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C2C2C2: PYTHAGOREAN STRUCTURES IN DESIGN Athanassios Economou Georgia Institute of Technology Abstract: The structure of the direct product group C2C2C2 is examined in detail. The foundations of the group structure in Pythagorean arithmetic are briefly presented, the realization of the group in three-dimensional space is discussed and illustrated, and some notes are suggested for its implication in music notation. A special emphasis is given in four different decompositions of the group and all subsets are enumerated and illustrated. Various applications of this structure are suggested for further systematic studies in the analysis and synthesis of the architecture of form. Introduction “Between two cube numbers there are two mean proportional numbers, and the cube has to the cube the ratio triplicate of that which the side has to the side” (Euclid, Elements, Book VIII, Proposition 12) The conceptual shift from two-dimensional space to three-dimensional space has always been a central theme in mathematics, philosophy, aesthetics and design. Things that work in two-dimensional space might work in three-dimensional space or might fail miserably and the opposite. And quite often these shifts are combined with different conceptual frameworks and the boundaries between geometry, arithmetic, spatial design, or music composition merge and dissolve. Euclid’s proposition above is one more statement in this world making; it is essentially a formalization of Plato’s Pythagorean argument in Timeaus where Plato asserts that between two planes one mean suffices, but to connect two solids two means are necessary. Planes and solids in Plato’s argument are really square and cube numbers and this is clearly stated in Euclid’s proposition (figure 1). Geometry and arithmetic were inexorably related and two-dimensional and three-dimensional shapes among other things had numbers associated with them and the opposite.

2:2:2 2:2:3 3:3:2 3:3:3 a. b. c. d.

Figure 1. The two Platonic means between two cubes. The two cube numbers here are the numbers 8 and 27 and are constructed as cubes with sides 2:2:2 and 3:3:3 respectively. The two means are the

numbers 12 and 18 and are constructed as square prisms with sides 2:2:3 and 2:3:3 respectively.

A proper exposition of a unique three-dimensional structure in design, the C2C2C2, that has traditionally played the role of the mean between two-dimensional space and three dimensional space is the purpose of this paper and some of the underlying themes related with this structure in arithmetic and geometry will be briefly presented. There is a

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tremendous wealth of lessons to be learned by the closer look in this shape and particularly by its study of parts and their constant redefinition in various design settings. The arithmetic of C2C2C2 Shapes and numbers were closely associated in Platonic and Pythagorean thought. From a wild taxonomy of numbers that encompassed triangular, square, pentagonal, hexagonal numbers, and cube, pyramidal, truncated pyramidal, parallelepiped numbers and so on, all attributed primarily to neoplatonic thought, the only ones that have survived to our days are the square and the cube numbers; the former a species of planar numbers, the latter a species of solid numbers. For a convenient summary, see (T L Heath, 1981), and for a recent discussion on its impact on architectural thought in renaissance architecture, see (L March, 1998). Few remarks about solid numbers follow below to prepare the ground for the fundamental relation of the rectangular prism with specific constructs in arithmetic and neoplatonic lore. Solid numbers are seen as the factors of a number in three distinct dimensions; Cubes (κύβοι) are the product of three equal numbers and they are visualized as pairs of squares that are separated by a length equal to the side of the squares. Beams (δοκίδες) or columns (στηλίδες) and tiles (πλινθίδες) are the product of two equal numbers and one unequal and are visualized as pairs of squares separated by a length greater than the side of the square and are of the form m2(m+n); tiles are visualized as pairs of squares separated by a length lesser than the side of the square and are of the form m2(m-n). For example, the first Platonic mean in Euclid’s proposition is a beam, and the second is a tile. Potentially both beams and tiles could be further differentiated if the length between the two pairs of squares were just one unit greater or lesser than the side of the square. Scalene (σκαληνοί) are the product of three numbers all unequal; these numbers were further differentiated with respect to the type of relation between ratios of faces and pairs of ratios of faces: Parallelepipeds oblong (παραλληλεπίπεδοι ετεροµήκεις) are those numbers whose factors correspond to rectangular faces of the form m(m+1), so that the two factors differ by unity; Parallelepipeds prolate (παραλληλεπίπεδοι προµήκεις) are those numbers whose factors correspond to rectangular faces of the form m(m+n) where n>1. A classic neoplatonic example in the visualization of solid numbers is the transformation of these numbers with respect to the products of subsets of their factors. Nicomachus discussed various transformations of the number 180 including a variety of scalene, beam and tile decompositions (M D’Ooge, 1938); Many other possible sorts of their products that include the unity are not included in this discourse because in neoplatonic thought unity was not considered a dimension or a proper part. For example, any prime number n could be visualized only as a line n rather than a rectangular number n.1 or a solid number n.1.1. Eleven of possible neoplatonic transformations of the number 180 are depicted in Figure 2.

2:2:45

a. b. c. d. e. f. g. h. i. k. l.

2:3:30 2:5:18 2:6:15 2:9:10 3:3:20 3:4:15 3:5:12 3:6:10 4:5:9 5:6:6

Figure 2. Transformations of solid numbers: a. beam b-e. scalene f. beam g-k. scalene l. tile

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Still another powerful tradition in this discussion of rectangular prisms with unequal sides comes from the same neoplatonic strand but this time the focus lies in the relationships between three numbers or three magnitudes. For any two numbers can be considered as greater and lesser extremes with a medial term in between. Among the infinite numbers that lie in between any two numbers, some medians are more privileged at least in platonic thought and certainly absolutely revered in neoplatonic thought. These medians are computed taking into account all possible one-to-one relations that may exist between the two extremes and the medial term as well as their emerging three differences; the difference of the greater extreme and the mean, the mean and the lesser extreme, and the greater extreme and the lesser extreme. Two accounts have survived from antiquity, one by Nichomachus ( M D’Ooge, 1938) and one by Pappus (T L Heath, 1981); both specified ten possible means missing one from the eleven means that are possible in this context. Nowadays the only means that have survived are the arithmetic mean, the geometric mean, the harmonic mean, and the division in extreme and mean ratio, known after renaissance as the golden section. For a thorough discussion on the underlying mathematics of the theory of means and its significance in neoplatonic thought, see (T L Heath, 1981), and for excellent commentary on the allegedly ubiquitous golden section, see (L March, 2001). All eleven means are illustrated in a series of scalene parallelepipeds in figure 3 and are constructed following Pappus’ algorithm for the generation of proportional triads using the smallest solution of the algorithm.

a. b. c. d. e. f. g. h. k. l. m.1:2:3

P1*N1

1:2:4

P2N2

2:3:6

P3N3

2:5:6

P4N4

2:4:5

P5N5

1:4:6

P6N6

1:2:3

P7*N10

3:4:6

P8-

2:3:4

P9N8

1:2:3

P10N9

2:3:4

-N7

Figure 3. 11 scalene parallelepipeds after the 11 means. The Pn series denote Pappus’ 10 means. The

Nn series denote Nicomachus’s 10 means. The shapes with (*) are not given by Pappus’ algorithm. A nice modern example of this neoplatonic tradition in design pedagogy is the design of the Froebel Building gifts, an indispensable part of the kindergarten method. This method has enjoyed much publicity in architecture circles because of its association with the formative years of F.L. Wright, and recently it has been extensively revisited after its formalization as a formal system to teach basic principles of algorithmic spatial composition (Stiny, 1981, Knight, 1992, March, 1995, Economou, 1999). The greatest tribute of Froebel to the tradition he was coming from is found in the dissections of the cubes of the Building Gifts 3 and 6, the two building gifts that defined the first and the last of the building blocks of his pedagogical method. The cube of gift 3 is dissected in eight smaller cubes and the cube of the Gift 6 is dissected in three types of solids, namely squares with proportion 1:4:4, pillars with proportion 1:1:4, and oblongs with proportion 1:2:4. The individual solid forms resulting from the dissection of the cube are shown in figure 4.

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Figure 4. The dissection of Gift 6 in terms of pillars, squares and oblongs; alternatively to shapes corresponding to beam, tile and scalene numbers.

The geometry of C2C2C2 In two-dimensional space there are only two algebraic structures that capture the structure of shapes with a center of symmetry; these two algebraic structures are the cyclic group Cn and the dihedral group Dn and they nicely correspond with two geometrical structures characterized by rotational and rotational-reflectional symmetries. In three-dimensional space, the two infinite types of algebraic structures become four and the corresponding geometrical structures they characterize become seven. In this space the cyclic groups Cn are realized by two geometric instantiations and the dihedral groups Dn are realized by three geometric instantiations. One of the two new types of algebraic structures that capture the structure of shapes with a center of symmetry in three-dimensional space is a structure composed by the combination of three smaller cyclic groups all ubiquitous in two-dimensional space. This structure corresponds to the structure of the rectangular prism and provides a wonderful setting for studies on how bigger groups can be generated from smaller ones as well as how complicated groups can be described in terms of smaller familiar groups. The rectangular prism is the first shape in three-dimensional space that requires three generators to be constructed; these generators correspond to three reflections about mutually perpendicular mirror planes through the centroid of the shape. Each of the reflections may be considered as the generator of a small cyclic group C2 of order two. In general, for G, H groups, the set G·H with the operation (g1,h1)·(g2,h2) = (g1g2,h1h2) is a group and is called the direct product group of the groups G, H. The emergent structure of the rectangular prism is then given by the abstract direct product group C2C2C2, a group that is created by the combination of three smaller groups cyclic groups C2 of order two each (M Armstrong, 1988). The set of eight isometries that leave the rectangular prism invariant are the identity e, three half-turns, h1, h2, h3, through the three facial rotation axes, N1, N2, N3, three reflections, m1, m2, m3, through the mid-facial planes, M1, M2, M3, and one inversion, z, through the centroid of the shape, O. These isometries form a group G, called the group of the rectangular prism. A catalogue of the symmetries of the group can be given pictorially with the representation of all elements of the group as the geometric loci of points that remain invariant under the repeated application of a specific symmetry transformation. In the former case symmetry elements are the isometries that leave the shape invariant. In the latter case symmetries are the geometric loci such as points, lines and planes, of the rectangular prism that remain invariant. An invariant point is the centroid of the shape, an invariant line is an axis of n-fold rotations, and an invariant plane is a mirror plane bisecting opposite faces or connecting opposite edges. A pictorial representation of the symmetries of the rectangular prism is given in figure 5.

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π/2

π/2

π/2

h1

h2

h3

m1

m2 m3

z

Figure 5. Pictorial representation of the symmetry elements of the rectangular prism: geometrical

loci of invariant points Still another illustration of the complete catalogue of the symmetries of the group can be given pictorially with a series of spatial labels that map the transformations that leave the structure invariant. These labels can be points, lines, planes, or solids and they can be combined with the shape under consideration to reduce or not its symmetry; typically spatial labels associated with the front or the back face of a shape are denoted as filled or outlined shapes respectively. Four alternative settings of labels are illustrated in figure 6.

a. b. c. d.

Figure 6. Pictorial representation of the symmetry elements of the rectangular prism: Spatial labels

a. points b. lines c. planes d. solids This method of pictorial representation of symmetry elements using spatial labels such as points, lines, planes or solids, can be further generalized the systems of spatial labels embedded in shapes with infinite order of symmetry; in this case the arrangements of labels reduce the infinite symmetry of the underlying structure to the symmetry of the arrangement of the labels. All prismatic groups can be pictorially represented upon the orthographic projection of a cylinder upwards; the inclusion of polyhedral groups requires the projection of a sphere. Four alternative settings of labels for the group of the rectangular prism are illustrated in figure 7.

a. b. c. d.

Figure 7. Pictorial representation of the symmetry elements of the rectangular prism: Orthographic projection of a sphere upwards. Spatial labels associated with the front or the back face of the sphere are denoted as filled or outlined shapes respectively: a. points b. lines c. planes d. solids The eight elements of the group and the corresponding labels yield 28 =256 possible subsets that all have a variety of different properties. Still some of these possible designs can be mapped one upon the other via a symmetry operation. The identification of all non-equivalent configurations of these subsets can be computed using Polya's theory of counting non-equivalent configurations (Polya, 1983). This theorem specifies the numbers

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of different ways k objects can be assigned to n vertices of an n-cornered figure without considering any two arrangements as different if they can be transformed one to another by a symmetry operation. A key element in the computation is the identification of all products of cycles of permutations under the elements of the symmetry group. All combinations or products of cycles of permutations under the elements of the symmetry group of the rectangular prism are shown in figure 8.

π π/2π/2

f24

c. d.

f18 f2

4

a. b.f2

4

Figure 8. Cycle of permutations under the symmetry elements of the rectangular prism

The sum of all combinations or products of cycles of permutations under the elements of the symmetry group divided to the total number of the elements in the group is the cycle index of the corresponding permutation group; the cycle index is used in a straightforward way to identify all possible non-equivalent configurations of parts permutated upon the structure. For detailed computations and recent applications of this theorem in spatial systems and sound structures see (Economou, 1998, 1999); the exact computation of all structural configurations of the eight parts of the rectangular prism is left to the interested reader. All possible non-equivalent configurations are shown in figure 9. There are altogether 46 non-equivalent designs that comprise the set: 1 8x-label, 1 7x-label, 7 6x-label, 7 5x-label, 14 4x-label, 7 3x-label, 7 2x-label, 1 1x-label, and 1 0x-label.

5+3

8+0 7+1 6+2

4+4

3+5

2+6 1+7

*

* * * * * * * *

*

* * * * * *

Figure 9. Non-equivalent subsets of the symmetry of the rectangular prism. The configuration (0+8), the empty shape, is not shown.

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The 46 non-equivalent designs based on the structure of the rectangular prism exhaust all possible non-equivalent configurations of the symmetry group of the rectangular prism. Still some of the subsets are qualitatively very different from others because they satisfy the group axioms and therefore capture an important property of the whole structure. These subsets are noted with an (*) in figure 9. The best tool for a first scanning of possible symmetry subgroups is given by the Lagrange's theorem that states that if H is a subgroup of a group G, and if the order of G is n, then the order m of H is a factor of n. In other words in the case of the rectangular prism, only subsets consisting of 1, 2, 4, and 8 elements could potentially form subgroups; all other subsets consisting of 3, 5, 6 and 7 elements cannot. Other scanning techniques involve existences of identities and inverses in each subset. Alternatively each subset of the group can be isolated to generate subgroups called the subgroups generated by the subset. For example given a group G and an element x of G, the set of all powers of x is a subgroup of G and is called the subgroup generated by x and is written as <x>; similarly, given a group G and two elements x, y in a subset H of G, the set of all powers of x and y and their combinations is a subgroup of G and is called the subgroup generated by H and is written as <H>. The idea of a group generated by one or two elements may be extended for any number of generators (I Grossman, W Magnus, 1964). Among the 512 possible subsets of the isometries of the rectangular prism, and the 46 possible non-equivalent subsets of these isometries, only 16 qualify as subgroups of the rectangular prism. And still this is the richest structure among all other decompositions of symmetry structures of order 8. The 16 possible subgroups of the rectangular prism are illustrated in figure 10 in a lattice diagram that shows all the possible connections of all parts to one another. The four rows of the lattice starting denote nested symmetry groups with an order of symmetry of 8, 4, 2 and 1 respectively. This lattice furthermore shows that the subgroups of the rectangular prism form a partial ordered set or poset because the part relation that orders the subgroups in the set cannot be established for all pairs of subgroups.

C2C2C2

C2C2 C2C2 C2C2 C2C2 C2C2 C2C2 C2C2

C2 C2 C2 C2 C2 C2 C2

C1

Figure 10. The 16 subgroups of the rectangular prism

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The structure of the rectangular prism can be still further scrutinized through the examination of the type of isometries that leave the structure of the rectangular prism invariant. The tool that permits this type of decomposition is the conjugacy relation, essentially an equivalence relation that partitions the sets of symmetry elements of a group into equivalent classes of isometries that are characterized by the same type, i.e., they impose the same type of transformation or rearrangement within a spatial structure. Formally, given elements x, y, of a group G, x is conjugate to y if g-1xg = y for some g∈G. The equivalence classes are called conjugacy classes and the elements within the same class must have the same order. The conjugacy class of an element x in G is found my calculating g-1xg for every g∈G. Similarly the conjugacy class of a power of x, say xm, is found by calculating g-1xmg for every g∈G. The 8 symmetries of the rectangular prism split naturally into 4 classes of isometries: a rotation consisting of no motion at all, a half-turn equivalent to any if the three half-turns about the three face axes of the rectangular prism, a reflection equivalent to any of the reflections about the three mirror planes, and the inversion about the centroid. The 16 subgroups of the rectangular prism are reduced to 8 conjugate classes that capture the conjugacy structure of the rectangular prism. And still these 4 types of symmetries can be further reduced to only 2 in Euclidean space for all half-turns, reflections, and the inversion produce a permutation of order 2 in the structure; in this later case the 8 subgroups are still further reduced to 4. All three versions of the lattice of the rectangular prism are shown in figure 11.

C2C2C2C2C2C2C2C2C2

C2C2C2C2 C2C2 C2C2 C2C2C2C2C2C2 C2C2 C2C2C2C2 C2C2

C2C2 C2 C2 C2C2C2 C2 C2 C2C2

C1C1 C1

Figure 11. Three settings for the partial order of the rectangular prism The sound of C2C2C2 The structure of the rectangular prism provides a wonderful setting for music notation because it captures transformations routinely used in music composition; for a convenient summary of modern compositional processes, transformations and notational devices see (Brindle, 1975). For simplicity only symmetry transformations of an initial motif will be discussed here. The one trivial transformation that leaves the statement of a music phrase completely unaltered along with three more possible symmetric transformations of a music idea, that is, its melodic mirror, retrograde and retrograde inversion exhaust all four possible symmetric variations of the musical shape. These transformations in pictorial representations in transformational geometry are equivalent respectively to the identity, horizontal mirror, vertical mirror and half-turn, a collection of four isometries that comprise the symmetry group C2C2 of the rectangle. All motifs in a musical setting may have some

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symmetry or not and these symmetries are completely described by some of the symmetry subgroups of the rectangle. Often these four symmetry transformations increase to eight when the theme is augmented with dynamics and in traditional music notation these transformations are depicted in four versions in two alternate settings characterized by symbols such as < or >. The structure of the rectangular prism could provide a nice visual representation of this complex setting. Briefly, in traditional western notation a sound is specified by the four attributes of duration, frequency, amplitude and spectrum. These attributes are translated in the perceptual domain and in music notation alike in time values, pitch content, dynamics and timber. Traditional western notation uses discrete symbols plotted on a graph whose x dimension denotes time and the y dimension denotes pitch content. The position of the symbol within the graph, in musical terms the stuff, denotes the pitch content of a sound. The time value is specified by variations of these symbols denoting quavers, semiquavers and other time values. The dynamics is specified by symbols such as pp, p, mf, f, ff or < and > above or below the stuff. The timber is often specified explicitly in the beginning of the score with the assignment of the instrument. A three-dimensional version of this system of notation is readily suggested. The dynamics, literally and metaphorically the third dimension of sound, can be described in the z-coordinate of a three-dimensional graph whose xy-plane is the same two-dimensional graph as before. The four isometric transformations of a thematic idea are increased to eight when they combine with dynamics to create the group C2C2C2, the symmetry group of the rectangular prism. The eight possible symmetric transformations of a motif are shown in Figure 12.

a. b. c. d.

e. f. g. h.

p

p

f

f pf pf pf

p f p f p f

Figure 12. The 8 possible isometric transformations of a musical shape in three-dimensional space.

a. motif b. melodic mirror c. retrograde d. mirror retrograde e. motif with opposite dynamics f. melodic mirror with opposite dynamics g. retrograde with opposite dynamics h. mirror retrograde

with opposite dynamics. The shape of C2C2C2 The structure of C2C2C2 provides a wonderful setting for systematic studies in analysis and synthesis of form. In both modes of inquiry designs are described and interpreted in terms of their compliance with the underlying schema of the C2C2C2 structure and its various manifestations in different conjugacy settings. Parts of existing designs, or parts of novel designs are correlated to the various lattices including the 512, 60, 16, 8, and 4 subsets.

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There is no need for parts of an existing design to comply with all established parts and of course there is no need for a novel design to utilize all parts. The structure is acting as a guideline for design action to deduce or infer an inner logic to the design. Moreover the apparent complexity of an existing design or the purposeful complexity of a novel design is controlled through the use of this underlying structure. And resulting designs are all variations of the shape of C2C2C2 providing a very controlled method for composition. A series of design exercises have been constructed for both purposes and they are given in a sequence at the classes on computational design at the College of Architecture at Georgia Institute of Technology. A sample of this approach in the construction of novel designs from scratch is illustrated in figure 13. All spatial relations instantiated in this study are constructed from scratch.

Figure 13.The lattice of a simple spatial composition based on the underlying schema of the C2C2C2

structure (Laura McLeod)

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Discussion Formal composition in spatial design typically relies in sets of vocabularies of shapes and sets of spatial relations between these shapes that specify how these shapes are to be combined; alternatively, formal composition often relies in sets of vocabularies of shapes and sets of underlying diagrammatic schemata that show how these shapes are transformed. In modern and post-modern spatial composition there is a definitive shift towards composition with three-dimensional elements and direct design manipulation of three-dimensional forms and models that describe some the properties of the design they stand for. Still, while elements in this mode of composition are quintessentially three-dimensional such as blocks and solids, rectangular or deformed in some way, the transformations and the ways that combine and condition this mode of composition typically are often two-dimensional. Here a first look was presented for one of the most paramount three-dimensional structures to provide the framework for a better understanding of the rudiments of formal composition in three-dimensional design spaces and reconnect some of the bridges that once tied those spaces in between arithmetic, geometry and music. References Armstrong, M. A.: 1988, Groups and Symmetry, Springer-Verlag, New York. Brindle, R. S.: 1975, The New Music: The avant-garde since 1945, Oxford University

Press, Oxford D’Ooge, M. L.: 1938, Nichomachus of Gerasa: Introduction to Arithmetic, University of

Michigan, Ann Arbor Economou, A.: 1999, The symmetry lessons from Froebel building gifts, Environment and

Planning B: Planning and Design 26: 75-90 Economou, A.: 1998, The symmetry of the equal temperament scale, in Javier Barrallo

(ed), Mathematics and Design 98: Proceedings of the Second International Conference", ed., University of the Basque Country, San Sebastian, Spain, pp. 557-566

Euclid [300AD], 1956, The Elements, (tr) Sir Thomas Heath, Dover Publication Inc, N Y Heath, T. L.: 1981, History of Greek Mathematics, Dover Publication Inc, New York Grossman, I., and W. Magnus: 1964, Groups and their Graphs, Random House Inc., New

York Knight, T. W.: 1992, Shape grammars and color grammars in design, Environment and

Planning B: Planning and Design 20: 705-735 March, L.: 1995, Sources of characteristics spatial relations in Frank Lloyd Wright's

decorative designs. In The Phoenix Papers, Volume 2, Natural Pattern of Structure, (ed) L Johnson, The Herberger Center for Design Excellence, Arizona Press, pp 80-117.

March, L.: 1998, Architectonics of Humanism: Essays on Number in Architecture, Academy Editions, London.

March, L.: 2001, Exit d’Or, in John Peponis, Jean Wineman, Sonit Bafna (eds), Proceedings of Space Syntax, 3rd International Symposium, University of Michigan, Ann Arbor, pp. 8.1-8.12.

Polya G., Tarzan R. E., Woods D. R.: 1983, Notes on Introductory Combinatorics, Birkhauser, Boston

Stiny, G.: 1981, Kindergarten grammars: designing with Froebel's building gifts, Environment and Planning B: Planning and Design 7: 409-462.