical presence over the millennia. Their connection in modern times hasbecome more literal than symbolic through literature such as Invisible Cities, byItalo Calvino, and “Coleridge’s Dream” by Jorge Luis Borges. The patterns of theirlabyrinths cannot be assessed as easily as the Cretan, Roman, and church labyrinths,but they are nevertheless part of Ariadne’s thread.

References

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Attali J (1999) The labyrinth in culture and society. North Atlantic Books, BerkeleyBradley H, Murray JAH, Craigie WA, Onions CT (1989) The Oxford English dictionary.

Clarington Press, OxfordHerodotus (1939) The histories – book II. Methuen, LondonHall HR (1904) The two labyrinths. J Hell Stud XXIV, p 320–337Heller JL (1946) Labyrinth or troy town. Class J 42:175–191Kern H (2000) Through the labyrinth. Prestel, MunichMorrison T (2009) Labyrinthine symbols in western culture: an exploration of the history,

philosophy and iconography. VDM Verlag, SaarbruckenPliny TE (1949) Natural history. Harvard University Press, Cambridge, MAYates FA (1966) The art of memory. The University of Chicago Press, Chicago