art and mathesis: mondrian's structures

Leonardo

Art and Mathesis: Mondrian's StructuresAuthor(s): Anthony HillSource: Leonardo, Vol. 1, No. 3 (Jul., 1968), pp. 233-242Published by: The MIT PressStable URL: http://www.jstor.org/stable/1571867 .

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Leonardo, Vol. I, pp. 233-242. Pergamon Press 1968. Printed in Great Britain

ART AND MATH ESIS: MONDRIAN'S STRUCTURES

Anthony Hill*

Abstract-The author introduces extracts from his study of the relationship of the works of Mondrian to mathematics. They are part of his research into mathematical thinking and abstract art. Concepts of structure, especially topological concepts of connectivity, are applied to the linear schemes in Mondrian's compositions. The schemes, to be regarded as networks, are then analysed in terms used in Graph Theory.

The idea of the linear infra-structure is introduced as well as the idea of the formal topological image. The notion of symmetry is restated for structures, where congruence is defined topologically; as a result, the discussion of symmetry and asymmetry as applied to Mondrian's compositions is broadened.

The possibility of computing the information content of linear infra-structures is discussed by the author. He investigates Mondrian's 'axioms' treating them as a restricted sub-set ofpossibilities allowed when composing on a square lattice.

The author hopes to have shown how we could enrich our pleasure and interest in looking at Mondrian's paintings-through asking different kinds of questions relating to their structure and geometry. Furthermore the approach may be used consciously by the artist in creating his works.

'Science says the first word on everything, and the last on nothing.'

Victor Hugo [1]

I. INTRODUCTION

In the first issue of Leonardo readers were presented with a number of definitions of visual fine art.

Now a question of kind 'What is Art?' can be construed as a directed question of a rather special kind; one could say that it is caused by a state of doubt (or unknowing) and answered with the tacit assumption that the doubt (or some of it) has been dispelled. Simpler questions raise no further questions, especially those of the kind that ask 'but is that really a question/answer?' To special questions, answers are often given by persons whose field of study includes the subject of the question. Now in the case of Art we can ask who best to answer 'What is Art ?' and there could be two quite different candidates for the best answer here:

(a) the artist, as he is engaged in making Art, (b) the philosopher, as he is engaged in framing

and answering questions. This leads us to ask the following: need the artist familiarize himself with philosophy and, conversely, need the philosopher familiarize himself with art?

* Artist living at 24 Charlotte Street, London W. 1, England. (Received 28 November, 1967.)

Clearly each has a need to be partially familiar with the other's work, and the very state of being 'partially familiar' becomes the link between the two.

I have called this paper Art and Mathesis and, as with any paper whose title employs the form 'Art and ...', the issue is presumably that of bringing into question the possible relationships existing between art and some other discipline.

Let us assume for the present purpose that examples tacitly cited here as Art promote no question of the kind 'but is it Art?' This leaves us with the question 'but what is Mathesis ?' (especially since this word does not figure in smaller dictionaries -such as the Concise Oxford.)

Now in defining something, one says first what the word means,-an explanation-and then what the explanation amounts to our saying. Had the word been mathematics instead of mathesis, the problem of whether it would be necessary to define mathematics would certainly be pertinent but perhaps not essential.* So why Mathesis?

Mathesis is a euphonious but rather grandiloquent word, it belongs with Ars magna and Ars combi- natoria and in the hands of Descartes it was Mathesis universalis.

Possibly the last author to use the word was Husserl, who spoke about 'pure Mathesis'-a

* If asked for a definition, I would choose that of N. Bourbaki (cf. Note E) for whom mathematics is none other than the study of abstract structures or formal patterns of connectiveness.

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Anthony Hill

universal science to include all the pure 'structures' of thinking [2].

My purpose in using the word Mathesis is deli- berate, although it is perhaps not as yet well defined, for I am persuaded that at least some part of the general point of view adopted in my approach can be described as phenomenological.

I am not too well acquainted with the inner workings of this philosophical tendency,* but have noticed that the subject of abstract art has recently been approached from the standpoint of phenomenology. Some time ago I chanced to read the following:

'If the adherents of phenomenology and Neo-positivism had paid any attention to it, their epistemological assumptions would have let them to a recognition of abstract art' [3].

More recently I came upon the following state- ments by the Swiss mathematician Paul Bernays:

'In the region of colours and sounds the phenomenological investigation is still in its beginning and is certainly bound with the fact that it has no great importance for theoretical physics, since in physics we are induced, at an early stage, to eliminate colours and sounds as qualities. In fact, what contrasts phenomenologically with the qualitative is not the quantative, as is taught by traditional philosophy, but the structural, i.e. forms of being aside and of being composite, etc. with all the concepts and laws that relate to them- Mathematics, however, can be regarded as the theoretical phenomenology of structures' [4].

In my analysis of Mondrian's compositions- that is to say their structure-the point of view, vis-a-vis the mathematics cited, can be taken as phenomenological in the sense expressed above.

I am interested to explore concepts of structure hitherto ignored or taken for granted-such as connectivity-when describing a compositional scheme. This means that attributes normally singled out as 'mathematical' (those dependent on measurement, such as ratios and the symmetry based on metrical congruence) are to be 'bracketed', i.e. put to one side, even where they seem to be part of the immediate data of perception.

The material in Section II is extracted from a much larger sketch that sets out to show how Mondrian's compositions can be classified, and also explores the basis and the application of the criteria used.

In the former part a detailed survey has been made based on available sources to date [5], while in the latter the researches have been wholly in the province of mathematics; i.e. topology, graph

* In psychology an unprejudiced attitude (i.e. without prior assumptions) is referred to as descriptive or phenomenological, and in philosophy, phenomenology attempts to analyse consciousness, taking as its sole absolute- 'objectivity.' The founder of modern phenomenology was Edmund Husserl.

theory, group theory and combinational analysis. The final aim would be to offer these findings to aesthetics, but an aesthetics phenomenologically orientated.

To give a specific example of what is envisaged, it would be necessary to examine around 130 of Mondrian's compositional schemes and to compute the topological information content of each scheme ... and, in invoking the theory of information, I am restricting this specifically to the linear network of the composition and not to the composition as a whole.

Ultimately findings from research, for example the topological information content, would become valuable data for a statistical account of the changes and of the stable factors (invariants) in Mondrian's structural syntax. And this in turn could lead to establishing the 'set of Mondrian axioms'.

Mondrian's 'axioms' allow for a very large range of syntactical usage, of which his own works drew upon only a significantly small range. I believe it would be valuable to explore fully the entire set of these lattice structures and, within them, the subset that was drawn upon by Mondrian. I believe this could provide us with part of the essential material with which to tackle the obvious but puzzling question-what lay behind Mondrian's choice ?

The remaining source material needed is, of course, Mondrian's recorded views; with particular attention to the account he gave of his working methods and ideas-which unfortunately does not amount to very much.*

It is generally accepted that Mondrian employed no calculation or measuring when composing his pictures; it follows from this that no 'laws of composition' were consciously applied by Mondrian. The main evidence for believing this to be true comes from two sources: (a) Mondrian's own assertions and (b) the evidence of the paintings themselves. Thus, those who would seek to prove that Mondrian did employ laws or rules, would have to take as their premise the idea that he did so unwittingly and/or unconsciously. This, too, must rest on the assumption that the compositions do in fact approximate to 'mathematical proportions'. As a consequence, we are left with the contention that Mondrian was a classical artist, without his being aware of the fact.

If it is the compositions that are taken as providing evidence for this contention (or for its refutation), we shall need to be quite objective in any demon- stration that seeks to show a valid correspondence

* Piet Mondrian (1872-1944). Since his death, the number of publications on Mondrian has remained small (see Note A) (there were none during his lifetime) and we still await publication of his complete writings and eventually an authoritative catalogue raisone-in chronological order. Due to the very high prices that his paintings commanded (as soon as he died) there are unquestionably forgeries at large and this is a further reason for hoping that Mondrian's works should be studied in the best traditions of disinterested scholarship. One could hope to see the foundation of a Mondrian museum or a Mondrian institute.

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Art and Mathesis: Mondrian's Structures

with such 'rules,' and to be equally objective regarding the validity of the 'rules,' 'systems' etc. that are invoked.

In the event of concluding one or the other contention to be 'proven,' we would still not be satisfied. The conclusion would still be spurious, since any judgment of this kind would not rest solely on our knowledge of the work and the artist alone (his psychology, etc. of which we are, clearly, not well acquainted), but tacitly we would be invoking as evidence the responses of both the investigation and his subjects....

I have looked into numerous published 'analyses' of Mondrian's compositions (see Note B) and, in my opinion, they contribute nothing to our under- standing of Mondrian's actual procedure. However, looked at from another viewpoint, these analyses command our attention as evidence of a significant trend within the field of'aesthetic analyses' (whether it be numerological mysticism, Gestalt psychology or, simply, empirical laboratory investigation.) The basis of my investigations is one of non-metrical considerations of Mondrian's compositions and, therefore, has little or nothing to say by way of overlap with researches into 'the painter's secret geometry,' be it Mondrian's or any other artist.

The analyses presented in Sections II and III have no recourse to 'number lore' and, although some numerical computation is required (things are counted, etc.), nothing is measured against a scale (a ruler) nor do ruler and compass constructions play any part.

However, let me make it quite clear in advance that the ideas presented here are in no way offered as being amongst those that Mondrian is known to have employed.

So, while the implications of this last statement indeed call for a detailed discussion, they remain outside the scope of the present article. It also follows that Mondrian's interest in mathematics (where this can be shown) and in philosophy, cannot be explored here, for example, his possible interest in authors other than Schoenmaekers, such as L. E. J. Brouwer, Note C, must be set aside as having no essential bearing on the procedures adopted here.

II. BASIC NOTIONS FOR THE ANALYSIS

Essentially, a graph is to be taken as an abstract structure but, in practice, particularly in Graph Theory, it is usually interpreted as a network. In the analysis that follows, a network is treated as something actually visualized, such as an elastic net made from elastic thread and held together with knots that can slide freely. Such an object should be kept in mind as a model.

An elastic network is a structure that can be given innumerable forms-it is a deformable structure allowing continuous deformation from any chosen form into any other; the other restrictions to bear in mind are that:

(1) We do not undo or add knots, or add or remove the threads.

(2) We do not allow cutting of the threads.

This kind of network consists only of knots and the lengths of thread spanning the knots. The knots are the nodes of the network and the lines which span the knots are called alternatively lines, edges, members or bonds, depending on the field of study that uses networks or graphical models.

In what follows, wherever possible, the procedures in Graph Theory are used, introducing terms as they are required. Networks, it should be emphasized, consist of an assemblage of connected lines and points and, in a diagram of this kind, the lines are obviously less 'abstract' than the points.

Three basic kinds of points are referred to in Graph Theory: nodal points (already discussed), terminal points, and what can be referred to as arbitrary points. A nodal point is therefore any point from which emanate three or more lines, arbitrary points-only two lines, and terminal points-only one line.

Only one kind of point figures in the 'elastic' model, the nodal point, so the arbitrary and terminal points have to be 'added' (in the form of 'beads' with which we could represent all the kinds of points).

For the mathematician the 'fundamental particle' of Geometry is the point and the foundation of modern Topology is the discipline known as Point Set Theory (the Theory of Sets was due to Cantor in 1879). For us the fundamental element in a network is the line, and there are no points as such.

'Graphical Models' can either present the lines as the objects (or elements), with points representing the relationships-or vice versa, the objects are the points and the lines the relationships, as with the atoms and bonds in molecular structures.

In the diagrams I have used, arbitrary points (called nodes of degree two) represented by small white circles. Points of degree n (n being any integer greater than two)-called nodes of degree n -and terminal points (terminal nodes) represented by small black circles.

A formal topological image is a graph drawn in such a manner as to visually emphasize the structure; the containing circuit is to be drawn in the form of a perfect circle and lines are drawn with equal length wherever possible and distributed with the greatest possible symmetry.

III. MONDRIAN'S LINEAR INFRA- STRUCTURES

During the two decades 1918-1938 Mondrian painted over one hundred and ninety pictures, of which at least a hundred exhibit the following 'mathematical feature':

the linear infra-structure is a polyhedral network. Since Mondrian's paintings present us with a

purely two dimensional structure, it must seem irrelevant at the outset to invoke a three dimensional image-the polyhedron. What do we mean by a polyhedral network? For our purposes a polyhedral

235



Anthony Hill

network is simply of the class of plane networks characterized by the following:

(1) It is a system or network (or structure) of connected lines that can be drawn in a plane. It is a planar graph, meaning that it can always be drawn so that none of the lines produce a cross-over.

(2) Every line in the network belongs to two and only two cells (regions, circuits, cycles) and, as a result, the graph is said to be triply connected, i.e. it is only possible to disconnect the graph (to product two connected graphs) by cutting at least three lines.

(3) Every cell (cycle or circuit) is made up of at least three lines, this means the graph can be conceived as a planar projection (a Schlegel diagram) of a simply connected polyhedra (i.e. a map on a plane or a sphere).

Thus a flag can be seen as presenting a network that is also the projection of a square based pyramid (Fig. I).

1 2 3 4

Fig. I.

If we imagine the thick cross in the flag to shrink to a thin linear cross and re-draw the figure as a graph, including only the nodal points, we arrive at a network which is essentially that of a polyhedron.

But 'seeing it' this way might require that we examine other 'ways' of seeing it (Fig. II).

I \ 2 3

X\ i3(S'"''^ 4

Fig. II. 5

The scheme of reduction in Fig. II shows that II (1) can be seen as composed of twenty 'lines' (instead of the eight shown in Fig. I); the sequence II (1), II (4), II (5), treats the figure as a map, which in turn can be considered to exhibit a linear structure where the sequence II (1), II (2), II (3) shows II (1) as essentially a graph but of a different kind from that shown in Fig. I.

The reduction shown in Fig. III takes III (1) to be interpreted as III (2) or III (3) i.e. with the interior lines crossing, and not meeting at a point, and, if as III (3), then it is again a polyhedron (the tetrahedron represented as the projection III (4) and as the solid III (5)).

2

3 4 5

Fig. III.

We can see what happens if we replace the flag by a version where the cross is a thinner form- where the flag has the cross suppressed by a counter- change pattern (Fig. IV).

We see, from the standpoint of a linear network, that IV (1) and IV (2) are indistinguishable, despite

43 2

Fig. IV.

the fact that in IV (1) we say that we see 'lines' and in IV (2) only squares.

Finally, in further pursuit of the question of lines, or more specifically the number of lines in a configuration, we could take as a criterion the traversability of different graphs (Fig. V).

2

II 5

8

2

Fig. V.

Figure V shows how the flag can be traced (or drawn) by a minimum of four continuous sweeps of the hand, that is, four necessarily discontinuous trajectories (if we trace each line once and only once). The first trajectory takes in lines 1, 2 and 3, and this configuration can be repeated in a clockwise direction three more times. (There are of course other ways of setting out and completing the tracing with only four trajectories.)

In deciding to make the reduction shown in Fig. I we arrive at a graph (or network) composed of eight lines and five points or, considered as a map, we would also refer to the regions as cells, in this case 1 four-sided cell divided into 4 three-sided cells. In calling them cells we retain the idea of two-dimensional areas; while in graph theory we would see them as one dimensional and call them cycles (or closed circuits). If we had decided on the reduction scheme in Fig. II, the outer cell or cycle

236




would be taken as twelve sided (as in II (2)) composed of twelve line segments, and twelve points, or in the case of II (4) eight sided.

Thus we would seem to have changed our terms considerably as we developed a way of 'looking at' partitions of rectangles (or squares) into rectangles and/or squares. We now show how a reconciliation can be affected.

Let us consider the network in Fig. VI. Figure VI (2) is to be taken as the formal topological image

0 0 0 0 0

FO8 C o t t0

O D

2 o 0 0 0

3

Fig. VI.

of VI (1); in mathematical languag homeomorphically irreducible graph c In reducing VI (1) to VI (2) we hav 'corners' of the 'square' VI (1) and ii have also ignored four of the points.

Before proceeding, we must c definition of reducing a graph to a t homeomorphically irreducible) graph

In Fig. VII the reduction VII (2) can as a representation of the continuous 1 could describe VII (1).

d d a

cd

dC a 2

Fig. VII.

;e VI (2) is a )n five points. re ignored the n doing so we

:omplete our

We are in a position to define F, now to be taken as a topological lattice network T.

The topological network becomes a network on the lattice and, as such, becomes an n x n or n x m array. The smallest array on the lattice with a topological network that is polyhydral has n=2, n2 being the number of cells enclosed. Arrays are necessarily convex, as are the cells in the interior.

In a lattice embedding F, all the points of F lie on lattice points and all the lines must span lattice points by lines of unit length, that is all lines will be of length one, two, three etc.

For any F to be a T, the containing cell (cycle) must be composed of points of degree three, and no independent cycles can be composed of less than four lines; a cell with only three edges must share at least one edge with the containing cell.

Our interest is in enumerating the number of distinct Fand then the number of distinct Temployed by Mondrian, with specific regard to symmetry and asymmetry and/or the topological information content.

Our first question might well be this: are there compositions where the linear network takes a lattice embedding requiring the minimal array n2 when n =2?

topological (or As a start, let us examine the possible networks l (Fig. VI). within the 22 array (Cf. Fig. VIII). be considered In Fig. VIII we show five possibilities allowing trajectory that for every kind of network (excluding all reflections,

rotations, etc., which would bring the total to fifteen).

O O O O We note that VIII (2) and VIII (3) are indis-

0

O I i O O (X4) (X4) (X2) (X4) (XI) 1 2 3 4 5

3

In the Graph VII (2) we ignore the four points d but not the point a. The d points are points of degree two, and the point a is a terminal point. In VII (1) there are seven points in all and eight lines.

The reduced graph VII (2) has three points and four lines; point a, a terminal point; c, a point of degree three and b of degree four. Thus a topological graph is simply a graph with no points of degree two. (In Fig. VII (3) we show VII (2) on a lattice.)

Incidental here, but of importance to us in Fig. VII, is that whereas VII (1) is a graph whose group is the identity-which means it is asymmetric, VII (2) is symmetric ... and not only because it is 'drawn' symmetrically.

Returning to Fig. VI, what we see is the 'square with a cross' VI (1) transformed into an elastic topological network presented as a formaltopological image VI (2)-and this is then laid over the lattice VI (3).

We shall call the operation of transferring a formal topological image F on to a lattice-making a lattice embedding of F.

18

Fig. VIII.

tinguishable as topological networks. As a set of lattice networks VIII (2) is ruled out, as one of the interior cells is not convex (Mondrian never used the bent line-the 'left or right hand bend in a path').

Of the examples that remain VIII (1) is not a graph we would call a network in our sense, as it contains a terminal line (Mondrian used such graphs, but not this minimal example). We are left with VIII (4) and VIII (5); VIII (4) is the minimal polyhedral network (Mondrian never employed it) while VIII (5) is a complete lattice.

It is this last example that we discover to be the smallest F employed by Mondrian but with this reservation, while it is a lattice network, Mondrian employed it in a non-lattice format, i.e. a lozenge. (Cf. Fig. IX).

The two paintings represented by the F in Fig. IX are 'Composition with Black and Blue', 1926, and 'Composition with Two Lines', 1931. (We note that for Mondrian the linear structure is conceived as comprising two 'lines' . ..)

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Anthony Hill

2 Fig. IX.

Amongst his compositions in a rectangular (and/or square) format-the majority of Mondrian's paintings-the smallest arrays employed are rectangular, 3 x 2, as shown in Fig. X (1) and (2).

o ::-o o6--*-,

I 2 Fig. X.

If we examine the relative sizes of arrays used by Mondrian between 1918 and 1944, we find that at the beginning he used 'complete' lattices (arrays of 162 and 15x16) but by 1930 had moved to lattices of half as much, 82, to rise slowly and almost attain the same density as in 1918, namely in the paintings called 'New York City', 1942 and 'Broadway Boogie Woogie', 1942-1943, each of which employ an array of 12x 13. (However in these final paintings it is no longer possible to see the lines as comprising a planar arrangement.)

Before examining in detail the polyhedral networks, we should briefly mention the exceptions, namely:

(a) graphs with terminal lines; (b) networks which are less than triply

connected. Of the first group, there exist at least ten compo-

sitions (6 of 1921 and 4 of 1922) and it is interesting to note that the terminal lines, if extended, would reach the containing cycle without crossing any other lines, thus they always paint 'outwards'.

Of the second group, there is only one example from the period ending in 1938. It is the 'Composi- tion with Blue and White', 1936, shown in Fig. XI (1) and (2).

G

0 a

Fig. XI.

This network is not polyhedral, for it can be disconnected by cutting the lines a and a', which separates it into two connected graphs.

Other features of this graph are: (i) that it is point-regular-all the points are of the same degree, and (ii) that the lines in the interior comprise two trees (a tree being a connected graph with no cycle).

Amongst the one hundred or so polyhedral networks, well over two thirds are networks con-

taining independent cycles. However between 1922 and 1936 there are at least 23 compositions where the networks contain no independent cycles-the interior lines forming a single tree. The earliest examples of a composition with no independent cycles is unique on two counts; it contains the largest tree employed by Mondrian (it has fourteen points).

In Fig. XII we show a sketch of the line structure of the 'Composition with Red, Yellow and Blue', 1922 (XII (1)); the lattice network XII (2) and the formal topological image XII (3).

From the size of the reproduction shown in Seuphor's catalogue, the linear infra-structure of

0I

c < -

r , d G 0 ;_f.~~~~~~~~~~~~~~~~ .,<

2 3

Fig. XII.

this composition would appear to be a cubic network. But on closer inspection we see that this is not the case, and that it has a free terminal line.

We find in all the linear infra-structures containing single trees that they reduce to only nine distinct trees; ranging from trees with seven points right through to fourteen points.

IV. SYMMETRY/ASYMMETRY

The major purpose of nearly all that was included in Section II has been to set out the essential background for the treatment of symmetry/asymmetry in a linear infrastructure.

I attempted in an article on structure [11] to explain briefly the concepts of symmetry/asymmetry in graphs and the related topic of the topological information content.

I chose there for my example a painting by Mondrian exhibiting the smallest asymmetric network on the lattice (Cf. Fig. XIII).

0

L 1

I

2 3

Fig. XIII.

It is not difficult to show that Mondrian's avoidance of symmetry was real, but this can only be done by a careful examination of a large enough quantity of his works. Once we have reached agree- ment on this, we will have 'defined' symmetry/ asymmetry in a manner that is quite acceptable.

a

238




It is my intention here to attempt a similar definition, i.e. by showing in this case how we arrive at a different concept of symmetry.

Let us start the attempt not with the examples in Scheme VIII but something different:

In Fig. XIV we show a lattice network XIV (2)

w- < v 2 3

Fig. XVI.

2 3

Fig. XIV.

employed in at least three of Mondrian's compositions made between 1930-1932 (XIV (1) indicates the kind of proportions used).

Now no lattice arrangement of XIV (3) (the formal topological image) can result in a symmetrical disposition of the lines. This becomes obvious in observing the structure of XIV (3), which at the same time we see drawn with its lines disposed symmetrically.

This means that when we take a network off the lattice we will often find that it is symmetrical.

Before focusing sharply on the distinction between symmetry/asymmetry on the lattice, as against that of the formal topological image, let us consider Fig. XV.

o -- -- --

2

Fig. XV. 3

The graph in Fig. XV (2) does not reveal the fact that it is symmetric. To show this, we must resort to a mapping of XV (2) into itself; the result of which XV (3) is plainly symmetric. However, as with XV (2), there is no possible arrangement of XV (3) on the lattice because one of the independent circuits contains only three points.

To return to the graphs XV (2) and XV (3), we note in Fig. XVI how the graph (2) in XVI has been mapped into itself, i.e. into the cycle of bb'c'c, which now becomes the 'containing cycle'.

The points a and a', b and b', c and c' and d and d' are said to be interchangeable; the remaining points are distinguishable, i.e. they cannot be interchanged with any others.

Thus for a graph to be asymmetric, we have to show that all of its points are distinguishable, and this is the case in the examples shown in Fig. XII (1)

(a non-topological graph) and in Fig. XVI (3) (a topological graph).

The subject under discussion is given its fullest treatment when we ask: 'what is the group of a given graph?' Obviously, we depend on just this treatment, which is perhaps too large a topic to pursue here any further.

However there is a further example that it would be useful to show. We pose the following problem:

Given two cubic asymmetric graphs, both containing the conjectured minimum number of points (namely twelve), we ask: which of the two provides us with the greater number of distinct lattice embeddings ?

The problem can be stated in the following terms: of the two trees (a) and (b) shown in Fig. XVII, which

A ,,i v (o) (b)

O 0 0

0 0 (0')

0 0 0 0 0

(a") (a )

Fig. XVII.

one has the greater number of distinct 'citings' on the lattice such that all the terminal points can lie on a convex cycle ?

In Fig. XVII we show the two distinct topological trees that give rise to networks whose group in each case is the identity (i.e. they are asymmetric).

In XVII (a) we show a possible lattice citing of XVII (a); and in XVII (a") a further example which

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Anthony Hill

requires that we lengthen three of the terminal lines, so as to arrive at the lattice in XVII (a"').

At a first count I have obtained 30 for XVII (a) (240, if we include all rotations, reflexions, etc.) and 36 for XVII (b) (288 with rotations, etc.)

There are two factors of interest here; first, that a network can be point regular and yet have all its points distinguishable so that it is asymmetric; and second, that two such networks can exist while being topologically distinct as seen in the example of the two cubic graphs on twelve points.

In constructing these two networks, each from a distinct tree, we notice that in the case of one, XVII (a), it has only one planar embedding while with the other, XVII (b) has two more embeddings as shown in Fig. XVIII.

> H Z K

such graphs all the respective components are interchangeable; as a result no points, lines or cycles are topologically distinguishable. These graphs are therefore completely symmetric and their information content is zero.

There are three graphs of this kind that Mondrian could have used. I show two of these. They are the networks of the tetrahedron and the cube shown in Fig. XIX (1) and (2).

" U

2 2 i

T

Fig. XVIII.

All distinct lattice citings of a tree are topologically congruent, as are the networks that can be made by joining their end points to a circuit. But distinct embeddings of a given tree in the plane (if it should have more than one)-prior to citing on the lattice- give rise to distinct networks (i.e. not topologically congruent).

With the networks derived from XVIII (1) and XVIII (2), there are at least 24 lattice embeddings that are distinct (producing 172 with all rotations) for the former, and 12 (76 with rotations) for the latter.

V. THE TOPOLOGICAL INFORMATION CONTENT

In an article [11] already referred to, I outline the application of the topological information content to networks. Broadly speaking, this amounts to giving a quantitative index to a qualitative feature in networks, namely the amount of symmetry/asymmetry.

We have seen that if all the points of a network* are distinguishable, then it is asymmetric and, in this case, the topological information content is maximal. Thus the network shown in Fig. XIII has an information content expressed by:

log2 9=3-17,

since there are nine points, all of them distinguishable. In the same article I pointed out that I could find no example where Mondrian employed a network whose information content was zero.

I have mentioned that on a lattice we can have only one kind of point-regular graph, the cubic. Now amongst planar graphs there is a class of polyhedral networks that are point regular and symmetric, that is point and line symmetric. In

* A 'strict' topological graph, connected and containing no multiple lines or loops (one- or two-sided cells).

Fig. XIX.

In Fig. XX we show the four distinct lattice embeddings of the network in XIX (2)-giving sixteen

* ? U--~--- o X , V V -

O . } I t ,

(X2) (X4) (X8)

3

(X2)

4

Fig. XX.

in all-(XX (3) being asymmetric on the lattice, it gives rise to seven other embeddings by permuting the reflexions, rotations, etc.).

VI. CONCLUSIONS

Up till now, the 'conventional' view of mathematics and its relevance to art has obscured the possibility of looking into Mondrian's compositions for any features other than those of an 'arithmetical' character (ratios, proportions, etc.) The search for a 'hidden geometry' in his works has so far found nothing of importance, therefore it must remain in the realm of conjecture. I hope in the analyses presented here to have shown how we can enrich our pleasure (and interest) in looking at Mondrian's paintings-through asking different kinds of questions relating to their structure and 'geometry'. Furthermore this approach is, in general, extendable to many other types of work and may be used consciously by the artist in creating his works.

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I




Of the questions discussed, the idea of Mondrian's 'axioms' seems to me probably the most worthwhile for further investigation. The idea of being able to perceive simple examples of topological symmetry and asymmetry also raises questions of interest in the study of visual perception.

NOTES A. The standard work on Mondrian to date is

Seuphor's Monograph [5]. The catalogue raissone (the first attempted) puts the works into categories of 'subject matter' and 'compositional features' (for the abstract works).

B. See Bouleau's The Painter's Secret Geometry [7], Biederman's The New Cezanne [8], and Ragghi- anti's Mondrain e 1' Arte del XX Siecle [9].

Bouleau's analysis of Mondrian's 'Composition with Two Lines', 1926-undertaken to show that the work utilizes the Golden Section-is worth a study in itself, on the basis of the errors in measurement and other inadmissible assumptions. In an article by Seuphor [6], drawings were repro- duced bearing the caption: 'Etude pour le tableau 24 Losangique,' these were mathematical-looking schemes in which the symbol 7r appeared. The reader was left to draw the conclusion that Mon- drian made such studies . . . and at least one se- rious journal (that of the Modular Society) was tricked into reproducing them, credited to Mon-

drian, as evidence of his manner of composing. However in a later edition of the magazine that carried Seuphor's article, Seuphor explained that they were not by Mondrian but by Georges Vanton- gerloo. In Seuphor's book he explains that Vantongerloo, who had made numerous analyses of Mondrian's compositions, finally abandoned these studies as he considered them to be quite incorrect!

C. See Joost Baljeu's 'Problem of Reality' [10]. In the present author's view the account that Baljeu gives of Brouwer's philosophy (and his assessment of Brouwer's position in Science) is deceptive, to say the least.

D. Bourbaki's concept of structure recognizes three basic kinds, from which all other kinds are to be derived:

(1) Algebraic structures whose prototype is the group.

(2) The order structures, one of whose principal forms is the lattice.

(3) Topological structures concerned with continuity.

(I found both their definition of mathematics and the above categories only recently and was interested to note the coincident, although superficial simi- larity, to my own categories in analysing the networks.)

REFERENCES

1. Victor Hugo, Things of the Infinite, essay in Hugo's Intellectual Autobiography (Post- criptum de ma Vie), translated by Lorenzo O'Rourke (New York: Funk & Wagnallis, 1907).

2. Armando Asti Vera, Epistemology, Science of Structures, Transformation 1/3 (New York: Wittenborn Schultz, 1952).

3. Wladyslaw Tatarikiewicz, Abstract Art and Philosophy, Br. J. Aesthet. II, 3 (1962). See also Irving L. Zupnik: Phenomenology and Concept in Art, Br. J. Aesthet. VI, 2 (1966).

4. Paul Bernays, Comments on Wittengenstein's Philosophy of Mathematics, Ratio II, No. 1 (1959).

5. Michael Seuphor, Piet Mondrian (London: Thames & Hudson, 1957). 6. Michael Seuphor, Piet Mondrian et les origines du Neo-Plasticisme, Art D'Aujourd'hui,

No. 5 (1949). 7. Charles Bouleau, The Painter's Secret Geometry (London: Thames Hudson, 1963). 8. Charles Biederman, The New Cezanne (Red Wing Minnesota: Art History, 1958). 9. Carlo Ragghianti, Mondrian e l'Arte del XX Siecle (Venice: Neri Pozza, 1963).

10. Joost Baljeu, Problems of Reality-Marginal notes on the Dialectic Principles in the Aesthetics of Mondrian, Tatlin etc., Lugano Review, 1 (1965).

11. Anthony Hill, Program: Paragram: Structure, in Data-Directions in Art, Theory and Aesthetics. (London: Faber & Faber, to appear in 1968).

Art et Mathesis: Les Structures de Mondrian

Resume-L'auteur presente des extraits de son etude sur les rapports existant entre les aeuvres de Mondrian et les mathematiques. Ces extraits font partie de ses recherches sur la pensee math6matique et l'art abstrait. Des concepts de structure, et en particulier des concepts topologiques de connexite, sont appliques aux compositions lineaires dans les oeuvres de Mondrian. Ces compositions, dans lesquelles il faut voir des reseaux, sont ensuite analysees d'apres la theorie des graphes.

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242 Anthony Hill

II nous propose l'idee d'une infrastructure lineaire, ainsi que celle d'une image topologiqueformelle. I1 precise la notion de symetrie dans les structures ou la congruence est definie topologiquement, generalisant ainsi les notions de symetrie et d'assymetrie dans les compositions de Mondrian.

L'auteur envisage la possibilite de calculer le contenu d'informations donne par les infrastructures lineaires. I1 etudie les 'axiomes' de Mondrian en tant que sous-ensemble limite des possibilites que permet une composition sur un treillis carre.

L'auteur espere montrer comment nous pouvons accroitre notre plaisir et notre interet devant les peintures de Mondrian, en nous posant diff6rentes questions a propos de leur structure et de leur geometrie. Et de plus, cette approche peut etre consciemment utilisee par l'artiste lors de la creation de ses ceuvres.



art and mathesis: mondrian's structures

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