celtic knotwork: mathematical art

Celtic Knotwork: Mathematical Art Peter R. Cromwell

The interlaced ornament produced by Celtic scribes and stone masons has fascinated people for many cen- turies. The designs, ranging from small individual knots to elaborate panels composed of many motifs, provide the geometrically minded mathematician with a rich source of examples. Many aspects of the interlaced patterns can be studied mathematically, and some of these are explored in this article. We begin with the geometry of the knots.

Constructing Interlaced Patterns

Underlying many of the Celtic knot patterns is a lattice. It is this lattice which imparts the distinctive propor- tions to Celtic knotwork. It is usually composed of squares, but is occasionally built from 3 x 4 rectangles. For reference purposes, it is convenient to regard this lattice as the union of two dual lattices whose mesh size is double that of the original lattice (Fig. lb). These

Figure 1. How to construct alternating plaitwork.

36 THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 1 • 1993 Springer-Veflag New York

Figure 3. The rule for eliminating crossovers.

Figure 2. A non-alternating Celtic pattern.

will be called the auxiliary grids. When laying out a design only the vertices of these two grids are drawn in (Fig. lc).

The knotwork created on these grids is all related to plaitwork the basic weaving pattern used in basket- work and other crafts. This is shown in Figure ld. Note how the crossovers in the pattern lie at the intersection points of the two auxiliary grids, and that the interlacing has an alternating quality: Each string goes alter- nately over, then under other strings. This is a characteristic feature in Celtic patterns, too, although one or two anomalies are known (Fig. 2).

Interlaced patterns formed from portions of plaitwork can be found in Egyptian, Greek, and Roman ornament and in the art of many other cultures. Yet for the Celt, these alternating plaits were merely the raw material on which the artist sets to work. To obtain the more elaborate interlaced designs, the regularity of this primal pattern must be interrupted. This is achieved by breaking two strings at a crossover and rejoining the ends as illustrated in Figure 3. (Operations such as this are currently used in combinatorial knot theory.) Note that eliminating crossings in this fashion preserves the alternating property of the interlacing. When suffi- ciently many crossings are removed, the underlying plait ceases to be the dominant feature in the design and a pattern composed of knot motifs appears. In this way, a bewildering assortment of interlaced designs can be created.

So far the only aids to laying out a design are the vertices of the auxiliary grids. Additional construction lines are drawn between these vertices to indicate which crossings are to be removed from the plait and how the ends are to be rejoined. These lines are called break-markers. Each break-marker is an edge of one of the auxiliary grids. At each crossing point of the plait, there are two such edges, and the edge chosen indi- cates which one of the two possible reconnections is to be used. A glance at the example in Figure 4 will make the convention clear.

To complete the design, the path of the strings is outlined and the background is filled in. This obscures all the construction lines. The pattern is then interlaced to produce the characteristic alternating weave. The construction of some elemental knot motifs is illustrated in Figure 5.

Figure 4. Break-markers aid the construction by indicating how the crossings are to be broken.

Figure 5. Examples of interlaced motifs together with their construction.

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. I, 1993 37

Lindisfarne Gospel

(a)

I !

(b) Figure 6. Irregularly shaped panels can be decorated using the same techniques.

Eliminating the crossings according to the simple rule indicated in Figure 3 does not always produce aesthetically pleasing results. Sometimes a better result is achieved if the path of the strings is allowed to de- viate from the path of the plait. At places where two break-markers meet to form a corner, the path of the strings can also be made angular. This has been done in the accompanying figures. In other situations, fairly sharp bends can be replaced with more gentle curves to produce a more graceful, freely flowing design. Fig- ures 7 and 8d exhibits arcs of several different curva- tures. These modifications help to disguise further the under lying plait structure on which the design is based. Another variation produces a very delicate form of interlacing: What are normally taken to be the two edges of an interlaced ribbon are themselves used as the strings to be interlaced. An example of such 'double interlacing' is shown in Figure 8j.

Once the technique for constructing Celtic ornament has been understood, the art itself loses some of its mystery. It becomes possible to copy the ancient designs fairly accurately and easily and to create designs of your own. (Do not be surprised, though, if you discover your creations elsewhere.) Any rectilinear area can be filled with knotwork by regarding its boundary as being composed of break-markers on a suitable lattice (see Fig. 6a). Patterns can also be mapped into curvilinear regions by dividing them into quadrilaterals (Fig. 6b).

Simple though this construction procedure may appear, the idea that the Celtic interlaced patterns were related to plaitwork took many years to mature. The methods used by the Celts themselves are no longer known, and the above technique was developed by J. Romilly Allen while he was surveying the patterns in the British Isles at the turn of the century. He records [1] p. xvii:

The theory of the evolution of Celtic knotwork out of plaitwork . . . is entirely original, and, simple as it appears when explained, took me quite twenty years to think out whilst classifying the patterns.

When we remember the large number of modifications the underlying plait can undergo and the ease with which it is disguised, perhaps this is not so surprising.

Figure 7. This pattern is not based on the standard grid.

38 THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 1, 1993

I n t e r p r e t i n g C e l t i c D e s i g n s A s F r i e z e s

Much Celtic knotwork is in strip form, either as part of a border or simply as a narrow rectangular panel. In many of these strip patterns, the crossings are eliminated systematically and the pattern of break-markers repeats at regular intervals. This produces a pattern which can be described as "locally periodic": A single motif is repeated side by side. In other forms of orna-

ment, such periodicity gives rise to translational symmetry when the pattern is regarded as a randomly chosen segment of an infinite strip. However, in knotwork, the transition from a small segment to an idealised frieze is not so immediate. Whereas in other forms of decoration, a pattern is simply truncated when it reaches the edge of the available space, knotwork is rarely terminated so abruptly. The pattern is adapted so that the otherwise free ends join up to form continuous strands.

In this article, the knotwork patterns are regarded as friezes in the conventional mathematical sense: as parts of patterns which extend indefinitely in both di- rections. In the under lying lattice, the pat tern is bounded by two parallel lines of break-markers and the other breaks are arranged so that the pattern as a whole has translational symmetry. Some examples are shown in Figure 8. The first two patterns are not constructed on a standard lattice. The pattern in Figure 8a is believed to be Scandinavian. Triangular motifs such as that used in Figure 8b are normally arranged in sets of four to form square patterns. An example based on the same motif is shown in Figure 7. I have excluded patterns that can be split up into other patterns. For example, if the knot design in Figure 2 is converted into a frieze, the resulting pattern comprises two parallel patterns which are not interlinked. An interlaced pattern is said to be connected if the unlaced path (or equivalently, the projection of the link onto the picture-plane) is connected. The frieze constructed from Figure 2 is not connected. For the moment, I shall as- sume that the friezes are connected and that the notion of connectedness captures to some extent our intuitive understanding of when a pattern can be split up. We shall, however, return to this question of separability later.

The Symmetry of Interlaced Friezes

The symmetrical nature of frieze patterns is analysed mathematically in terms of symmetry operations: isometries which carry a pattern onto itself. It is well- known that there are four isometries of the plane: rotation, translation, reflection, and glide-reflection. For planar frieze patterns, the only possible symmetries are 2-fold rotation, reflection in the centre-line, reflection in a line perpendicular to the centre-line, and glide-reflection along the centre-line. These symmetries can be combined in different ways but the number of combinations is limited to seven by the rigidity of the geometry. Examples of patterns exhibiting these seven different symmetry types are shown in Figure 9.

When the knotwork friezes are compared with these planar patterns, it becomes apparent that we do not interpret the two kinds of pattern in the same way. The interlaced patterns are not confined to lie in the plane.

At the crossovers, the strings appear to extend behind and in front of the picture-plane. We perceive a three- dimensional object composed of continuous strings lying in a neighbourhood of the plane, not a collection of arcs lying in the plane. We expect to see the obverse pattern on the back.

I n t e r p r e t i n g the i n t e r l a c e d f r iezes as th ree - dimensional patterns means that there are additional isometries which can act as symmetries. Two of these are compound motions like the glide-reflection: a screw is a rotation followed by a translation along its axis; a rotatory-reflection is a rotation followed by a reflection in a plane perpendicular to its axis. Both of these isometries can act as symmetries of an interlaced pattern.

The complete set of possible symmetries of 2-sided friezes is described by reference to a set of standard axes: the a-axis runs along the band, the b-axis lies in the plane of the band perpendicular to a, and the c-axis is normal to the band. The possible symmetries are a 2-fold rotation about any of the three axes, reflections in the planes orthogonal to each of the three axes, glide-reflections in the planes orthogonal to a and b, a screw motion along a, and a 2-fold rotatory-reflection about c. This last symmetry is the same as reflection in a point or inversion (because it is a 2-fold symmetry).

There are many different ways in which these symmetries can be combined. As these 2-sided friezes are less familiar than the seven 1-sided ones, patterns il- lustrating the 31 symmetry types [5] are shown in Fig- ure 10. The motif in each of the patterns is a scalene triangle which, in most cases, is coloured black on one side and white on the other. When reflection in the plane of the strip is a symmetry, then both sides of the triangle must be the same colour: in this case the tri- angles are coloured gray. Next to each pattern is a label of the form P[313U] which encodes the symmetries present in the pattern. After the symbol P (which denotes that the pattern is periodic in one direction) the symbols 2, 2', 2, m, and a are used to indicate that an axis of 2-fold rotation, 2-fold screw, 2-fold rotatory- reflection, or the normal vector of a mirror plane or glide plane coincides with one of the reference axes. The first, second, and third symbols after the P corre- spond to the a-axis, b-axis, and c-axis respectively. If an axis and a plane of symmetry coincide with the same reference axis, both symbols are given; if no symmetry element corresponds, then the symbol I is used as a place marker.

Trying to identify which symmetries are present in a particular knot pattern is not trivial. The reader is en- couraged to experiment on the patterns in Figure 8. A useful observation is that the crossings in the Celtic friezes all have the same alignment and that they come in two kinds. Furthermore, each kind is stabilised by all the 2-fold rotations (see Fig. 11). Thus, direct symmetries carry a crossing to one of the same kind; indi- rect symmetries interchange the two kinds.

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 1, 1993 39

Sutton Hoo Buckle Pla l

(a)

Rossie Priory Stone Plal

(b)

Book of Kells P2'1 1

(c)

Book of Durrow P121

(d)

Lindisfarne Gospel P121

(e)

Figure 8. Examples of Cel t ic f r ieze pa t te rns ( th rough p. 42).


Lindisfarne Gosoel P222

(f)

Maiden Stone. Aberdeenshire P222

(g)

St. Vioean's Stone, Tayside P211

(h)

I i n H i . ~ f ~ r n ~ Gosoel P2'22

(i)

F i g u r e 8 i s c o n t i n u e d o n p . 42.


Figure 8. Continued.

4 2 THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 1, 1993

The Celtic Frieze Groups

After analysing a few examples, one is led to ask how many of the 2-sided groups can arise from Celtic knot patterns. Can they all occur, and if not, then which ones? The slightly different question of which ones actually occur in practice can also be considered.

The seven "gray groups"-- those which have the picture-plane as a symmetry element---cannot occur for knot patterns because the crossing points do not obey this symmetry. Therefore, we can eliminate all the groups whose labels have an m as part of the last symbol. The observation that virtually all interlaced ornament are alternating allows further groups to be eliminated.

Suppose that a knot motif has mirror symmetry. An example is shown in Figure 12a. To convert this knot into an alternating one, it is necessary to add an extra string which lies in the mirror plane as shown in Figure 12b. Those groups which contain reflections in planes orthogonal to the a-axis cannot arise from alternating knotwork patterns because the strings that lie in the mirror planes would run directly from the top edge of the band to the bot tom-- they could never join up with anything else. Thus, those groups whose labels have an m in the first position cannot be found in Celtic patterns. The groups whose labels have an m as part of the second symbol can arise from alternating patterns: such patterns must have a string that runs straight down the centre-line of the band. There are two of these groups: Plml and p121. They are marked with a dagger (t) in Figure 10. Patterns of this form are not consistent with the standard Celtic grid, however, and so these symmetry groups cannot arise in Celtic patterns either.

There is one more class of group labels that can be eliminated: those which contain an a as part of the third symbol (indicating that the picture-plane is also a glide-plane). The reason for this is that to create an alternating design, straight strands must run across the band as in the Pm[3D case, and these strings can never join up with anything.

There remain ten possibilities. These are marked with an asterisk (*) in Figure 10. Examples of Celtic patterns exhibiting each of these symmetry groups are shown in Figure 8. Where I have been able to find ancient designs I have used them; a few are my own creations.

The Relative Abundance of Symmetry Types

In my search for examples of Celtic designs, I discov- ered that the abundance of the different symmetry types varied greatly. Some groups (Pl12, P222, P2'22) were very common; others were rare. In fact, the only patterns I could find which have group Plal are not constructed on the standard lattice described above.

RRRRRRR

PbPbPb

, AAAAAAAA,

}BBBBBBBB

/AVAVA { q qHHH H

Figure 9. The seven 1-sided frieze patterns.

For the two groups Pl12 and p121, I found no examples at all. Are there any features of these patterns that make them difficult to obtain?

Before resolving this puzzle we shall consider another question: Is there any correlation between the (2-sided) symmetry group of an interlaced design and the (1-sided) symmetry group of its underlying pattern of break-markers? The answer to this is yes. To see why, note that the unlaced path of the strings and the distribution of break-markers have the same symmetry type. The correspondences are listed in Table 1. Ob- serve how each of the three rare groups is paired with another group which appears to be the preferred op- tion. In fact, this preference is not a matter of a choice having been made by the designer, consciously or otherwise. It is a consequence of the fact that the ancient patterns terminate and are not true friezes.

Study the two sections of plaited friezes in Figure 13. Do you notice any differences? Structurally, they are the same; symmetrically, they are not. This is seen most easily along the edges. In Figure 13b, the outer- most crossovers are opposite each other; in Figure 13a they are staggered with those on one edge lying between those on the other. The symmetry types of these two plaits are, therefore, different.

The symmetry type of a plaited frieze depends on its width. Those plaits that have an even number of lattice cells between the two edges have symmetry type P222; those plaits that span an odd number of cells have type p121. This observation allows us to resolve the mystery of the missing groups. It also provides an alternative method for enumerating the Celtic groups; because eliminating crossovers can only destroy symmetry, the Celtic groups must be subgroups of P222 or p121.


Figure 10. The thirty-one 2-sided frieze patterns.

Table 1 can now be refined to show how the symmetry type depends on the width of the frieze as well as the underlying markers. The result is shown in Ta- ble 2. We now have at most one group per box of the table: The symmetry type of the interlacing is completely determined by the geometry of the underlying break-markers. Furthermore, all the rare groups are associated with friezes of odd width. The only frieze I found that has odd width is shown in Figure 8d. Its symmetry type is P121--a group which is independent of the width of the frieze.

It is not difficult to discover why odd-width friezes are uncommon. It is not, as has been suggested [1, p. 260], that the patterns look lopsided. Rather it has to do with the fact that the original Celtic patterns are finite designs with no loose ends. At the end of a row of motifs, the strings are paired up and joined--a pro- cess which requires an even number of strings. The number of strings is related to the width of the frieze: The parity of the width equals the parity of the number of strings. The Celts could only use odd-width friezes in situations where continuity could be ensured, such

4 4 THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 1, 1993

1-sided

2-sided

R

P l l l

~P

Pla l

P2'l 1

A

P121

B

P211

H Pl l~ Pl12

VA p121

P2'22 P222

Table 1.

1 -sided

2-sided (odd)

2-sided (even)

P l l l

P l l l

Pla l

P2'l 1

A

P121

P121

B

P211

P l l~

Pl12

VA

Pla21

P2'22 P222

Table 2.

Figure 11. Each of these 2-fold rotations preserves the crossing.

(a) (b)

Figure 12. A pattern with bilateral symmetry can be made alternating by adding an extra strand.

(a) (b)

Figure 13. Alternating braids with five and six strings.

, % f % f % f ~ f ~

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 1, 1993 4 5

4x4 3x5 4x6 5x5

Figure 14. Can the number of components in a rectangular motif be determined from the dimensions of the rectangle?

as in complete borders. In some places, loose ends occurred naturally. These are often in zoornorphs: strange creatures whose elongated tails, limbs, necks, or tongues are intertwined in fanciful patterns. How- ever, these patterns are never large enough to have repeated motifs and so will not provide examples of friezes.

Cont inu i ty , Transi t iv i ty , and Separabi l i ty

One problem faced by the designer of interlaced patterns is that of determining the number of components in the completed design. In early forms of Celtic ornament, it is clear that continuity of the path was sought. It was important as a symbol of eternity. Even ex- tremely intricate patterns have just one string arranged in an endless loop. In some examples, the regularity of a pattern has been deliberately abandoned and the pattern modified to ensure that a unique path was ob- tained. In later times, this rule was followed less strictly, but small rings in a pattern were still avoided. This raises the question of whether there is a simple way to determine the number of strings in a design from the underlying pattern of break-markers.

When the interlacing is a rectangular portion of plaitwork (Fig. 14), then the answer can be expressed in terms of the bounding rectangle. If the lattice underlying the plait contains n x rn cells, then the pattern will have a single component if h.c.f. (m, n) ~ 2. If the pattern is square (n = m), then the number of components is [1/2n]. In fact, if we count closed loops rather than components, then the number I/2n can be regarded as correct. When break-markers are added and the plaitwork is broken up, these rules are no longer valid. What rules replace them?

How about interlaced friezes: How many strings do they have? The pattern in Figure 8f is a chain composed of infinitely many closed loops. All the other friezes in Figure 8 have a finite number of infinitely long strings. Most of the patterns from Celtic sources have an even number of strings; only the nonstandard patterns a and b, and my designs l, m, and n of Figure 8 have a single strand; Figure 8d has three.

For patterns with more than one component, we can investigate whether the symmetry group acts transitively on the strings: Can each string be carried onto any other string by some symmetry of the pattern? In the context of layered patterns and fabrics, a pattern which is string-transitive is said to be isonemal. Adopt- ing this terminology, we can say that of the patterns in Figure 8, c and e are not isonemal, and all the others except i and j are isonemal by translation. In cases i and j, rotations are required to achieve complete transitiv- ity.

There is a simple test to check whether an interlaced frieze pattern is isonemal. Construct the quotient link by joining the two ends of a fundamental region of the frieze together. If this quotient link has a single component, then the frieze is isonemal by translation.

Another problem arising in the mathematical study of fabrics is that of determining whether a layered pattern falls apart: Can the strings be separated into two or more sets which are not interlinked? We assumed above that friezes are connected. One consequence of this is that Celtic interlaced friezes never fall apart. If an interlaced frieze is alternating, connected, and sep- arable, then so is the quotient link constructed from it. However, alternating diagrams of links represent split links if and only if they are not connected [4].

A Finer Class i f icat ion

Classifying the multitude of Celtic interlaced patterns into only ten classes is, in some respects, not very satisfactory especially when we realise that the resulting three-dimensional symmetry type is completely determined by the underlying two-dimensional pattern of break-markers. It would be nice to have some kind of classification by pattern type [3] rather than merely by symmetry type.

Intuitively, a pattern is a collection of motifs arranged in a systematic fashion. This regularity is mod- elled mathematically by requiring that the symmetry group of the pattern acts transitively on the motifs. Classification by pattern type depends on three factors: the symmetry group G of the pattern; the stabiliser


stabG(M) of a motif M; and the set of motif transitive subgroups of G. Two patterns are said to have the same pattern type if all these features coincide.

When we try to apply this kind of analysis to Celtic friezes, however, we immediately run into difficulties: The patterns are not discrete, so there is no obvious or natural way to split them up into motifs. The continuity of the strings makes it impossible to choose a motif in an unambiguous way, and the choice made will affect the resulting pattern type. Ironically, the ar- rangement of break-markers associated with an interlaced frieze pattern is discrete but not necessarily transitive.

Coda

In this study of Celtic knotwork I have concentrated on one particular theme: symmetry. That an analysis in terms of symmetries is possible is due partly to the lattice structure underlying the construction of the designs. This inbuilt regularity and the imposition of periodicity by the artist means that many of the patterns have nontrivial symmetry. The reader may feel that this same rigidity would lead to a dull and sterile art form. A nonmathematician confronted with the mathematical classification of patterns according to their symmetry type wrote [2, p. 70],

There is no danger that the resources of the pattern maker will be exhausted by the constraints of geometry.

The remark seems appropriate in this context, too. A glance at any of the illuminated manuscripts produced by Celtic scribes will easily convince you that a geo- metric framework in no way hinders the artist. There is still room for imagination and creativity to express themselves.

I. Bain, Celtic Knotwork, London: Constable (1986). A. Meehan, Celtic Design: knotwork, London: Thames and

Hudson (1991).

Related Topics

H. Arneberg, Norwegian Peasant Art: men's handicrafts, Oslo: Fabritius & Son (1951).

K. M. Chapman, The Pottery of San Ildefonso Pueblo, School of American Research, monograph 28, Albuquerque: Uni- versity of New Mexico Press (1970).

D. W. Crowe and D. K. Washburn, Groups and geometry in the ceramic art of San Ildefonso, Algebras, Groups and Geometries (2) 3 (1985), 263-277.

B. Grfinbaum, Periodic ornamentation of the fabric plane: lessons from Peruvian fabrics. Symmetry I (1990), 48-68.

B. Gr~nbaum and G. C. Shephard, The geometry of fabrics, Geometrical Combinatorics (F. C. Holroyd and R.J. Wil- son, eds), Pitman (1984), 77-97.

B. Grfinbaum and G. C. Shephard, Interlace patterns in Is- lamic and Moorish art, Leonardo (in press).

B. Grfinbaum, Z. Grfinbaum, and G. C. Shephard, Symme- try in Moorish and other ornaments, Comp. & Maths. with Appls., vol 12B, Nos. 3/4 (1986), 641-653.

A. Hamilton, The art workmanship of the Maori race in New Zealand, Wellington: New Zealand Institute (1896).

I. Hargittai and G. Lengyel, The seven one-dimensional space-group symmetries in Hungarian folk needlework, J. Chem. Educ. 61 (1984), 1033.

G. H. Knight, The geometry of Maori art. Part I: rafter patterns, New Zealand Math. Mag. (3) 21 (1984), 36--40.

G. H. Knight, The geometry of Maori art. Part 2: weaving patterns, New Zealand Math. Mag. (3) 21 (1984), 80--86.

D. K. Washburn and D. W. Crowe, Symmetries of Culture: Theory and Practice of Plane Pattern Analysis, Seattle: Uni- versity of Washington Press (1988).

Department of Pure Mathematics The University of Liverpool PO Box 147 Liverpool, L69 3BX England

References

1. J. Romilly Allen, Celtic Art in Pagan and Christian Times, London: Methuen (1904).

2. E. H. Gombrich, The Sense of Order: a study in the psychol- ogy of decorative art, Ithaca, NY: Cornell University Press (1979).

3. B. Gr~nbaum and G. C. Shephard, Tilings and Patterns, New York: Freeman (1987).

4. W. W. Menasco, Closed incompressible surfaces in alternating knot and link complements. Topology 23 (1984), 37-44.

5. A. V. Shubnikov and V. A. Koptsik, Symmetry in Science and Art (translated from the Russian by G. D. Archard), New York: Plenum (1974).

Further Reading

Celtic Knotwork G. Bain, Celtic Art: the methods of construction, London: Con-

stable (1977).


celtic knotwork: mathematical art

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