Unraveling Complexity
A Computational Approach for the Generation of All Underlying
Structures of Three-Dimensional Shapes with an n-Fold Symmetry
Axis
Athanassios Economou and Thomas Grasl
Georgia Institute of Technology, USA
A computational approach for the generation of all underlying structures of threedimensional shapes with an n-fold symmetry axis is briefly discussed and an
automated environment for the complete generation of all partial lattices of threedimensional shapes with an n-fold symmetry axis is presented in the end.
Introduction
Contemporary architecture discourse is driven by extensive research on
issues of three-dimensional patterns, space packing, non-periodic threedimensional tilings, parametric space modules, three-dimensional Voronoi
tessellations and so on. This is not an accident; three-dimensional spatial
vocabularies and transformations have already been in the center of design
inquiry in twentieth century architectural discourse and the recent emphasis on CAAD related three-dimensional descriptions of architectural form
could only foreground this trajectory even more. Still, an inherent difficulty in the systematic exploration of all these classes of designs and configurations lies in the fact that the representations that the architects use to
explore these spaces have the same dimensionality of the space they intend
to explore; in other words, there is a lack of a vantage point from which an
architect can see and control the whole spatial composition. What exacerbates the problem even more is the lack of a body of formal knowledge
that could effectively help architects to explore systematically the possibilities afforded in a given setting and specifically within the threedimensional design space.
J.S. Gero and A.K. Goel (eds.), Design Computing and Cognition ’08,
© Springer Science + Business Media B.V. 2008
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Among various competing formal methods that are typically employed
to address these problems, group theory plays a prominent role. Classic
expositions of group theory and their application in visual arts can be
found in the work by Weyl [1], and Shubnikov and Kotpsik [2] and various
applications in architecture especially for two-dimensional designs can be
found in March and Steadman [3]. Extensions of these methods for point
and linear designs in three-dimensional space have been given by Economou [4]. Some first steps towards the extension of the tools of group theory to explore asymmetry –or in fact complexity, if complexity can be associated with the lack of symmetry– have been taken by March [5] and
Park [6]. A computational environment for the complete enumeration of all
nested symmetries in planar designs has been given by Economou and
Grasl [7].
The work here looks at a specific class of designs in three-dimensional
space, namely the three-dimensional designs with an n-fold symmetry axis,
and provides a computational approach to a) enumerate all their repeated
parts; b) depict their relationships in a graph theoretical manner; and c) illustrate all shape correspondences with pictorial visualizations for each
distinct class of designs. The specific sets of symmetry groups that are examined here are the four infinite types of the point space groups, namely,
the cyclic groups, the dihedral groups and their direct product groups with
a cyclic group of order two. These four types of groups can capture the
symmetry of any three-dimensional shape or design with an axis of symmetry of an order n. The complexity of these structures can be staggering
and it is suggested here that their graph theoretical representation and pictorial representation can contribute to a better understanding of problems
of spatial complexity in architectural design. The paper outlines the computational approach for the generation of all partial order lattices of these
shapes and illustrates some of these ideas with consistent mappings of
these lattices to a language of diagrams to visualize their part to whole
relationships.
Parts and wholes
In three-dimensional space the symmetries of any finite three-dimensional
shape can be captured by any of the fourteen possible types of symmetry
groups [8]. These fourteen types of symmetry groups split into two types:
seven finite types and seven infinite types. The symmetries of the seven finite classes of symmetry are described by the seven polyhedral groups that
provide the symmetries of the platonic solids and their variations. The
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symmetries of the seven infinite classes of symmetry are described by the
seven infinite types prismatic groups that provide the symmetries of any
solid that has a primary axis of n-fold rotation and does not have a greater
than a two-fold rotation axis perpendicular to this major axis. Both types of
these geometric symmetries, the finite and the infinite types, can be succinctly described algebraically by abstract groups whose elements are not
described in some concrete way [9]. More specifically, the seven finite
geometric types can be captured by six abstract groups, and the seven infinite geometric types can be captured by four infinite abstract groups [10].
These last four structures, namely, the abstract cyclic group Cn of order n,
the abstract dihedral group Dn of order 2n, and their direct products with
the abstract cyclic group C2, that is, the group CnC2 of order 2n and the
group DnC2 of order 4n, comprise the object of study here and provide the
corpus of configurations to be systematically explored.
The key idea explored in this work is the part relation (≤) as it is manifested in group theory and exemplified in spatial composition. Briefly a set
G together with a rule on G becomes a group when it satisfies three axioms: a) The composition of the elements in G is associative, that is to say,
(xy)z = x(yz) for any three, not necessarily distinct elements in G; b) There
is an element e in G, called an identity element, such that xe = x = ex for
every x G; and c) Each element in G has an inverse x–1 which belongs to
the set G and satisfies x–1x = e = xx–1. Formally, for any group G with an
operation (.), a subset H of G is a subgroup if H forms a group under the
operation (.). Not every subset of G is a subgroup; for example any subset
of G that does not contain the identity element is bound not to be a group
because it does not satisfy the group identity axiom. Still, subgroups are
extraordinarily common; every group G is a subgroup of itself, and the
identity element e is also a subgroup because it satisfies all group axioms.
The enumeration of all possible subgroups for a given group is a very difficult task and has been carried through only for selected few finite groups.
The corresponding task of enumeration and illustration of all cyclic and
dihedral groups for a given group n is straightforward but the computation
is not trivial.
The set of the symmetry subgroups of a particular symmetry group can
be further sorted by a relation that orders all the subgroups in the set. If this
relation can be established for all pairs of elements in the set then this relation is called total or strict order and the set is called chain. For instance,
the relation “less than or equal to” is a total order on integers, that is, for
any two integers, one of them is less than or equal to the other. If this relation is defined for some, but not necessarily all, pairs of items, then the order is called partial order and the set is called a partial ordered set. For
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instance, the sets {a, b} and {a, c, d} are subsets of {a, b, c, d}, but neither is
a subset of the other. In other words, the relation “subset” is a partial order
on sets. Typically graphs are used to represent such order and show the
nested relations of the subgroups diagrammatically in lattice diagrams. The
graph representation of all symmetry subgroups of a configuration suggests a complex but rewarding insight in the symmetry structure of a spatial configuration.
Sieve
Sieve is a computational tool that has been developed to address the inquiry discussed above and to enumerate and illustrate all possible subgroups of a given three-dimensional finite shape. The tool has been built
upon an earlier version designed to compute and illustrate all possible subgroup relationships of two-dimensional shapes [7]. This earlier version of
the software, code-named, Sieve 1.0, produces all dihedral groups Dn of
order 2n and all cyclic groups Cn of order n are found embedded within
each corresponding graph. The new version of the software, code-named
here Sieve 2.0, produces analogously all possible direct product groups
DnC2 of order 4n consisting of a dihedral group Dn of order 2n with a cyclic
group C2 of order 2, and all four types of three-dimensional groups with an
axis of symmetry n, namely, a) the cyclic groups Cn of order n; b) the dihedral groups Dn of order 2n; c) the direct product groups CnC2 of order 2n,
consisting of the cyclic group Cn of order n and a cyclic group C2 of order
2; and d) the direct product group DnC2 of order 4n itself, consisting of the
dihedral group Dn of order 2n and a cyclic group C2 of order 2, are found
embedded with the computed graph.
The theory for the computation here relies on a sorting based on two
theorems proved by Lagrange and Sylow respectively: Lagrange theorem
identifies a very precise numerical relationship between subgroups and
groups, namely that the order of a subgroup always divides the order of a
group. Sylow’s theorem proposes that if a number m is a power of a prime
k and divides the order of a group n, then the group has a subgroup of order m. Here the automation relies on a routine to generate the complete list
of all prime factors for a given number n. The simplest, albeit not most efficient, algorithm to generate the primes is the sieve of Eratosthenes. Since
computing time is not a key feature for the relatively small magnitudes
dealt with in this project, algorithmic simplicity has been chosen over efficiency. Once the primes are extracted all possible distinct products are
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computed and tested to produce the possible lists of factors and the corresponding cyclic and dihedral subgroups.
The graph representation of the symmetry groups and subgroups is a
straightforward task of iterating through the factors and generating the
nodes noting each time which set of operations the node represents. The
completion of the illustration of the structure of the graph is done with the
pictorial representation of the edges of the graph deduced from each label
iterating over the nodes and arranging the nodes and the edges in a hierarchical manner for different orders of symmetry. The application was created using the Java Universal Network/Graph Framework (JUNG) [11].
All complete subgroup graphs of the direct product groups DnC2 consisting
of dihedral groups Dn with a cyclic group C2 of order 2 for order n ≤ 12,
are shown in Figure 1.
The stunning complexity found even in the simplest of structures is
overwhelming. The series is illustrated here for an order up to 12 to compare this rising complexity with that of the five polyhedral shapes – the infamous platonic solids and their derivatives. For example, the last shape in
Figure 1 has an order of symmetry 48, that is, an order of symmetry equal
with that of the cube. Still, their respective structures are quite different;
the lattice of the 48-gon prism here has 125 subgroups and the lattice of the
cube has 98 nested subgroups [12].
Sieve 2.0 produces all possible symmetry structures of threedimensional space as embedded graphs within a super graph of order 4n
for an n-fold symmetry axis. For example, a graph depicting an order of 12
in the DnC2 type will have embedded subgroups within it to exemplify all
possible four types of three-dimensional symmetry with a three-fold axis
of symmetry, namely, a) C3; D3; C3C2; and D3C2 with orders of symmetry
3, 6, 6, and 12 respectively. These nested graphs are shown in Figure 2.
The numeric relation of all four types of n-fold prismatic symmetry to
the group DnC2 is clearly defined. Any specified order of symmetry n is
represented at least by one symmetry group which has this order n and
every order of 4n is represented by all four types of symmetry structures.
Interface
The interface of Sieve 2.0 is straightforward and combines typical conventions found in graphical applications as well as specific characteristics and
functions unique for the computation of the lattices of symmetry groups.
The key characteristic of the application is that it combines two interfaces,
one for the computation of the lattice representation of the structure in-
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quired, and a second one for a pictorial representation of all the nested
symmetries corresponding to each node of the lattice. Some of the characteristics of these two interfaces are described below.
Graph output
The default output of the program has been designed to provide a consistent layout for all graphs. This layout allows for a visual inspection and
comparison and of the rising complexity of all nested subgroups. All lattices are ordered from top to bottom from higher to lesser degrees of symmetry; the top node has the maximum symmetry and the last node in the
bottom shows the identity element with a symmetry of 1. All in-between
rows pick up the different orders of symmetry for a specific shape. For example, the lattice of the symmetry of a shape with symmetry 12 will consist of nodes arranged in six horizontal lines, each picking up in a descending order the symmetries of 12, 6, 4, 3, 2, 1. Furthermore, this distribution
of all horizontal lines along the y axis of the graph shows spatially the metric difference of the nested symmetry classes found in a shape. The visual
arrangement of the graph in this way show especially nicely the relationship of odd and even factors in the graph picking up polyrhythmic relationships in the arrangement of the nodes. For example, in the lattice of the
symmetry of a shape with symmetry 12, the horizontal line that hosts the
nodes that represent the symmetry equal to 6, is 6 units below the horizontal line of 12, 2 units above 4, 3 units above 3, 4 units above 2 and 5 units
above 1. Figure 3 shows diagrammatically the distribution of horizontal
orders in the symmetry lattice of the triangular prism D3C2.
The distribution of all vertical lines along the x axis shows spatially the
qualitative difference of different types of symmetry classes found in a
shape. These groupings pick up all the rotational, reflectional, screw rotational and rotor reflectional types and their combinations and show their
interrelationships. Still subgroup relationships between the nodes cut
across these initial visual groupings. One of the most exciting characteristics of the design of the interface of the application is that the complete
partial order lattice for any node of the graph can be computed and illustrated automatically in a set of weighted lines and nodes. Any node in the
graph once triggered by a mouse-over function show visually the complete
set of subgroup below the node as a set of subgraphs that are linked from
the node to the identity element itself. Figure 4 shows diagrammatically
the distribution of subgroups within the symmetry lattice of the triangular
prism D3C2; each subgraph unfolding from any node can be highlighted.
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AQ: Please check for
alphabatical order (J is missing)
Fig. 1. The lattices of the DnC2 symmetry groups for n ≤12.
a) 4; b) 8; c) 12; d) 16; e) 20; f) 24; g) 28; h) 32; i) 36; k) 40; l) 44; m) 48
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Fig. 2. The four types of prismatic symmetry found within the structure of the triangular prism D3C2. a) C3; b) D3; c) C3C2; and d) D3C2
Fig. 3. Vertical arrangement of nodes in the graph
Fig. 4. Horizontal arrangement of nodes in the graph
A second characteristic of the design of the interface of the application
is that it allows for repositioning of the nodes while keeping the topology
of the graph invariant. All structures computed by the system can be repositioned and reshuffled in any setting and degree desired and all transformations of the graph retain the number of the nodes and links between the
nodes. A typical application of this functionality is that the graph can be
transformed in meaningful ways to foreground specific relations that are
possibly hidden in the complexity of the original output. Figure 5 shows
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the same subgraph unfolding from the node in Figure 2(d) repositioned to
clarify visually the given subgroup relation within the structure of the triangular prism D3C2.
Fig. 5. A part of the structure of the triangular prism repositioned
A nice by-product of this functionality of reshuffling is that the application allows for an immediate appreciation of topology and the aesthetic
possibilities of graph rearrangement. It is characteristic of all these lattices
that there is no known algorithm to distribute the nodes and their connections in some common aesthetic mode. Drafting graphs can be as aesthetic
and complex as design itself! Figure 6 shows a different repositioning of
the structure of the triangular prism D3C2. The repositioning here considers
the identity element in the center and all subgroups are unfolded from this
central point.
Fig. 6. The structure of the triangular prism repositioned
All structures can be inspected at any desired degree of closeness. Each
time the user zooms in or out of the graph, all spatial relationships are
scaled to simulate the closeness, but the nodes, the width of line, and the
discursive notation of the nodes are all remaining constant to allow a
common framework for illustration. The zoom is most effective when it is
performed on a node; everything around it scales to de-clutter the drawing
and show the computed part. Figure 7 shows a scaled version of a part of
the structure of the triangular prism, D3C2.
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Diagram output
The diagram interface of Sieve 2.0 provides the most immediate visual
output to the structure of any finite three-dimensional shape with one axis
of symmetry. This interface provides a pictorial representation of the
symmetry of the shape in terms of labels in the form of an open, closed and
a dashed circle that are all associated with the faces of the shapes to break
down their symmetry. The set of conventions for this representation includes: a) a set of closed, open and grey labels; b) a set of lines representing the symmetry axes of the shape; and c) a set of lines representing
Fig. 7. The structure of the triangular prism scaled
the outline of a regular n-prismatic shape. More specifically, in this notation labels associated with the front face of a shape towards the viewer are
represented as closed and labels associated with the back face of a shape
are open. A grey label is used only as a placeholder to show the comparative disposition of all possible symmetries in a configuration. All labels are
combined here with the lines of the outline of the shape to break down the
symmetry of the shape; the labels themselves being circular would be ambiguous to break down the symmetry because any axis of the whole configuration traversing through their center would render the shape as at least
possessing one reflectional symmetry. Moreover the positions of the labels
are asymmetric with respect to the length of the line – edge of the regular
polygon. Finally the symmetry axes in the notation are primarily used for
clarity of presentation and visual foreground of the reflectional parts in all
computed configurations and they are not needed for the reduction of
symmetry. Figure 8 shows the three rules of combination of lines of the
outline of the shape with the three different labels to reduce the symmetry.
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Fig. 8. Three rules of combining lines of the outline of the diagram with labels to
reduce the symmetry of the configuration
The complete diagrams of all subgroups found in the structure of the triangular prism D3C2 are shown in Figure 9. The first diagram in the upper
left corner has the complete symmetry of the triangular prism and it is
composed by twelve labels, six closed and six open, and the final diagram
to the lower right has the identity element and it consists of a single closed
label. All the other fourteen diagrams represent the total number of possible subsymmetries of the rectangular prism and they can be all automatically retrieved in Sieve 2.0 by mouse-over functions over the nodes of the
graph. Significantly, the visual computation of all diagrams nicely illustrates the equivalent classes of symmetry within each symmetry class and
shows spatially some of the groupings in the output of the graph. For example, the diagrams in Figure 9(b) and (d) are equivalent because they are
produced by a single generator and they are both described by one algebraic group, the C6. The diagrams in Figure 9(e)-(g) are all equivalent because they are all produced by two generators, rotations and reflections
through the vertices of the triangle, and are all described by the group D2.
Less intuitively, the diagrams in Figure 9(i)-(p), are all produced by two
generators and are all described by the group D1 but these generators are
different in kind here; three include reflections and rotations, three include
rotations along different axis and one includes rotor reflections and rotations.
Discussion
A major motivation behind this work is the hypothesis that any asymmetric shape or configuration can be understood as a sum or subtraction of
configurations that have discerned orders of symmetry. The key idea that
underlies this work is that asymmetry –and uniqueness and complexity and
so forth– can be understood as a sum or subtraction of several parts that in
themselves may have some symmetry but their products or spatial relations, render the design asymmetric. Needless to say, any superimposition
or any subtraction of spaces nonesoever will not immediately render inter-
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esting a space nor will say an interesting story – most probably not. What
most probably will say an interesting story is the ability to reasonably discuss the possibilities that each system allows. The compositional and organizational processes that are immediately available to the designer during
any design inquiry can be determined; whether all these will be used is irrelevant; whether they all known and are readily available is significant.
The answers to all that can be taken from group theory and particularly
from its construct of the part relation that structures the whole hierarchy of
algebraic subgroups and supergroups and its spatial counterpart in configurational possibilities.
Fig. 9. Diagrammatic representation of all subgroups embedded within the structure of the triangular prism
Here a computational approach for the generation of all partial lattices
of three-dimensional shapes with an n-fold symmetry axis has been presented to provide an interactive catalogue with the architecture of form of
all possible structures with a center of symmetry. Future work will explore
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a) the automatic generation of all possible graphs for all finite shapes in
three-dimensional space including those of the seven polyhedral shapes; b)
a design refinement of the interface to include the automatic generation of
all geometric structures in terms of any algebraic structure; and c) the design of a new interface to correlate the diagrammatic language of the existing subgroups with shape grammars rules [13] [14] to generate designs that
will be described by the given symmetry configurations.
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