THE DECORATIVE SYSTEM OF THE LAPITA POTTERS OF SIGATOKA, FIJI

Size: px
Start display at page:

Download "THE DECORATIVE SYSTEM OF THE LAPITA POTTERS OF SIGATOKA, FIJI"

Transcription

1 Part 2 THE DECORATIVE SYSTEM OF THE LAPITA POTTERS OF SIGATOKA, FIJI by S. M. Mead INTRODUCTION In this section, I analyse the decorative system inferred from a study of pottery fragments recovered from two sites near Sigatoka, Fiji, by Mr and Mrs L. Birks. The fragments from the deepest levels of the two sites exhibit a style of decoration which appears, by inspection, to be closely related to decorated sherds from sites as far west as New Guinea and as far east as Tonga. The style has come to be known familiarly as the Lapita style and the pottery as Lapita Ware. Such pottery fragments are normally the earliest known at each of the sites from which they have been recovered, though pottery sites of equal age belonging to other cultural traditions are also known. The similarity of the decorations on Lapita Ware has been noted by many investigators, and tentative comparisons have been made by some of these authors.(1) Comparisons of a more controlled and rigorous nature have not yet been possible, however, partly because archaeological investigations in Lapita-bearing sites are comparatively few, and to a greater extent because detailed descriptions of the decorative systems from known sites are lacking. In this situation, the occurrence of decorated sherds in one of the two Sigatoka sites in sufficient quantity to warrant a full scale analysis presents an opportunity to apply a method of treating the decorative system which I believe will open the waiy for comparisons between sites on a more formal and systematic basis. This article, then, goes beyond a mere listing of elements and motifs to a structural approach in which an attempt is made to reveal the steps and rules by which patterns were constructed. As such, it differs from the approach used by Poulsen for the decorative system on Tongan pottery,(2) the only other case of a reasonably full description, because it yields not only an inventory of design elements and motifs but also a set of design processes and pattern-making rules. 1. Specht 1968; Golson 1968,1971; Poulsen 1967; Palmer 1966, Poulsen (1967) provides a full listing of design elements and design motifs. 19

2 THE LAPITA STYLE OF FIJI & ITS ASSOCIATIONS The importance of this approach in the wider field of Oceanic prehistory is to shift the basis of comparisons between assemblages of sherds decorated in the Lapita style from the concentration on simple identities in content and technique to a consideration of similarities and differences in their structural organisation as well. However, as broad-scale comparisons will depend on the recovery from other sites of a representative sample of decorated sherds and their adequate analysis and rapid publication conditions which are now lacking such comparisons as are attempted here must necessarily be very tentative and subject to revision. The emphasis, therefore, will be on setting out and demonstrating the techniques to be employed. METHOD OF ANALYSIS The assumptions and termsused in this analysis require some clarification. It is assumed that the decoration applied by potters to their pottery consists not of elements haphazardly arranged but rather of elements arranged in a systematic manner. The decorative systems may be broken down into three main parts: a set of techniques by which design elements can be given visual form; an inventory of design elements and units; and a set of design processes which may be couched in the form of rules. All three parts yield data which are susceptible to comparison for the purpose of inferring cultural relationships. The absence of prehistoric informants from Sigatoka makes it impossible to isolate and confirm culturally recognised design elements, that is to say, elements of an emic character, for which the potters and pattern-makers probably had special linguistic terms. Since informant confirmation is impossible, reliance will be placed on one important criterion, namely that of frequent occurrence. Each design element or unit which is isolated as a necessary part of the design inventory must occur more than once, either on the same sherd or on different sherds and either from the same site or from different sites. The justification for this point of method is that confirmation of the discreteness of an element is necessary before it can be given the status of a design element. Without confirmation the element might represent only an accidental slip of the potter s hand. A further assumption made here is that a decorative system can exist independently of the artefacts to which it is applied. The design elements used on Sigatoka pottery were not necessarily applied only to ceramics but could have been applied to textiles, woodcarving and tattooing as well. While it is true that some design elements might appear only in one medium and not in another, it is assumed that the system of decoration, nevertheless, remains much the same. THE CORPUS As mentioned earlier, the corpus of sherds comes from two sites excavated by Lawrence and Helen Birks in 1965 and 1966.(3) The analysis which follows is based primarily on the cave-shelter material from Yanuca Island, there being only six decorated sherds from the other 3. Details are described in Parti, above. 20

3 S. M. MEAD, L. BIRKS AND E. SHAW site.(4) For convenience the corpus is henceforth referred to as the Yanuca material. Technical details about the shapes of pots, manufacturing technique, texture and size are given in Part 1. THE DESIGN FIELD Yanuca pots are decorated on the outside walls, sometimes on the underside, and frequently on that part of the everted dm which can be seen by an observer. Favourite design fields are the inside of the everted rim and the outside walls. In the majority of cases, the design field is divided into zones which may or may not receive further decoration. Typical in Yanuca pottery decoration is subdivision of the outer surface of the pot into transverse bands one below the other. The wider bands may be further divided into squares and rectangles and these in turn further reduced either by adding internal squares or by drawing in diagonals which subdivide the space into triangles. Zoning into circles does not occur in Yanuca pottery, though thereisevidenceof at least one motifwhichis based on a circular arrangement. The zoning of the design field in Yanuca decoration is a very important feature of the decorative system. It would appear that the entire outer surface of a pot was never regarded by Yanuca potters as one large design field. Instead, it was conceived as a space which could receive so many transverse bands of decoration. Associated with the idea of arranging the patterns in bands was the notion that a pattern ought to be confined within a clearly demarcated zone and not left hanging in space. Strictly speaking, the boundary markers which confine patterns into a restricted space are not design elements. Their primary function is to mark the boundaries of a design field or zone, but Yanuca potters have added a decorative dimension to these markers by using a decoration-producing technique to give them visual form. In some cases, too, zone markers comprise the only decoration on a pot. Thus, although the basic element is really a straight line, the techniques used to produce such a line result in visually different kinds of decorative division, which justifies their being treated as design elements. In the following analysis, boundary markers are analysed as design elements. A classification of design zones is given in the next section following the description of zone markers. A rule stated for constructing a pattern does not include an instruction to produce a specified type of zone. However, where the sherds show clearly the type of zone in which a pattern is placed, the type may be indicated. PROCESSES, RULES AND TERMINOLOGY A rule by which any particular pattern is produced can be isolated and stated as a formula. Any rule given in this analysis is not necessarily the same rule which the prehistoric potters applied when they were decorating pots. The only claim made here is that the rule given will produce the same result. In some cases, it is possible to retrace the steps taken by a decorator in constructing a pattern with a fair degree of accuracy and in others it is 4. At the time of writing, a full count of the decorated sherds from Yanuca had not been made. 21

4 THE LAPITA STYLE OF FIJI & ITS ASSOCIATIONS possible to follow the stages of development which led up to the creation of a motif. When developmental stages are described, the rules tend to be very complicated because all design-making processes which are applied to a simple element in order to transform it into a motif need to be isolated and described. Such developmental stages are described in a number of cases, but it is neither economical nor profitable to do this in all cases. The rules tend to be much simpler when they are stated for a given motif. A general aim of the description is to choose the simplest rather than the most complicated rule. To accomplish this aim, it is often necessary to isolate a unit which the native potter may not actually have visualised. But, as pointed out earlier, repetition of the arbitrarily isolated unit nevertheless produces the pattern. The statement of rules in iconics A rule for constructing a pattern is stated as follows: P = M8.1 -> ConR/E-W/CLS. That is to say, alloform M8.1 of motif 8 is subjected to continuous repetition in a transverse or east-west direction*6*in such a way that coalescence occurs at each junction where one unit meets the next. The key to setting out a rule is as follows: P =/1/2/-*/3/4/5. P equals pattern, slot 1 identifies the element or motif, slot 2 identifies the process applied to 1, slot 3 identifies the major pattern-making process, slot 4 identifies the direction in which the element or motif is repeated and slot 5 identifies the secondary process applied. Not all slots need be filled; for example, slots 2 and 5 are often not used. Abbreviations grouped under numbered slots where they are most likely to be used are explained below. Slot 2, rep repose, i.e. lie on its side; rev reverse; rd round; com compress; db double; sq square; tri triangular. Slot 3. ConR continuous repetition; DisR discontinuous repetition; D M arrange in half-drop mesh. Slot 4. E-W east-west direction; N-S north-south direction; Multi multi-directional. Slot 5. INT interlock; SUP superimpose, place over; MR present a mirror image; DECR decrease in size; CLS coalesce at union. Incidental abbreviations and signs, n unspecified number; # separate by space; apply processes which follow; ( ) regard as one unit of repetition; regard the patterns above and below the line as one complex pattern. The formula P = M M18.1 (rev/cls) -> ConR/E-W/CLS is to be read as follows: the unit to be repeated consists of Ml8.1 against which another M18.1 is placed in reverse position. Coalescence occurs at the junction of the two. Then this composite unit is subjected to continuous repetition in an east-west direction, with coalescence occurring at each junction. 5. In stating directions in this way, I am following the practice introduced by William Watt in his Nevada Cattlebrands reports (1966,1967). 22

5 S. M. MEAD, L. BIRKS AND E. SHAW Rules for complex patterns, as in the case above, tend to be more complicated than others but are relatively few in number. THE CLASSIFICATION OF DESIGN ELEMENTS To facilitate description, design elements are classified according to the following criteria: 1. On whether they are three-dimensional or two-dimensional. 2. On whether they are simple and discrete or whether they are complex and contain other elements. 3. On the function of an element in the building up of patterns. The justification for classifying the elements is to ensure that elements of like type are kept together both in the description and in the comparative tables to follow. Yanuca design elements may be divided into two broad categories. In the first category are those elements which are actually modelled in clay and then attached either to the inside surface of the pot or, as is more common, to the external surface. All other elements are two-dimensional and are given visual form by impressing, incising or stamping the surface of the pot. Each of the broad groups of elements can be further subdivided into other categories: namely, simple design elements which cannot be further analysed into smaller elements and complex elements which consist of more than one element. Complex elements are here regarded as being the same as motifs and are treated separately. Also within the second broad division of elements is a category distinguished by its specialised function. This category includes what have been previously referred to as boundary or zone markers. Each element or motif analysed and isolated is given an identifying letter or letters and a numerical label. Each significant variant form or alloform is identified as a decimalised number. Alloforms are regarded simply as variants of a motif or element and do not necessarily appear in a different kind of design zone from other alloforms. CLASS 1. THREE-DIMENSIONAL DESIGN ELEMENTS As already stated, a three-dimensional design element is a feature which is actually added to the shape of the pot. The feature is modelled separately and is then attached to the wall of the pot, either on the upper part of the everted rim or below the rim, or around the body of the pot. Another type of Class 1 element is actually part of the pot but this is discussed in Appendix 1. Analysis of Yanuca sherds reveals three groups of three-dimensional design elements which are summarised below in Table 9. T able 9 Summary of Three-dimensional Design Elements Nubbins Vertical bars Transverse bars Figures Figures Figures N l.l (cone) VB2.1 (rd) TB3.1 (sq) N1.2 (sq) VB2.2 (sq) TB3.2 (rd) N1.3 (com/sq) TB3.3 (tri) Nl.4 (com/rd) N1.5(rd) 23

6 THE LAPITA STYLE OF FIJI & ITS ASSOCIATIONS Each group is described in detail in Appendix 1. Here it is sufficient to point out that the alloforms listed result from the application of design processes, such as squaring (sq), compressing (com), or rounding (rd), which change the shape of the basic element. The ideal or basic shape for a nubbin is assumed to be a cone shape (N l.l) which is then modified in various ways according to the effects desired by the designer. There are two basic shapes for vertical bars: the rounded bar (VB2.1) and the squared bar (VB2.2,) and in transverse bars there is, in addition, a third shape which results when a triangular cross-section is desired (TB3.3). When Class 1 elements are arranged into patterns the following rules apply: (a) For nubbins :P =N1 -+ DisR/E-W. That is to say, each nubbin is repeated discontinuously in an east-west direction so that each nubbin is separated by an interval of space. (b) For vertical bars: P = VB2 -> DisR/E-W. (c) For transverse bars: P = TB 3 ConR/E-W. A transverse bar normally forms a continuous ring around a pot, and it is thus the ideal example of continuous repetition in an east-west direction. CLASS 2. ZONE MARKERS The zone markers used by Yanuca decorators may be classified into two main categories, namely general zone markers (Figure 2.11) and restricted zone markers (Figures 2.12, 2.13). The results of analysis are summarised in Table 10. Detailed descriptions are in Appendix 2. General (Figure 2.11) T able 10 Summary of zone markers Restricted (Figures 2.12,2.13) GZ1 (double dentate) RZ1 (multiple, transverse) GZ2 (single dentate) RZ2 (multiple, oblique) GZ3 (single incised) RZ3 (multiple, diamond) GZ4 (single rubbed) When zone markers are applied to the surface of a pot, the result is a rearrangement of the design field into a number of characteristic zones which are listed below. Types of design zone (Figures 2.14,2.15) Zone Type Al. A single continuous border with upper and lower boundaries clearly defined by zone markers. Zone Type A2. A series of continuous borders one below the other in which the lower boundary of the uppermost border becomes the uppermost boundary of the next one down and so on. Note that Al and A2 are basically the same type, hence both are classified as Type A. Zone Type B. A single border which is divided into rectangular or square areas by the use of vertical zone markers. Zone Type C. A series of Type B borders one below the other forming a 24

7 S. M. MEAD, L. BIRKS AND E. SHAW o _o o Fig. 2.1 N 1.1 Fig. 2.2 N1.2 Fig. 2.3 N 1.3 Fig 2.4 N1.4 Fig. 2.5 V B 2.1 Fig. 2.6 VB 2.2 O Q O Fig. 2.7 VB 2 + N1 Fig.2.9 TB 3.2 Fig.2.10 TB 3.3 Three-dimensional design elements 25

8 THE LAPITA STYLE OF FIJI & ITS ASSOCIATIONS GZ QZ (incised) GZ 4 (rubbed) Fig General zone m arkers Fig RZ3 as decoration. P Fig.2.15Typcs of design zone E -H Fig Types of design zone A - D Zone markers and types of design zone 26

9 S. M. MEAD, L. BIRKS AND E. SHAW design grid. Note, however, that Type B is suitable for a border pattern and Type C for an all-over pattern. Zone Type D. Small triangular areas which are the result of tertiary zoning. Zone Type E. Borders of varying widths in which one of the boundaries is defined by the lower structural limits of the vessel. Zone Type F. Narrow borders on the inside of the lip in which the lower boundary is defined by a zone marker or by a continuous transverse bar or by both. Zone Type G. The flat edge of the lip or of a structural transverse bar (TB3.1) in which both boundaries are structurally defined. Zone Type H. A border in which the upper boundary is defined by the rim and the lower boundary may be marked either by a zone marker or by a sequence of nubbins, vertical bars or transverse bars. In some cases, the lower boundary is not defined. It does not follow that a design zone of any type must be decorated. There are many examples among the sherds recovered of clearly marked zones without any patterns in them. Although one zone type only may be applied to a pot, there are many examples of one zone type being combined with another or with several others, as shown in Figure F ig u r e 2.16 Types of design zone in relation to vessel form 27

10 THE LAPITA STYLE OF FIJI & ITS ASSOCIATIONS a DE 2.1 -TT 'I T7- I' TT I) DE 1.2 Fig 2.17 Design element 1 b DE 2.2 DE 2.3 Fig 2.18Design element 2 Q a DE 3.1 Q Q Q Q Q Q a DE v //is III // n o' in /«r v> y Mm Fig.2.19 Design elem ent 3 Fig. 2.20Design elem ent 4 DE 5 Mil 11 *»Ml I l I I I ill _llu! I! 11! M Fig Design elem ent 5! r,, _, I I A / / ^ J I' I I Xs\ V / / / s II ll «----- \ \ ^ M ii i / V i1 1 a bu NV \ / ' [ i' D E 7.1 D E 7.2 Fig. 2-23Design elem ent 7 // // '/ // // ^ y - -~ a DE 6.1 // '/ ;/ // // / // // / / / / b * /'/ DE 6.2 / / / Fig Design clem ent 6 DE 8.3 Fig 2-24 Design element 8 ^ A /^. /' V_ i a M 1.11 < "Ā ā / ^ y \, ^ b M 1 pattern Fig.2.25 M o tif 1 xx! y \ i y " - s v. A. A. - - V. / x ^, - X.. ' X., ' x y ' Ā I > 1 b '(M 2.2) Fig-2-26 M otif 2 28

11 S. M. MEAD, L. BIRKS AND E. SHAW CLASS 3. TWO-DIMENSIONAL DESIGN ELEMENTS Class 3 design elements are flat, two-dimensional elements which are discrete and stand alone in a pattern arrangement. They may also enter into combination with other elements to form motifs. The eight recognisable design elements in this corpus of material are described briefly below. Full details are given in Appendix 3. DEL This element consists of a single crescent which is placed in a vertical position within a design zone. It is alloform DEI. 1 if single lines are used and alloform DEI.2 if the crescents are doubled (Figure 2.17). When combined in a motif it may be reposed (Figure 2.17c) or reversed. DE2. This element consists of two crescents placed one above the other end to end. The single line version is alloform DE2.1, the double line version is alloform DE2.2 and the triple line version is alloform DE2.3 (Figure 2.18). DE3. This is a small circle which appears to be stamped into the clay either with a hollow cylindrical stamp or with a solid cylindrical stamp (Figure 2.19). Thi6 element is often used as a part of a motif as in M20 and M33. DE4. This element is in the form of an oval in which the ends tend to be sharp pointed (Figure 2.20). DE5. This element is a straight line which is stamped along its length with a stamp often thicker than those used for zone markers. It is used invariably in a vertical position. The element is usually clustered in groups of from two to five(figure2.21). DE6. This element is an oblique line. Its most common alloform is represented by a double line (DE6.1). Favoured zones for its occurrence are Types A1 and E, in which the element is usually drawn across the zone from the lower to the upper boundary. The element appears also in Type A2, where it is usual for it not to touch the upper zone (Figure 2.22). DE7. This element takes the shape of a chevron which may terminate in fingers as in DE7.2. Alloform DE7.1 is a plain chevron represented by a double dentate line (Figure 2.23). DE8. Also functioning as a discrete element is DE8, a triangular shape which is best represented by notching on vessel rims (Figure 2.24). Alloform DE8.1 represents a single triangular notch, alloform DE8.2 consists of two triangular notches placed side by side and DE8.3 is a more complicated form. THE RULES FOR DESIGN ELEMENTS For the majority of alloforms of design elements, there is a simple pattern-making rule. The element is repeated discontinuously in an eastwest direction. Discontinuous repetition is characteristic of design elements, DE8 being the only one in which the elements may actually touch one another so as to form a continuous repetition. Reference to Table 11 will show that there are a few instances, such as in DEI, DE2 and DE5, where the rules appear to be rather complicated. In each case, the last part of the rule is straightforward. The complication arises from what is done to the basic design element before it is repeated to form a border pattern. For example, rule (b) for DEI. 1 appears complicated 29

12 THE LAPITA STYLE OF FIJI & ITS ASSOCIATIONS because the decorator has placed a mirror image of DE1.1 against itself and then regarded the pair of elements as the unit to be repeated. The rule for DE2.1 reflects a complexity which the decorator has introduced by grouping DE2.1 elements together in a group of five against which another group of five is placed as a mirror image of the first. The unit for repetition consists of these two groups of five elements. There are two basic pattern-making rules used in Yanuca: Rule 1 which creates a continuous border pattern and Rule 2 which creates a discontinous border pattern. It will be seen in Table 11 that the main rule applied to design elements is Rule 2. The use of Rule 1 is rare. The application of rules is discussed more fully below. T able 11 Design elements and their rules Rule Element Alloform Rule No. Class 3 elements DEI DE1.1 (a) P = DE1.1 -> DisR/E-W 2 (b) P = DE1.1 + MR/DE1.1 DisR/E-W 2 DEI.2 P = DEI.2 DisR/E-W 2 DE2 DE2.1 (a) P = ((DE2.1 5/E-W) -> MR/E-W) DisR/E-W 2 (b) P = (DE2.1 -> DisR x N/alt with MR/DisR x N) -* DisR/E-W (where N = variable 2 number from 5 to 10) DE2.2 P = DE2.2 -* DisR/E-W 2 DE2.3 P = DE2.3 DisR/E-W DE3 DE3.1 P = DE3.1 -* DisR/MULTl 4 DE4 DE4.1 P = DE4.1 -> DisR/E-W 2 DE5 DE5.1 (a) P = DE5.1 DisR/E-W 2 (b) P = (DE5.1 x 4 or 5) fl-* DisR/E-W 2 DE6 DE6.1 P = DE6.1 DisR/E-W 2 DE6.2 P = DE6.2 - DisR/N-S 3 DE7 DE7.1 P = DE7.1 -*> DisR/N-S/DECR 3 DE8 DE8.1 P = DE8.1 -> DisR/E-W 2 DE8.2 P = DE8.2-* ConR/E-W 1 DE8.3 P = DE8.3 - DisR/E-W 2 Class 1 elements N1 P = N1 -* DisR/E-W 2 VB2 P = VB2 DisR/E-W 2 TB3 P = TB3 -> ConR/E-W 1 THE INVENTORY OF DESIGN MOTIFS The majority of patterns applied to Yanuca pottery are based on sequences of motifs. There are 33 motifs (Table 12 and Figures ). The last three motifs in the list, M31, M32 and M33, are unconfirmed and tentative because there are no sherds in the corpus which show either a full pattern sequence, or even a complete unit of the sequence. Nevertheless, what is preserved on the sherd fragments available is sufficiently clear to enable one to make an intelligent guess as to what is missing. Detailed descriptions of the motifs and some of the problems associated with their definition are in Appendix 4. 30

13 S. M. MEAD, L. BIRKS AND E. SHAW \ A / \ A A A!'< >Ct >' _V / - 1" - a T--- < b Fig M o tif 3 1 _! _.>/ T. '=C 1V 'f '~ - " 'j'"1''. b Fig M o tif 4 nn 'V'--- *i " ' x r > c x : : > o < b Fig M o tif 5 71 ' *. 1 N^ ~ - a x'' 'w V '. y*l ' "'JIS - ~ x' "v_ A * x b... Fig 2.31 M otif 7 i i i [! i {! J <i, v» 1 1 > ; ' i!! :!! b > j ' A*» ^ j * Fig.2.33 M qtif 8 ( M 8.2 ) i i ;; HI I! 11 I I I ; i'j'a ( M 9. 1 ) (M 9.2) i *i i1 11 J 1 i i 11i i ; ;! ; j i ; ; j i b ' ' i»i i. i i i, 11t 111M 1I 11 M i I 111 ctm 9.V ) in'. i - %~ *>' a /av ^/\s b Fig.2.30 M otif 6 /a% 1 1 ~tt i! i! j!! :!.A, A. A.A..A.A. b F ig 2.32 M o tif 8 (M 8.1 )!! J J i!! i!! } } i i ; i j! i i j j!!! I! O i i i 1i! 1i ' a!!!!!! b c Fig.2.34M otif 8 (M 8.3 ) \ [!! Vj a M 10.1 I 'l J 1(1»II 1M M il II /1 1III II j 1111 II \>»,-XV - X ' ' Fig.235 M o tif 9 Fig 2.36 M o tif 10 31

14 THE LAPITA STYLE OF FIJI & ITS ASSOCIATIONS Q ( M 11.1 ) ytn / I 1'. IN ^ ^ ^ ' 1 \ 'v1ll_ ~ Li L /Tin Vi 1/ " ^ - - ' 7 TT\ \ 11 / *" Fig 2.37 M o tif 11 (M11.1) b o o r M 121 & M M M 12.2 Fig M otif 12 A /V A A A <,X, * / T r M? 4^ v " ^ - ^ < '\ > < '\ > < A / (M 14.2) { \i- > O ic < ) b Fig.2.41 M o tif 14 V j'll - " a " (M 11.2) _ ^ " i i i " n r ; n j ^ ^ T jtn b Fig 2.38 M otif 11 (M11.2) -K l \ '* i \ '-? bis - i u >*\ >7 i '- A ' t v? \ ' V j_ [_ ^ '_j _ j j S j. J S i. * 'a M 13.1a b' M 13.1b i~rc * 7* 7 r/' N^ V \ '/ --x ^ ' \ v' A i V/A, Vy/\ V ; U c M 13-2 Fig M o tif 13 i;!:!! I!i i M 11 1, :! i 1 1 * *»!!!!!'!!!.!!!! i'' i : : : : : ^ - Y J - L l l I - b Fig 2.42 M o tif 15 T \------,'V---- 7,- T t \ \ / A\ \ ' / / ' \ \. \ ' 7 N \ \ ' \ 7 \ \ ' " V v V L _Vi. V w».vy 'jwr " " '* 1 M16.1 a b M Ī6.2 c > T ~.~ r M 16 3 N5>fc' 1. JZnf J W Fig.2.43 M o tif A \~/ A. A\ X. / A \* '/ K ///) \ \^ / / A \''\ //l//\ x v j j i, o A lv 'A a b c M 18.1 M 18.2 Fig.2.45 M o tif '\N\>! 1 ' 'V. 0 M i* pattern 1 M17.1.'! m { 1 / NV \ii 11/ y> 'J-., \i J y <u jic. ^11 b p attern 2 M ~ ^ v s~ 7'7/^A- V1/4 / w - - / / w / / W / / S\ C /t \ ',/ \ \ / / N\ / / W c pattern 3 M 17.2* Fig.2.44 M otif 17 32

15 S. M. MEAD, L. BIRKS AND E. SHAW /, \ ~ \\\>!/// \N \\ /// //-,\\ /// \ \ SA ' / / ; A \ \ - / \ /v J ' J 1 i\ ' l \ x i _-V \ ' >2 \VV M19.1 a b - r, r \ \ I i 7 V / / \ \ / / \\«i / / v // \v / d «'V.f// ^ M 19.2 ^ Fig. 2.46M otif 19 ~7\~" M 'A Fig.2.47 M o tif 20 / '\ \ / / nn Fig a M21.1 NNv W N' / / y v W ' ' / ' ' / \ \ S s N X M o tif 21 a M 22.1 Fig M o tif 22 A\ A <-i^ As A \ /! \ / \ Y /l\ > / \ * t? I / \ ' / v y 9 V a M 23.1 b M 23.2 c d Fig M otif 23 Fig \ ' W V. ' \ /'N * X v. XN / N ^ N >N )< V A /< V V ' ' \ N\ /X nv / A O / V V ' * N/ - ' ' ' \ / ^ ' / \ X M otif 24 33

16 THE LAPITA STYLE OF FIJI & ITS ASSOCIATIONS 7 1 i i > \ / \ Fig a M 25.1 T -T ' I! S) N\ /* " ^ V ' / ''N '. / 'A \ ' s \. v s \ I tf ' 'I 11 I > ll t l_ i i i L. b M 25.2 c M 25.3 M otif 25 \ Fig M otif 26 Fig. 2.54! ii i i -----i!i r M otif 27 niinnr*" /\\/\/\/\/\ _/_\l / \/ j V[i _y_ 1 M 28.1 Fig.2.55 / 111«11 ; 11 f MI11II i i if y v i b M otif 28 j! AAA r t u r n c M 28.2 ( I I I a h a ' /V IV \/ ' M M i V 1 Fig.256 a M 29.1 M 29.1 V M M29.2 c M o tif 29 34

17 S. M. MEAD, L. BIRKS AND E. SHAW T v 7 - i ' ' v ' ' ^ V l!! \ * / X \! ^ N i 1 \ * / / /-N ' s N ; Is -*L _l/ -J \ ila M 30.1 b M 30.2 Fig M otif 30 f " I Fig M o tif 31 (uncon firm e d) i j 1! ii - ii ii i' j 1!j i 1 11!!!, !* I I! L-1- - Ll Fig M otif 32 (unconfirm e d) / ^ \ Fig M o tif 33 T able 12 Motifs and Associated Rules Motif Alloforms Rules Rule No. Ml P = Ml -> ConR/E-W M2 M2.1 P = M2.1 -> ConR/E-W M2.2 P = M2.2 -* ConR/E-W M3 P = M3 ConR/E-W M4 P = M 4 ^ ConR/E-W M5 P = M5 ConR/E-W M6 P = M6 ConR/E-W M7 P = M7 -> ConR/E-W M8 M8.1 P = M8.1 ConR/E-W/CLS M8.2 P = M8.2 -> ConR/E-W/CLS 35

18 THE LAPITA STYLE OF FIJI & ITS ASSOCIATIONS T able 12 Continued Motif Alloforms Rules Rule No. M8.3 P = M8.3 ConR/E-W/CLS 1 M9 M9.1 P = M9.1 -> ConR/E-W 1 M9.2 P = M9.2 DisR/E-W 2 M9.3 P = M9.3 ConR/E-W/CLS 1 M10 P = M10-* ConR/E-W/CLS 1 M il M ll.l P = M ll.l -> DisR/E-W 2 Ml 1.2 P = M11.2 -> DisR/E-W 2 M12 M12.1 P = M12.1 -* DisR/E-W/# 2 M12.2 P = M12.2 -> DisR/E-W 2 M13 M13.1 P = M13.1 ^ ConR/E-W 1 M13.2 P = M13.2 -* ConR/E-W 1 M14 M14.1 P = M14.1 -> ConR/E-W 1 M14.2 P = M14.2 ^ DisR/E-W 2 M15 M15.1 P = M15.1 DisR/E-W 2 M16 M16.1 P = M16.1 ConR/E-W 1 M16.2 P = M16.2 -> ConR/E-W 1 M16.3 P = M16.3 ConR/E-W 1 M17 M17.1 P = M17.1 (rep) -> DisR/N-S 3 P = M17.1 (rep) + MR/M17.1 -> DisR/E-W 2 M17.2 P = M17.2 # alt with M 17.2 rev # DisR/E-W 2 M18 M18.1 P = M18.1 -* ConR/E-W 1 P = M M18.1 (rev/cls) -> ConR/E-W/CLS 1 M18.2 P = M18.2 ConR/N-S 5 M19 M19.1 P = M19.1 ^ ConR/E-W 1 P = M19.1 (rev) -> ConR/E-W 1 M19.2 P = M19.2 ConR/E-W 1 M20 P = M20 ConR/E-W 1 M21 P = M21 ^ ConR/E-W 1 P = (M21) -> ConR/E-W 1 (M21) M22 P = M22 -* DisR/E-W 2 M23 (a) P = (M23) DisR/E-W 2 (M23) (b) P = (M22) (M23) (M21) alt. with (M23) 6 (M21) (M20) M24 P = M 24^ ConR/E-W 1 M25 M25.1 P = M25.1 ConR/E-W/CLS 1 M25.2 P = M25.2 -> ConR/E-W/CLS 1 M26 P = M26-y ConR/E-W/CLS 1 M27 P = M27 -> ConR/E-W/CLS 1 M28 M28.1 P = M28.1 -> ConR/E-W/CLS 1 M28.2 P - M28.2 -> ConR/E-W/CLS 1 M29 P = M29 DisR/E-W 2 P = M29 + JMR of M29 DisR/E-W 2 P = M29 DisR/E-W/INT 2 M30 M30.1 Rule not identifiable M30.2 Rule not identifiable M31 P = M31 DisR/E-W 2 M32 P = M32-» DisR/E-W 2 M33 P = M33 + DisR/E-W(?) 2 Eleven of the motifs are based on the curved line and produce patterns which can be described as curvilinear. Fourteen of the motifs are based on the straight line and these produce patterns which can be described as rectilinear. But eight motifs contain both curved and straight line elements 36

19 S. M. MEAD, L. BIRKS AND E. SHAW and thus produce patterns which cannot strictly be described as one or the other. The overall evidence shows only a slight bias towards rectilinear patterns. Nearly all of the motifs are based on geometric shapes rather than on natural shapes. It would thus be fair to characterise the decorative system of Yanuca and the Sigatoka Dunes as geometric. Patterns are arranged in borders; that is, one horizontal layer at a time. The pattern-making rules, which will be described in detail later, reflect this manner of building up a total pattern. Rules are provided for one horizontal layer at a time. The system as a whole may thus be characterised as a horizontal zoning system, or as either a layer or laminated system, indicating that a total pattern is composed of layers placed one below the other. The analysis of motifs by the methods applied here is not without its problems. The view adopted here is that it is better to make an attempt at resolving these problems than it is to ignore them. Decisions as to whether two particular items are different motifs or two alloforms of the same motif are often difficult to make. For example, it can be argued with considerable conviction that M8 and M9 represent alloforms of the same motif. Moreover, one could further argue that M10 is also an alloform of the same motif. To argue this way is to recognise that they are elaborations evolving from the one original motif. I have adopted the view that although M8, M9 and M10 are clearly related and represent an evolutionary sequence from M8 to M10, they are different motifs because the visual effect of each is different in some significant way from either of the other two and because the rules applicable to each are not identical. A similar problem was encountered in the cases of M25 and M26; again, a case of two related motifs. The decision taken to separate M26 from M25 will probably meet with approval since the order of visual difference is sufficient to warrant separation. However, if this case is accepted, the case for separating M8, M9 and M10 must also be accepted. The essential problem is how different units ought to be in order to warrant classification into either the motif or the alloform category. It has been demonstrated in this analysis that many motifs evolve from other motifs, as in the cases of Ml to M7. If an evolutionary development was accepted by decorators as representing a new motif, one should find evidence that the new idea was used by other decorators. Such evidence can be found for many of the motifs. In many cases, however, the sample of sherds is too limited to permit the discovery of the evidence. During the analysis of motifs evidence can frequently be found of instances where an idea was being developed and elaborated. Besides the examples mentioned above there are also M13 and Ml4. While the two motifs are generally related, the relationship is of a different order from that represented in M8, M9 and M10. When a motif such as M13 is arranged into a pattern it produces a negative shape which, if sufficiently arresting, would catch the attention of a decorator. Apparently, this did happen and the results of attempts to elaborate this negative shape are 37

20 THE LAPITA STYLE OF FIJI & ITS ASSOCIATIONS represented by the alloforms of Ml4. Thus a new motif can be developed by elaborating the negative aspect of some other motif. Sometimes a new form resulted from combining several motifs in a complex pattern sequence. An example is M23, which is really a negative space created when M20, M21 and M22 are combined. The decorator, however, noticed the diamond shape and elaborated it. What is lacking in the sherd sample is confirmation that the idea was adopted as a dominant theme in a pattern. Confirmation is, however, available from the site at Natunuku, so that it is highly probable that given a larger sample of sherds the evidence would have been found at Yanuca. Meanwhile, this motif must remain a bound form. Again, probably because of deficiencies in the sherd sample, there are some motifs which do not occur as a dominant motif in any pattern sequence. Examples are M20, M21, M22 and M23. These motifs occur several times but only in one complex pattern. Thus M20, M21, M22 and M23 are in association and, according to present evidence, are not free forms in the sense that they are free to enter into other associations or free to appear as dominant motifs. A larger sample of sherds would probably have supplied the missing evidence which would show that each of the above motifs is an independent and free motif. Comparative evidence from other sites indicates a high probability that this is indeed the case. Because of deficiencies in the sample and the manner in which sherds are broken and scattered, the list of motifs given in this analysis is probably short of a few items. There are many sherds which are too fragmented for recognition of the motif to be feasible. Notwithstanding these contingencies, the inventory presented in this analysis can probably be taken to be fairly representative of the possible motifs used at Sigatoka. DESIGN ZONES The rules for composing patterns are associated with particular design zones. Eight design zone types are used in this analysis. Where the visual evidence is clear, the design zone type into which a pattern is composed is isolated and mentioned in the description. The distribution of design motifs according to design zone type is summarised in Table 13. It will be seen from the distribution table that Type A1 is by far the most popularly used zone. Four design elements and 29 of the 33 motifs can be placed in a Type A1 design zone. Next in popularity is Type A2, but it should be pointed out that a Type A2 zone is really a series of Type A1 zones, thus elements and motifs which can be placed in a Type A1 zone can theoretically occur in a Type A2 zone, hence the classification of both as Type A. The distribution evidence for Type A zones indicates a strong preference on the part of Yanuca decorators for patterns which are continuous either in the sense of units running into each other or of units placed one against the other. Design zone Types B and C are also related, the former consisting of one vertically divided border and the latter consisting of several layers of zone Type B. However, as shown in the table, it does not necessarily mean 38

21 S. M. MEAD, L. BIRKS AND E. SHAW T able 13 Distribution of Elements and Motifs in Design Zones Zone Type Al Totals Elements: NI, DEI, DE2, DE6 4 Motifs: *Ml-7, *M9-10, *M12-16, M17, *M18 *M19, *M20-28, M31, M32, M33 29 Zone Type A2 Elements: DEI, DE2, DE3, DE5, DE6, DE7 6 Motifs: Ml-5, M7-8, M10-12, M16, M24-25, M28 14 Zone Type B Elements: DE4 1 Motifs: M il, M17, M30? 3 Zone Type C Motifs* *M11,*M16.3, M17 3 Zone Type D DE7 1 Zone Type E Elements: DE5, DE6? 2 Motifs: M8, M16, M27 3 Zone Type F Elements: DE6 1 Motifs: M19 1 Zone Type G DE2, DE8 2 Zone Type H Elements: DEI, DE5 2 Motifs: M18, M24, *M29, M30? 4 This motif is particularly likely to occur in this zone. that what can be placed in Type B must also be placed in Type C. These zones are not favoured for arranging design elements in patterns because only DE4 can be composed in a Type B zone. It is theoretically possible to place Ml 1 in a Type B zone, but the favoured and logical zone for patterns based on this motif is Type C because this is an all-over pattern. Most alloforms of M16 are placed in a Type A zone except alloform M16.3, which must be placed into a Type C zone because it is used, characteristically, as an all-over pattern. Alloforms of M17 are similarly spread over different zones, alloforms M l7.2 occurring in a Type A zone and M17.1 in a Type B zone. It is also possible that M30 is applied to a Type B zone but the evidence for this is not strong. A Type C zone is favoured for all-over patterns. A Type D zone is associated only with DE7 and nothing else. Type E is associated with DE5 and DE6 to a limited extent and with motifs M8, M l6 and M27. Element DE5 and motifs M8, M16 and M27 occur near the lower part of a vessel. Type F is also a zone of restricted use. When this zone is decorated, however, the most likely patterns to occur there are based on either DE6 or Ml9, the latter being more popular. Use of Type G is also restricted, as patterns applied to it consist only of sequences of DE2 or DE8. Associated with zone Type H are elements DEI and DE5 and motifs Ml8, M24, M29 and possibly M30. This zone, however, is the favoured one for patterns based on M29, which hardly ever occurs anywhere else. It is similarly a favoured zone for M24. Though M18 can be composed into a pattern in a Type H zone, it is more usual to find it in a Type A zone. 39

22 THE LAPITA STYLE OF FIJI & ITS ASSOCIATIONS Thus, while there are indications of associations of particular zones with particular elements and motifs, there is rarely a hard and fast rule which operates in all cases. Many of the associations do, however, occur in a large number of cases and can be useful in predicting what type of zone a given pattern is likely to occur in. RULES AND PROCESSES A relationship appears to exist between the highly favoured zones for composing patterns and the most highly favoured pattern-making rule. It is not surprising to find, therefore, that the majority of patterns are composed according to the rule P = M or DE -> ConR/E-W. That is, a pattern is formed by subjecting a motif or design element to continuous repetition along an east-west direction. Design elements TB3 and DE8 follow this rule and so do the following motifs and alloforms: Ml, M2, M3, M4, M5, M6, M7, M8, M9.1, M9.3, MIO, M13.1, M13.2, M14.1, M16, M18, M19, M20, M21, M22, M24, M25, M26, M27, M28 and M29. Discounting alloforms, the rule can be applied to no fewer than 24 of the 33 motifs in the inventory. This rule is thus the main one which is applied to motifs. It is rarely applied to design elements. The next most popular rule is P = M or DE -» DisR/E-W in which the units are separated by space or by a vertical zone marker. That is, the motif or design element is subjected to discontinuous repetition in an east-west direction. Patterns constructed according to this rule may occur in either a Type A or Type B zone. This is the main rule for arranging design elements into patterns as it is applicable to N l, VB2, DE1.1, DE1.2, DE2.1, DE2.2, DE2.3, DE4, DE5, DE6, and DE8. No other rule can be applied to as many elements as this one. As the first rule, Rule 1, is characterised by its general applicability to motifs, this second rule, Rule 2, can be characterised by its general applicability to design elements. However, Rule 2 can also be applied to 12 motifs, namely M9.2, M il, M12, M14.2, M15, M17, M22, M23, M29, M31, M32 and M33. That is, it can be applied to half as many as Rule 1. The 2 rules together can be applied to about 30 of the 33 motifs and to 7 of the 8 two-dimensional elements. It follows that all other rules are applicable to a very limited number of cases and cover the exceptions and the unusual compositions. Rule 3 is really only a slight modification of Rule 2 to accommodate the exceptional cases where the decorator has repeated a series of units in a vertical border. Rule 3 states that P = M or DE DisR/N-S. The rule can be applied to DE6, DE7 and Ml7.1. Such an arrangement is rare. A further modification of Rule 2 accommodates cases where a sequence of units can be placed in different directions, hence Rule 4 is P = M or DE -> DisR/MULT, that is discontinuous repetition in several directions. The rule is applicable to RZ3 and DE3. Though only design elements are listed here, it is quite possible that a motif could be arranged in series in a direction other than east-west or north-south. It is a simple matter to alter Rule 2 to increase its generality in the matter of direction and so encompass Rules 3 and 4. Direction is the only point of difference. However, the more specific rule is probably to be preferred over 40

23 S. M. MEAD, L. BIRKS AND E. SHAW the more general one in this instance as it is useful to draw attention to the unusual cases. Rule 5 accommodates cases where Rule 1 has to be applied in a vertical zone. Its use is extremely rare. Rule 6 is applied to very complex compositions in which two or more motifs placed one above the other are alternated with another group similarly positioned, as for combinations of M20, M21, M22 and M23. Not all the rules are as straight forward as those given above. Some require that several steps be taken in the construction of a pattern. In cases such as these the stages are indicated by bracketing, for example: P = (DE2.1 - DisR X N/alt with MR/E-W) -> DisR/E-W, or P = (DE2.1 -> DisR X 5/E-W + MR/E-W) -> DisR/E-W. That is, DE2.1 is repeated five times or some other specified number (N) discontinuously in an east-west direction. Then all of this is repeated as a mirror image. Both sets of elements are now treated as a single unit and the unit is repeated as a discontinuous series in an east-west direction around the pot. The basic rule is, nevertheless, still Rule 2. The rules tend to be complex when a pattern is constructed from the basic element rather than from the motif. While this is plainly uneconomical as it produces an involved rule, it is, nevertheless, a useful exercise to analyse the design processes which are applied to an element. It is especially useful in the case of a simple element. For example, Ml to M7 are elaborations of DEI. Motif 1 is composed into a pattern as follows: P = DEI (rep) ConR/E-W. An M2 pattern contains two or three chains of Ml arranged in a half-drop mesh, hence the rule P = DEI (rep) -> JDM x 2/ConR/E-W. A further design process comes into focus with M3. Now a pattern is composed by superimposing two continuous chains of DEI so that the second chain is shifted half a place to the side. I have called this process superimposing although, in fact, it is a kind of half-drop, except that in this case the shift is horizontal rather than vertical. A pattern composed from M3 is thus really P = DEI (rep) -> SUP x 2/Con R/E-W. In an M4 pattern a mirror image is posed against a chain of DE1. Rounding the points of the chains changes the character of the pattern drastically. An M5 pattern is produced by applying the rounding process to the superimposed chains of M3. In the case of an M6 pattern a second chain of DEI is opposed to the first so that the full opposition presented by a mirror image is not achieved. Instead, the second chain is half-opposed to the first. This results from shifting the mirror image chain half a place to the side. To create an M7 pattern the element is reposed, rounded and doubled before being subjected to continuous repetition. Besides employing the design-making processes described above, the Yanuca potters also compressed units when required. They rounded elements and motifs and they squared them. They composed decreasing series of units including decreasing series of concentrically arranged oblongs and circles. They reversed units or simply turned them 90 degrees to the side. They varied their lines by doubling or trebling them, by incising, 41

24 THE LAPITA STYLE OF FIJI & ITS ASSOCIATIONS rubbing, dentate stamping or impressing, and by widening them when necessary. These design processes tend to be universal wherever they are used by cultures which practise a fairly sophisticated system of decorative art. The decorative system could be applied with ease to the various types of vessels made by the potters. In some cases, as in vessel Type B, the shape of the vessel was modified to facilitate application of decoration. For example, the everted rim of a vessel was lengthened and flattened, as in Figure 2.16, to accommodate a Type F design zone. There is thus some evidence that vessel form is modified by the decorative system in the same way that the decorative system might be modified by the shapes of vessels. Taking as an example the Type B vessel illustrated in Figure 2.16,1 shall set out a sequence of steps by which the vessel was decorated. For convenience I shall assume that the decorator works from the top of the vessel to the bottom. Step 1. Divide the surface into design zones. Zone Types F and G are structurally defined. Below the rim stamp in a Type C zone following the measurements given in Figure From the shoulder downwards stamp in a Type A2 zone to the measurements given in Figure The vessel surface is now arranged from top to bottom in four zones, F, G, C and A2. Step 2. Decorate zone F to the rule P = M19.1 -» ConR/E-W. Step 3. Decorate zone G to the rule P = DE8 -> ConR/E-W. Step 4. Decorate zone C to the rule P = M ll.l -» DisR/E-W in each C zone. Step 5. Leave plain the first Al zone after this. Step 6. Decorate the next A1 zone according to the rule P = M32 -> DisR/E-W. Step 7. In the final Al zone put the pattern P = M2.1 (rev) -> ConR/E-W, beginning on the lower zone marker. The result of these steps as applied by one or more decorators can be seen in vessel 16/81/2-3/3/6 (Plate 1). CONCLUSION The Yanuca potters employed a zoning system of decorative art based on patterns composed layer by layer. They used 18 design elements and 33 design motifs which were transformed into many kinds of patterns by applying a variety of compositional devices which included the following. 1. Linking in chains (Ml, M2, M3, M4, M5, M6, M8, M9, M10). 2. Arrangement in half-drop mesh (M2). 3. Mirror image opposition (M4). 4. Half mirror image opposition (M6). 5. Superimposition (M3, M5). 6. Coalescence (M9, M10, M25, M26, M27). 7. Arrangement in diminishing series (DE7). 8. Arrangement in concentric series (M32, M33). 9. Doubling (DEI, DE2, M9, M10, M l3, M32, M33). 10. Arrangement for all-over effect (M11,M 16). 11. Trebling (DE3). 12. Arrangement in continuous unbroken series (M1-7, M10, M13). 42

25 S. M. MEAD, L. BIRKS AND E. SHAW 13. Infilling of shapes (M9, M10, M19, M20, M21, M22, M25). 14. Clustering of elements (M15). 15. Combining several motifs to form very complex patterns (M20-23). 16. Squaring elements and motifs which could be rounded (N1.2, N1.3, TB3.1, TB3.3, M25, M26). 17. Rounding elements and motifs which could be squared (N1.1, VB2.1, TB3.2, M8, M9, M10, M33). 18. Compressing elements and motifs where required by the design zone (N1.3, N1.4 and possible for all elements and motifs). 19. Cursive rounding (M7, M28, M29). 20. Arrangement of the design field into horizontal layers. Patterns were constructed by repeating elements and motifs in accordance with two basicpattern-making rules. These two rules account for the majority of patterns but exceptions are covered by slight modifications of the main rules. Most of the exceptions can be included in the main rules by so phrasing the rules as to increase their generality, but this course has been avoided in order to show more specifically how patterns were composed. Rule 6 is applied to highly complex patterns. In the majority of cases, patterns are given visual form by dentate stamping. Incised or rubbed lines are extremely rare. The Yanuca decorative system is a relatively sophisticated system in which both curved and straight line elements were being developed into patterns. Although many of the patterns appear to be very ornate and regular, the compositions are not as complicated in all respects as those found in Watom Island and New Caledonia. As will be shown by comparisons, however, the system as a whole is similar to Lapita decorative systems found in other parts of Oceania. 43

/ \ / \ / M i / i %/ II %# I 1 ' I II %/ M / M \ I II '/Mi/ II If!»f 1 ' I II II I 'I I II I I I* 'I I 'I >1 >» M 11 I 11 * 11 1

/ \ / \ / M i / i %/ II %# I 1 ' I II %/ M / M \ I II '/Mi/ II If!»f 1 ' I II II I 'I I II I I I* 'I I 'I >1 >» M 11 I 11 * 11 1 /\ \ \ /\ /\ \ POTTERY STYLE AND ITS ASSOCIATIONS " A----- A" ~ A----- C > /* \ /*-A / A /0 / / 1/ 1'\\ \ ' \ //I I l \ ' \ ///I \ ' \ / / lll\ / I l'\\ \ X\ /// l \ ' \ ///I \'v ///»l''\.'//i,7/1 l \

More information

The Degree of Standardisation in the An Sơn Ceramic Assemblage

The Degree of Standardisation in the An Sơn Ceramic Assemblage 7 The Degree of Standardisation in the An Sơn Ceramic Assemblage Introduction: Methodology for the study of standardisation The level of standardisation within an assemblage of pottery is used as an indirect

More information

EW MOVEMENT NOTATION: THE SYSTEM OF REFERENCE

EW MOVEMENT NOTATION: THE SYSTEM OF REFERENCE EW MOVEMENT NOTATION: THE SYSTEM OF REFERENCE Noa Eshkol and John Harries Abstract A survey is offered of the structure of the system of reference of Eshkol Wachman Movement Notation. It is not possible

More information

Non-independence in Statistical Tests for Discrete Cross-species Data

Non-independence in Statistical Tests for Discrete Cross-species Data J. theor. Biol. (1997) 188, 507514 Non-independence in Statistical Tests for Discrete Cross-species Data ALAN GRAFEN* AND MARK RIDLEY * St. John s College, Oxford OX1 3JP, and the Department of Zoology,

More information

Testing Research and Statistical Hypotheses

Testing Research and Statistical Hypotheses Testing Research and Statistical Hypotheses Introduction In the last lab we analyzed metric artifact attributes such as thickness or width/thickness ratio. Those were continuous variables, which as you

More information

Chapter 1: Logic systems

Chapter 1: Logic systems Chapter 1: Logic systems 1: Logic gates Learning Objectives: At the end of this topic you should be able to: identify the symbols and truth tables for the following logic gates: NOT AND NAND OR NOR XOR

More information

Linear Algebra. The analysis of many models in the social sciences reduces to the study of systems of equations.

Linear Algebra. The analysis of many models in the social sciences reduces to the study of systems of equations. POLI 7 - Mathematical and Statistical Foundations Prof S Saiegh Fall Lecture Notes - Class 4 October 4, Linear Algebra The analysis of many models in the social sciences reduces to the study of systems

More information

Chapter 13. On the special properties of coarse and subtle matter.

Chapter 13. On the special properties of coarse and subtle matter. 60 Chapter 13. On the special properties of coarse and subtle matter. 98. A body can only be placed into a smaller space if its pores are compressed; also only the apparent size of a body can be altered,

More information

On Objectivity and Models for Measuring. G. Rasch. Lecture notes edited by Jon Stene.

On Objectivity and Models for Measuring. G. Rasch. Lecture notes edited by Jon Stene. On Objectivity and Models for Measuring By G. Rasch Lecture notes edited by Jon Stene. On Objectivity and Models for Measuring By G. Rasch Lectures notes edited by Jon Stene. 1. The Basic Problem. Among

More information

KIRCHHOFF S LAWS. Learn how to analyze more complicated circuits with more than one voltage source and numerous resistors.

KIRCHHOFF S LAWS. Learn how to analyze more complicated circuits with more than one voltage source and numerous resistors. KIRCHHOFF S LAWS Lab Goals: Learn how to analyze more complicated circuits with more than one voltage source and numerous resistors. Lab Notebooks: Write descriptions of all of your experiments in your

More information

TRIPLE-SYSTEMS AS TRANSFORMATIONS, AND THEIR PATHS

TRIPLE-SYSTEMS AS TRANSFORMATIONS, AND THEIR PATHS TRIPLE-SYSTEMS AS TRANSFORMATIONS, AND THEIR PATHS AMONG TRIADS* BY H. S. WHITE A triple-system has been of interest as offering a puzzling problem in construction. When various methods of construction

More information

Lapita and Later Archaeology of the Malolo and Mamanuca Islands, Fiji

Lapita and Later Archaeology of the Malolo and Mamanuca Islands, Fiji Journal of Island & Coastal Archaeology, 2:245 250, 2007 Copyright 2007 Taylor & Francis Group, LLC ISSN: 1556-4894 print / 1556-1828 online DOI:10.1080/15564890701622896 Lapita and Later Archaeology of

More information

Linear Programming in Matrix Form

Linear Programming in Matrix Form Linear Programming in Matrix Form Appendix B We first introduce matrix concepts in linear programming by developing a variation of the simplex method called the revised simplex method. This algorithm,

More information

Slope Fields: Graphing Solutions Without the Solutions

Slope Fields: Graphing Solutions Without the Solutions 8 Slope Fields: Graphing Solutions Without the Solutions Up to now, our efforts have been directed mainly towards finding formulas or equations describing solutions to given differential equations. Then,

More information

Summer Solutions Common Core Mathematics 8. Common Core. Mathematics. Help Pages

Summer Solutions Common Core Mathematics 8. Common Core. Mathematics. Help Pages 8 Common Core Mathematics 6 6 Vocabulary absolute value additive inverse property adjacent angles the distance between a number and zero on a number line. Example: the absolute value of negative seven

More information

Map Projections. Chapter 4 MAP PROJECTION

Map Projections. Chapter 4 MAP PROJECTION Map Projections Chapter 4 Map Projections What is map projection? Why are map projections drawn? What are the different types of projections? Which projection is most suitably used for which area? In this

More information

Today we begin the first technical topic related directly to the course that is: Equilibrium Carrier Concentration.

Today we begin the first technical topic related directly to the course that is: Equilibrium Carrier Concentration. Solid State Devices Dr. S. Karmalkar Department of Electronics and Communication Engineering Indian Institute of Technology, Madras Lecture - 3 Equilibrium and Carrier Concentration Today we begin the

More information

Three More Examples. Contents

Three More Examples. Contents Three More Examples 10 To conclude these first 10 introductory chapters to the CA of a two-way table, we now give three additional examples: (i) a table which summarizes the classification of scientists

More information

Gabriel Kravitz Department of Anthropology, McGill University

Gabriel Kravitz Department of Anthropology, McGill University Has God Signed My Dollar Store Mug? The search for aesthetic ratios in the relationship between body and handle in assemblages of dollarware drinking vessels Gabriel Kravitz Department of Anthropology,

More information

Chapter 9: The Perceptron

Chapter 9: The Perceptron Chapter 9: The Perceptron 9.1 INTRODUCTION At this point in the book, we have completed all of the exercises that we are going to do with the James program. These exercises have shown that distributed

More information

Structure of Materials Prof. Anandh Subramaniam Department of Material Science and Engineering Indian Institute of Technology, Kanpur

Structure of Materials Prof. Anandh Subramaniam Department of Material Science and Engineering Indian Institute of Technology, Kanpur Structure of Materials Prof. Anandh Subramaniam Department of Material Science and Engineering Indian Institute of Technology, Kanpur Lecture - 5 Geometry of Crystals: Symmetry, Lattices The next question

More information

Geographic coordinate systems

Geographic coordinate systems 1 Geographic coordinate systems In this chapter you ll learn about longitude and latitude. You ll also learn about the parts that comprise a geographic coordinate system including Spheres and spheroids

More information

Year 8 Teaching Programme

Year 8 Teaching Programme Year 8 Teaching Programme TERM 1 # of weeks 1. Integers and Whole Numbers 3 2. Geometry of Polygons 2 3. Fractions and Decimals 3 4. Percentage 2 TERM 2 5. Algebra 2 6. Expansion and Factorisation 2.5

More information

Introduction to Basic Proof Techniques Mathew A. Johnson

Introduction to Basic Proof Techniques Mathew A. Johnson Introduction to Basic Proof Techniques Mathew A. Johnson Throughout this class, you will be asked to rigorously prove various mathematical statements. Since there is no prerequisite of a formal proof class,

More information

13.7 ANOTHER TEST FOR TREND: KENDALL S TAU

13.7 ANOTHER TEST FOR TREND: KENDALL S TAU 13.7 ANOTHER TEST FOR TREND: KENDALL S TAU In 1969 the U.S. government instituted a draft lottery for choosing young men to be drafted into the military. Numbers from 1 to 366 were randomly assigned to

More information

CMSC 425: Lecture 11 Procedural Generation: Fractals and L-Systems

CMSC 425: Lecture 11 Procedural Generation: Fractals and L-Systems CMSC 425: Lecture 11 Procedural Generation: ractals and L-Systems Reading: The material on fractals comes from classic computer-graphics books. The material on L-Systems comes from Chapter 1 of The Algorithmic

More information

The new physiology of vision-chapter The colours of interference. SIR C V RAMAN Received October 14, 1965

The new physiology of vision-chapter The colours of interference. SIR C V RAMAN Received October 14, 1965 Proc. Indian Acad. Sci. A62 243-248 (1965) The new physiology of vision-chapter The colours of interference XXVII. SIR C V RAMAN Received October 14, 1965 The characteristic features and properties of

More information

MATH2206 Prob Stat/20.Jan Weekly Review 1-2

MATH2206 Prob Stat/20.Jan Weekly Review 1-2 MATH2206 Prob Stat/20.Jan.2017 Weekly Review 1-2 This week I explained the idea behind the formula of the well-known statistic standard deviation so that it is clear now why it is a measure of dispersion

More information

M13/5/MATME/SP1/ENG/TZ2/XX/M MARKSCHEME. May 2013 MATHEMATICS. Standard Level. Paper pages

M13/5/MATME/SP1/ENG/TZ2/XX/M MARKSCHEME. May 2013 MATHEMATICS. Standard Level. Paper pages M1/5/MATME/SP1/ENG/TZ/XX/M MARKSCHEME May 01 MATHEMATICS Standard Level Paper 1 17 pages M1/5/MATME/SP1/ENG/TZ/XX/M This markscheme is confidential and for the exclusive use of examiners in this examination

More information

Probability and Statistics

Probability and Statistics Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be CHAPTER 4: IT IS ALL ABOUT DATA 4a - 1 CHAPTER 4: IT

More information

(df (ξ ))( v ) = v F : O ξ R. with F : O ξ O

(df (ξ ))( v ) = v F : O ξ R. with F : O ξ O Math 396. Derivative maps, parametric curves, and velocity vectors Let (X, O ) and (X, O) be two C p premanifolds with corners, 1 p, and let F : X X be a C p mapping. Let ξ X be a point and let ξ = F (ξ

More information

X X (2) X Pr(X = x θ) (3)

X X (2) X Pr(X = x θ) (3) Notes for 848 lecture 6: A ML basis for compatibility and parsimony Notation θ Θ (1) Θ is the space of all possible trees (and model parameters) θ is a point in the parameter space = a particular tree

More information

CHAPTER FIVE THE LITHIC ASSEMBLAGES

CHAPTER FIVE THE LITHIC ASSEMBLAGES CHAPTER FIVE THE LITHIC ASSEMBLAGES The lithic assemblages of Tshisiku Shelter, Balerno Main Shelter and Balerno Shelter 2 provide most of the data for this study of hunter-gatherers and interaction since

More information

California 5 th Grade Standards / Excel Math Correlation by Lesson Number

California 5 th Grade Standards / Excel Math Correlation by Lesson Number (Activity) L1 L2 L3 Excel Math Objective Recognizing numbers less than a million given in words or place value; recognizing addition and subtraction fact families; subtracting 2 threedigit numbers with

More information

ELECTROMAGNETISM The study of the relationship between electricity and magnetism is called

ELECTROMAGNETISM The study of the relationship between electricity and magnetism is called ELECTROMAGNETISM The study of the relationship between electricity and magnetism is called Electromagnetism Before, 1819 it was believed that there was no connection between electricity and magnetism.

More information

Analysis of California Mathematics standards to Common Core standards-grade 3

Analysis of California Mathematics standards to Common Core standards-grade 3 Analysis of California Mathematics standards to Common Core standards-grade 3 Strand CA Math Standard Domain Common Core Standard (CCS) Alignment Comments in reference to CCS 1.0 Number Sense 1.0 Students

More information

Math 350: An exploration of HMMs through doodles.

Math 350: An exploration of HMMs through doodles. Math 350: An exploration of HMMs through doodles. Joshua Little (407673) 19 December 2012 1 Background 1.1 Hidden Markov models. Markov chains (MCs) work well for modelling discrete-time processes, or

More information

Stratigraphy: Establishing a Sequence from Excavated Archaeological Evidence

Stratigraphy: Establishing a Sequence from Excavated Archaeological Evidence Stratigraphy: Establishing a Sequence from Excavated Archaeological Evidence THE PRINCIPLES OF STRATIGRAPHY In order to explain how and why cultural and social forms change through time, archaeologists

More information

Pre-Algebra (6/7) Pacing Guide

Pre-Algebra (6/7) Pacing Guide Pre-Algebra (6/7) Pacing Guide Vision Statement Imagine a classroom, a school, or a school district where all students have access to high-quality, engaging mathematics instruction. There are ambitious

More information

Chapter 2 Polynomial and Rational Functions

Chapter 2 Polynomial and Rational Functions SECTION.1 Linear and Quadratic Functions Chapter Polynomial and Rational Functions Section.1: Linear and Quadratic Functions Linear Functions Quadratic Functions Linear Functions Definition of a Linear

More information

Name: Date: Period: #: Chapter 1: Outline Notes What Does a Historian Do?

Name: Date: Period: #: Chapter 1: Outline Notes What Does a Historian Do? Name: Date: Period: #: Chapter 1: Outline Notes What Does a Historian Do? Lesson 1.1 What is History? I. Why Study History? A. History is the study of the of the past. History considers both the way things

More information

The Number System (NS) 8.NS.1 Standards for Mathematical Practice (MP): Connections

The Number System (NS) 8.NS.1 Standards for Mathematical Practice (MP): Connections Domain: The Number System (NS) Cluster: Know that there are numbers that are not rational, and approximate them by rational numbers. Standard: 8.NS.1. Know that numbers that are not rational are called

More information

KNOT CLASSIFICATION AND INVARIANCE

KNOT CLASSIFICATION AND INVARIANCE KNOT CLASSIFICATION AND INVARIANCE ELEANOR SHOSHANY ANDERSON Abstract. A key concern of knot theory is knot equivalence; effective representation of these objects through various notation systems is another.

More information

Chapter 3: The Derivative in Graphing and Applications

Chapter 3: The Derivative in Graphing and Applications Chapter 3: The Derivative in Graphing and Applications Summary: The main purpose of this chapter is to use the derivative as a tool to assist in the graphing of functions and for solving optimization problems.

More information

2.0 Research Project Design

2.0 Research Project Design 2.0 Research Project Design This chapter is presented in two parts. Firstly, Section 2.1 summarises the research methodology concerning the collecting and immediate processing of data for this research

More information

Interactive Chalkboard

Interactive Chalkboard 1 Interactive Chalkboard 1 Table of Contents Unit 1: Energy and Motion Chapter 1: The Nature of Science 1.1: The Methods of Science 1.2: Standards of Measurement 1.3: Communicating with Graphs 1.1 The

More information

(Refer Slide Time: 04:21 min)

(Refer Slide Time: 04:21 min) Soil Mechanics Prof. B.V.S. Viswanathan Department of Civil Engineering Indian Institute of Technology, Bombay Lecture 44 Shear Strength of Soils Lecture No.2 Dear students today we shall go through yet

More information

Boolean Algebra and Digital Logic

Boolean Algebra and Digital Logic All modern digital computers are dependent on circuits that implement Boolean functions. We shall discuss two classes of such circuits: Combinational and Sequential. The difference between the two types

More information

Grade 8 Chapter 7: Rational and Irrational Numbers

Grade 8 Chapter 7: Rational and Irrational Numbers Grade 8 Chapter 7: Rational and Irrational Numbers In this chapter we first review the real line model for numbers, as discussed in Chapter 2 of seventh grade, by recalling how the integers and then the

More information

(Refer Slide Time 02:20)

(Refer Slide Time 02:20) Soil Mechanics Prof. B.V.S. Viswanathan Department of Civil Engineering Indian Institute of Technology, Bombay Lecture 33 Stress Distribution in Soils Lecture No. 6 Students once again we meet. Today s

More information

Simple Neural Nets For Pattern Classification

Simple Neural Nets For Pattern Classification CHAPTER 2 Simple Neural Nets For Pattern Classification Neural Networks General Discussion One of the simplest tasks that neural nets can be trained to perform is pattern classification. In pattern classification

More information

Experimental Uncertainty (Error) and Data Analysis

Experimental Uncertainty (Error) and Data Analysis Experimental Uncertainty (Error) and Data Analysis Advance Study Assignment Please contact Dr. Reuven at yreuven@mhrd.org if you have any questions Read the Theory part of the experiment (pages 2-14) and

More information

Applications of Differentiation

Applications of Differentiation MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) Module9 7 Introduction Applications of to Matrices Differentiation y = x(x 1)(x 2) d 2

More information

Pasinetti s notion of a unit of vertically integrated productive capacity : a comment

Pasinetti s notion of a unit of vertically integrated productive capacity : a comment Pasinetti s notion of a unit of vertically integrated productive capacity : a comment Jaime L. del Valle Caballero Ensayos y Monografías Numero 87 marzo 1998 Pasinetti s notion of a unit of vertically

More information

1 Introduction to shells

1 Introduction to shells 1 Introduction to shells Transparent Shells. Form, Topology, Structure. 1. Edition. Hans Schober. 2016 Ernst & Sohn GmbH & Co. KG. Published 2015 by Ernst & Sohn GmbH & Co. KG Z = p R 1 Introduction to

More information

THE GEOMETRY OF THALES

THE GEOMETRY OF THALES THE GEOMETRY OF THALES BY PERCY J. HARDING. In this paper there will, of course, be found no new facts ; but merely a review of well-known historical statements and an attempt to connect them. It has often

More information

II. An Application of Derivatives: Optimization

II. An Application of Derivatives: Optimization Anne Sibert Autumn 2013 II. An Application of Derivatives: Optimization In this section we consider an important application of derivatives: finding the minimum and maximum points. This has important applications

More information

Archdiocese of Washington Catholic Schools Academic Standards Mathematics

Archdiocese of Washington Catholic Schools Academic Standards Mathematics 7 th GRADE Archdiocese of Washington Catholic Schools Standard 1 - Number Sense Students understand and use scientific notation* and square roots. They convert between fractions and decimals. MA.7.1.1

More information

N14/5/MATME/SP1/ENG/TZ0/XX/M MARKSCHEME. November 2014 MATHEMATICS. Standard Level. Paper pages

N14/5/MATME/SP1/ENG/TZ0/XX/M MARKSCHEME. November 2014 MATHEMATICS. Standard Level. Paper pages N4/5/MATME/SP/ENG/TZ0/XX/M MARKSCHEME November 04 MATHEMATICS Standard Level Paper 7 pages N4/5/MATME/SP/ENG/TZ0/XX/M This markscheme is the property of the International Baccalaureate and must not be

More information

Principal Component Analysis

Principal Component Analysis Principal Component Analysis Laurenz Wiskott Institute for Theoretical Biology Humboldt-University Berlin Invalidenstraße 43 D-10115 Berlin, Germany 11 March 2004 1 Intuition Problem Statement Experimental

More information

PHY 123 Lab 1 - Error and Uncertainty and the Simple Pendulum

PHY 123 Lab 1 - Error and Uncertainty and the Simple Pendulum To print higher-resolution math symbols, click the Hi-Res Fonts for Printing button on the jsmath control panel. PHY 13 Lab 1 - Error and Uncertainty and the Simple Pendulum Important: You need to print

More information

Computational Tasks and Models

Computational Tasks and Models 1 Computational Tasks and Models Overview: We assume that the reader is familiar with computing devices but may associate the notion of computation with specific incarnations of it. Our first goal is to

More information

Introduction. Chapter 1 : Introduction l 1

Introduction. Chapter 1 : Introduction l 1 Chapter 1 : Introduction l 1 1 Features 1. Definition. 2. Sub-divisions of Theory of Machines. 3. Fundamental Units. 4. Derived Units. 5. Systems of Units. 6. C.G.S. Units. 7. F.P.S. Units. 8. M.K.S. Units.

More information

The Evolution and Discovery of the Species of Equality in Euclid s Elements

The Evolution and Discovery of the Species of Equality in Euclid s Elements From the SelectedWorks of Lee T Nutini 2010 The Evolution and Discovery of the Species of Equality in Euclid s Elements Lee T Nutini Available at: https://works.bepress.com/nutini/2/ Nutini 1 The Evolution

More information

Chapter 10 Momentum, System of Particles, and Conservation of Momentum

Chapter 10 Momentum, System of Particles, and Conservation of Momentum Chapter 10 Momentum, System of Particles, and Conservation of Momentum 10.1 Introduction... 1 10. Momentum (Quantity of Motion) and Impulse... 1 10..1 Average Force, Momentum, and Impulse... 10.. Non-Constant

More information

Definition of geometric vectors

Definition of geometric vectors Roberto s Notes on Linear Algebra Chapter 1: Geometric vectors Section 2 of geometric vectors What you need to know already: The general aims behind the concept of a vector. What you can learn here: The

More information

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay 51 Module 4: Lecture 2 on Stress-strain relationship and Shear strength of soils Contents Stress state, Mohr s circle analysis and Pole, Principal stressspace, Stress pathsin p-q space; Mohr-coulomb failure

More information

YEAR 10 PROGRAM TERM 1 TERM 2 TERM 3 TERM 4

YEAR 10 PROGRAM TERM 1 TERM 2 TERM 3 TERM 4 YEAR 10 PROGRAM TERM 1 1. Revision of number operations 3 + T wk 2 2. Expansion 3 + T wk 4 3. Factorisation 7 + T wk 6 4. Algebraic Fractions 4 + T wk 7 5. Formulae 5 + T wk 9 6. Linear Equations 10 +T

More information

Late Bronze Age to Middle Iron Age Research topics and priorities

Late Bronze Age to Middle Iron Age Research topics and priorities Methodological approaches and cross-cutting themes These issues are relevant to several of the topics and priorities to be considered and discussed during the workshops. They will not form the focus of

More information

FACTORIZATION AND THE PRIMES

FACTORIZATION AND THE PRIMES I FACTORIZATION AND THE PRIMES 1. The laws of arithmetic The object of the higher arithmetic is to discover and to establish general propositions concerning the natural numbers 1, 2, 3,... of ordinary

More information

Math Analysis Chapter 2 Notes: Polynomial and Rational Functions

Math Analysis Chapter 2 Notes: Polynomial and Rational Functions Math Analysis Chapter Notes: Polynomial and Rational Functions Day 13: Section -1 Comple Numbers; Sections - Quadratic Functions -1: Comple Numbers After completing section -1 you should be able to do

More information

CHOOSING THE RIGHT SAMPLING TECHNIQUE FOR YOUR RESEARCH. Awanis Ku Ishak, PhD SBM

CHOOSING THE RIGHT SAMPLING TECHNIQUE FOR YOUR RESEARCH. Awanis Ku Ishak, PhD SBM CHOOSING THE RIGHT SAMPLING TECHNIQUE FOR YOUR RESEARCH Awanis Ku Ishak, PhD SBM Sampling The process of selecting a number of individuals for a study in such a way that the individuals represent the larger

More information

THE TEACHER UNDERSTANDS THE REAL NUMBER SYSTEM AND ITS STRUCTURE, OPERATIONS, ALGORITHMS, AND REPRESENTATIONS

THE TEACHER UNDERSTANDS THE REAL NUMBER SYSTEM AND ITS STRUCTURE, OPERATIONS, ALGORITHMS, AND REPRESENTATIONS THE REAL NUMBER SYSTEM C O M P E T E N C Y 1 THE TEACHER UNDERSTANDS THE REAL NUMBER SYSTEM AND ITS STRUCTURE, OPERATIONS, ALGORITHMS, AND REPRESENTATIONS This competency section reviews some of the fundamental

More information

Anthropology 1030: Introduction to Archaeology. Mapping and Taphonomy

Anthropology 1030: Introduction to Archaeology. Mapping and Taphonomy Anthropology 1030: Introduction to Archaeology Project Based Research Assignments #2 Mapping and Taphonomy Author: Steven Gregerson 6/20/2011-6/20/102011 Date: 6/20/2011 Site Location: Living Room at house

More information

On the Arbitrary Choice Regarding Which Inertial Reference Frame is "Stationary" and Which is "Moving" in the Special Theory of Relativity

On the Arbitrary Choice Regarding Which Inertial Reference Frame is Stationary and Which is Moving in the Special Theory of Relativity Regarding Which Inertial Reference Frame is "Stationary" and Which is "Moving" in the Special Theory of Relativity Douglas M. Snyder Los Angeles, CA The relativity of simultaneity is central to the special

More information

THE TEACHER UNDERSTANDS THE REAL NUMBER SYSTEM AND ITS STRUCTURE, OPERATIONS, ALGORITHMS, AND REPRESENTATIONS

THE TEACHER UNDERSTANDS THE REAL NUMBER SYSTEM AND ITS STRUCTURE, OPERATIONS, ALGORITHMS, AND REPRESENTATIONS The real number SySTeM C O M P E T E N C Y 1 THE TEACHER UNDERSTANDS THE REAL NUMBER SYSTEM AND ITS STRUCTURE, OPERATIONS, ALGORITHMS, AND REPRESENTATIONS This competency section reviews some of the fundamental

More information

Geology for Engineers Sediment Size Distribution, Sedimentary Environments, and Stream Transport

Geology for Engineers Sediment Size Distribution, Sedimentary Environments, and Stream Transport Name 89.325 Geology for Engineers Sediment Size Distribution, Sedimentary Environments, and Stream Transport I. Introduction The study of sediments is concerned with 1. the physical conditions of a sediment,

More information

Law of Trichotomy and Boundary Equations

Law of Trichotomy and Boundary Equations Law of Trichotomy and Boundary Equations Law of Trichotomy: For any two real numbers a and b, exactly one of the following is true. i. a < b ii. a = b iii. a > b The Law of Trichotomy is a formal statement

More information

The phenomenon: complex motion, unusual geometry

The phenomenon: complex motion, unusual geometry Part I The phenomenon: complex motion, unusual geometry Chapter 1 Chaotic motion 1.1 What is chaos? Certain long-lasting, sustained motion repeats itself exactly, periodically. Examples from everyday life

More information

The Australian Curriculum Mathematics

The Australian Curriculum Mathematics The Australian Curriculum Mathematics Mathematics Table of Contents ACARA The Australian Curriculum Version 2.0 dated Monday, 17 October 2011 2 Number Algebra Number place value Fractions decimals Real

More information

FIBONACCI SEARCH WITH ARBITRARY FIRST EVALUATION ABSTKACT

FIBONACCI SEARCH WITH ARBITRARY FIRST EVALUATION ABSTKACT FIBONACCI SEARCH WITH ARBITRARY FIRST EVALUATION CHRISTOPHWITZGALL Mathematics Research Laboratory, Boeing Scientific Research Laboratory ABSTKACT The Fibonacci search technique for maximizing a unimodal

More information

Notes for course EE1.1 Circuit Analysis TOPIC 4 NODAL ANALYSIS

Notes for course EE1.1 Circuit Analysis TOPIC 4 NODAL ANALYSIS Notes for course EE1.1 Circuit Analysis 2004-05 TOPIC 4 NODAL ANALYSIS OBJECTIVES 1) To develop Nodal Analysis of Circuits without Voltage Sources 2) To develop Nodal Analysis of Circuits with Voltage

More information

Notes on Complexity Theory Last updated: December, Lecture 2

Notes on Complexity Theory Last updated: December, Lecture 2 Notes on Complexity Theory Last updated: December, 2011 Jonathan Katz Lecture 2 1 Review The running time of a Turing machine M on input x is the number of steps M takes before it halts. Machine M is said

More information

PREFACE. Synergy for Success in Mathematics 7 is designed for Grade 7 students. The textbook

PREFACE. Synergy for Success in Mathematics 7 is designed for Grade 7 students. The textbook Synergy for Success in Mathematics 7 is designed for Grade 7 students. The textbook contains all the required learning competencies and is supplemented with some additional topics for enrichment. Lessons

More information

Enumeration Schemes for Words Avoiding Permutations

Enumeration Schemes for Words Avoiding Permutations Enumeration Schemes for Words Avoiding Permutations Lara Pudwell November 27, 2007 Abstract The enumeration of permutation classes has been accomplished with a variety of techniques. One wide-reaching

More information

Getting Started with Communications Engineering

Getting Started with Communications Engineering 1 Linear algebra is the algebra of linear equations: the term linear being used in the same sense as in linear functions, such as: which is the equation of a straight line. y ax c (0.1) Of course, if we

More information

MATH 521, WEEK 2: Rational and Real Numbers, Ordered Sets, Countable Sets

MATH 521, WEEK 2: Rational and Real Numbers, Ordered Sets, Countable Sets MATH 521, WEEK 2: Rational and Real Numbers, Ordered Sets, Countable Sets 1 Rational and Real Numbers Recall that a number is rational if it can be written in the form a/b where a, b Z and b 0, and a number

More information

GCSE MARKING SCHEME SUMMER 2016 GCSE MATHEMATICS LINKED PAIR APPLICATIONS UNIT 2 HIGHER WJEC CBAC Ltd.

GCSE MARKING SCHEME SUMMER 2016 GCSE MATHEMATICS LINKED PAIR APPLICATIONS UNIT 2 HIGHER WJEC CBAC Ltd. GCSE MARKING SCHEME SUMMER 2016 GCSE MATHEMATICS LINKED PAIR APPLICATIONS UNIT 2 HIGHER 4362-02 INTRODUCTION This marking scheme was used by WJEC for the 2016 examination. It was finalised after detailed

More information

COPYRIGHTED MATERIAL INTRODUCTION CHAPTER 1

COPYRIGHTED MATERIAL INTRODUCTION CHAPTER 1 CHAPTER 1 INTRODUCTION 1.1 INTRODUCTION We currently live in what is often termed the information age. Aided by new and emerging technologies, data are being collected at unprecedented rates in all walks

More information

(Refer Slide Time: 0:21)

(Refer Slide Time: 0:21) Theory of Computation Prof. Somenath Biswas Department of Computer Science and Engineering Indian Institute of Technology Kanpur Lecture 7 A generalisation of pumping lemma, Non-deterministic finite automata

More information

Vector Spaces in Quantum Mechanics

Vector Spaces in Quantum Mechanics Chapter 8 Vector Spaces in Quantum Mechanics We have seen in the previous Chapter that there is a sense in which the state of a quantum system can be thought of as being made up of other possible states.

More information

THE TEACHER UNDERSTANDS THE REAL NUMBER SYSTEM AND ITS STRUCTURE, OPERATIONS, ALGORITHMS, AND REPRESENTATIONS

THE TEACHER UNDERSTANDS THE REAL NUMBER SYSTEM AND ITS STRUCTURE, OPERATIONS, ALGORITHMS, AND REPRESENTATIONS The real number SySTeM C O M P E T E N C Y 1 THE TEACHER UNDERSTANDS THE REAL NUMBER SYSTEM AND ITS STRUCTURE, OPERATIONS, ALGORITHMS, AND REPRESENTATIONS This competency section reviews some of the fundamental

More information

Generalized Pythagoras Theorem

Generalized Pythagoras Theorem Generalized Pythagoras Theorem The Pythagoras theorem came from India through Arab mathematicians to the Greeks. It claims that if we draw squares on the sides of a right angle triangle, then the two smaller

More information

Limits and Continuity

Limits and Continuity Chapter Limits and Continuity. Limits of Sequences.. The Concept of Limit and Its Properties A sequence { } is an ordered infinite list x,x,...,,... The n-th term of the sequence is, and n is the index

More information

a (b + c) = a b + a c

a (b + c) = a b + a c Chapter 1 Vector spaces In the Linear Algebra I module, we encountered two kinds of vector space, namely real and complex. The real numbers and the complex numbers are both examples of an algebraic structure

More information

THE INVERSE PROBLEM FOR DIRECTED CURRENT ELECTRICAL NETWORKS

THE INVERSE PROBLEM FOR DIRECTED CURRENT ELECTRICAL NETWORKS THE INVERSE PROBLEM FOR DIRECTED CURRENT ELECTRICAL NETWORKS JOEL NISHIMURA Abstract. This paper investigates the inverse problem for the directed current networks defined by Orion Bawdon [1]. Using a

More information

Checking Consistency. Chapter Introduction Support of a Consistent Family

Checking Consistency. Chapter Introduction Support of a Consistent Family Chapter 11 Checking Consistency 11.1 Introduction The conditions which define a consistent family of histories were stated in Ch. 10. The sample space must consist of a collection of mutually orthogonal

More information

UNIT 2 KINEMATICS OF LINKAGE MECHANISMS

UNIT 2 KINEMATICS OF LINKAGE MECHANISMS UNIT 2 KINEMATICS OF LINKAGE MECHANISMS ABSOLUTE AND RELATIVE VELOCITY An absolute velocity is the velocity of a point measured from a fixed point (normally the ground or anything rigidly attached to the

More information

Efficient packing of unit squares in a square

Efficient packing of unit squares in a square Loughborough University Institutional Repository Efficient packing of unit squares in a square This item was submitted to Loughborough University's Institutional Repository by the/an author. Additional

More information

Lecture 12: Arguments for the absolutist and relationist views of space

Lecture 12: Arguments for the absolutist and relationist views of space 12.1 432018 PHILOSOPHY OF PHYSICS (Spring 2002) Lecture 12: Arguments for the absolutist and relationist views of space Preliminary reading: Sklar, pp. 19-25. Now that we have seen Newton s and Leibniz

More information