Shape Signature

Shapes as collections of maximal lines upon construction lines carrier lines

Registration points mark the intersections of two or more construction lines and produce regions in each construction line a No registration point--One region b One registration point--3 regions in two orientations c Two registration points--5 regions

Maximal lines are bounded by points in the regions defined by the registration points The 13 ways of placing lines in a construction with 2 registration points can be ordered in a partial order set by inclusion

The shape signature of a shape is the ordered sequence of pairs of boundary elements embedded in distinct regions of the construction lines

Counting shapes--For k construction lines--l registration points and m maximal lines a 303 b 313 c 323 and d 333 The type of valency of intersection counts too

The number of ways to place one maximal line on a construction line with k registration points is the sequence of centered square numbers

The number of ways to place n maximal lines on a construction line with k registration points is given by the sequences of 2n dimensional orthoplex numbers

Counting shapes--Enumerate all maximal lines on each arrangement of construction lines There are 125 shapes for the starconfiguration There are 2197 shapes in a triangular configuration

Counting nonequivalent shapes given a configuration of construction lines using groups of symmetries--Burnsides lemma and Polyas theorem of enumeration Six symmetrical copies of the same shape

Partially ordered set of the 19 distinct shapes on a 3star arrangement after symmetry

Encoding information with the shape signature

Some shapes consisting of 4 lines among the possible 317065 4line shapes

Athanassios Economou, Josephine Yu and James Park
2019

Keywords: Lines; Matroids; Chirotopes; Polya’s theorem of Enumeration; Figurative numbers; Symmetry; Mathematica

Shape Signature provides a new and uniform specification of parametric shapes consisting of points, lines, planes and solids. The model extends the shape grammar definition of a shape consisting of maximal elements of lines, planes and solids to the specification of parametric shapes consisting of maximal elements of lines, planes and solids to ultimately ensure that a specification of a parametric shape is unique - and even more, captured by a numerical formula. The method proceeds from considering a shape consisting of a collection of maximal lines upon an arrangement of hyperplanes to calculate and visually represent any parametric shape consisting of lines in CAD parametric models.

A key idea in the shape signature is the indefinite number of parts in a maximal element. This number is only defined with respect to the context where a maximal element is. For example, a line when it is in no spatial relation with some other line, it has 1 unlimited number of parts; the same line may have up to 5 parts when it is in a spatial relation with another line; 13 parts with 2 lines; 41 parts with 4 lines; and so forth, following the sequence of centered square figurate numbers and their corresponding powers in higher dimensions for more elaborate combinations of number of parts per line.

The very same parts combine one with another to create a visually staggering world rising from the singularity of the 1-line shape, to the 8 2-line shapes, to the 519 3-line shapes (among which there is the triangle and its parametric variations) and to the 317,065 4-line shapes (among which there is the square and its parametric variations). Checking these numbers, it is easy to see that any parametric shape consisting of 4 lines or strokes in the plane is an instance of the 317.593 possible 4-line shapes. The numbers for larger sets of lines increase dramatically to further the key insight of the formalism on a unique description of any shape out there.

The algebraic calculations of figure inventories for up to 4-lines so far are done in Mathematica. The calculation of the chirotopes for the underlying matroids are done by hand. The specification of the shapes is modeled in Grasshopper and visualized in Rhino. The project has been carried through with substantial help from Cvetelina Hill, May Cai and Nicholas Liao, doctorate and undergraduate students, respectively, at the School of Mathematics, College of Sciences at Georgia Tech.

Threeness

Pictorial catalogue of the 390 3line shapes 1-35

Pictorial catalogue of the 390 3line shapes 36-70

Pictorial catalogue of the 390 3line shapes 71-105

Pictorial catalogue of the 390 3line shapes 106-140

Pictorial catalogue of the 390 3line shapes 141-175

Pictorial catalogue of the 390 3line shapes 176-210

Athanassios Economou, Josephine Yu and James Park
2018

Keywords: Lines; Shape grammars; Burnside’ s lemma; Polya’s theorem of counting

We all can recognize simple shapes and relations effortlessly, for example, triangles of all sorts, no matter their specific geometric characteristics, be them scalene, equilateral, isosceles, Pythagorean, 30–60–90, or what-have-you. After all, all triangles consist of three lines (edges) connected in three points (vertices). Still, if someone asks the question about how many shapes can be made from three straight line segments that may be connected to one another or not, intersecting one another or not, floating to another or not, the answer is not straightforward. An unambiguous way to classify, retrieve and instantiate 3–, 4–, 5–, 6– and more generally, n-line arrangements to construct shapes will potentially have a great impact in computer shape recognition and CAD applications. The complete calculation of all possible parametric shapes made of 3 lines is given below without counting parallelism and/or intersections of valency greater than 2. All 390 shapes are unique, or otherwise, non-equivalent with respect to the symmetry group of the triangle, that is, there is no symmetry transformation of the Euclidean plane that can map one configuration to another.

Subsymmetry Lattices

Catalogue of the symmetry lattices of the dihedral groups D01-D10

Catalogue of the symmetry lattices of the dihedral groups D10-D20

Catalogue of the symmetry lattices of the dihedral groups D20-D30

Athanassios Economou and Thomas Grasl
2009

Keywords: Polya’s theorem of counting; Symmetry; Point groups; Graphs; Fruchterman-Reingold algorithm; Rhythm

One of the most interesting aspects of symmetry theory and pattern analysis and composition is the part relation (<) of symmetry groups; the elaborate and complex hierarchies of symmetry groups and subgroups, all nested one within the other, point to their direct correspondence with complex compositional structures of spatial patterns and suggest a precise methodology for formal analysis and composition in architectural design. This work looks closely at a specific set of symmetry groups - the two infinite types of the planar groups, the cyclic and the dihedral ones - and provides an automated environment to enumerate and represent the subgroups and their relationships with lattices in a graph theoretic manner. The complexity of these structures can be astonishing, and it is suggested here that its graph theoretical representation can contribute to a better understanding of problems of spatial complexity in architectural design. The computational approach outlined in this work may indeed be used in formal analysis to identify all spatial repetitions and spatial correspondences that can be observed in a design. Alternatively, the approach may be used in formal composition to structure the design choices and bring to the foreground the whole range of spatial relationships available to the designer at any level of the design inquiry.

The computation and manual illustration of all possible subgroups for a given group n is not a trivial task; 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 h is a power of a prime and divides the order of a group, then the group has a subgroup of order h. Here the automation relies to 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 in this project, algorithmic simplicity has been chosen over efficiency. Once the primes are extracted all possible distinct products are computed and tested to produce the possible lists of factors and the corresponding cyclic and dihedral subgroups. The generation of the graphs is then a straightforward task of iterating through the factors and generating the n/f 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 and iterating over the nodes. The pictorial representation of the graphs is given in two ways: The first representation shows all possible subgroups and maps all possible relationships between them; this representation offers the most detailed view in the inner structure of any design and illustrates nicely the stunning complexity found even in the simplest of structures. The second representation foregrounds the qualitative difference between rotational and reflective symmetries and classifies all subgroups and their relations in terms of two distinct classes, the cyclic and the dihedral groups respectively; this representation relies on a specific construct from group theory, the construal of conjugacy or equivalence relationships for elements of the group and essentially produces graphs where all identical instances of a dihedral group of some order n are collapsed to a single node; this preserves the relations between the various subgroups and produces a leaner graph. The application uses the Fruchterman-Reingold algorithm for the collapsed representation of the graphs.

Rod Symmetry

Instances of the the four nfold point groups--for n4--from which the seven symmorphic groups are derived

The four types of linear spaces based on the class na--for n 6 The fourth type is the enantiomorphic version of the third given above

The single type of a linear space based on the class 2na--for n 3

The two types of linear spaces based on the class nma--for n 6

The four types of linear spaces based on the class n2a The fourth type is the enantiomorphic version of the third given above

The two types of linear spaces based on the class 2nma for n 3

The three types of linear spaces based on the class nmma

Athanassios Economou
2006

Keywords: Shape studies; Symmetry; Linear Growth; Symmorphic groups

The project looks closely at a specific class of three-dimensional designs that have one axis of growth and presents all possible algebraic structures that capture the symmetries of these designs. A specific set of seven types of designs is discussed, the symmorphic designs, and is used as a framework to derive the non-symmorphic designs and to complete all three-dimensional linear structures. The complete catalogue of all nineteen space structures that may be generated in this manner is presented for n-fold screw rotations, n <=6.

Frit: An Enumeration of n^2-Cell Structures, for n≤4

Catalogue of all possible 102 bichromatic configurations of a 3x3 square grid

Catalogue of bichromatic configurations of a 4x4square grid--a1 configuration 016 3 configurations 115 21 configurations 214 77 configurations 313

Catalogue of 252 bichromatic configurations 412 of a 4x4square grid 1-128

Catalogue of 252 bichromatic configurations 412 of a 4x4square grid 129-252

Catalogue of 567 bichromatic configurations 511 of a 4x4square grid 1-128

Catalogue of 567 bichromatic configurations 511 of a 4x4square grid 129-256

Catalogue of 567 bichromatic configurations 511 of a 4x4square grid 256-384

Catalogue of 567 bichromatic configurations 511 of a 4x4square grid 384-512

09 Catalogue of 1051 bichromatic configurations 610 of a 4x4square grid 1-128

Catalogue of 1051 bichromatic configurations 610 of a 4x4square grid 128-256

Athanassios Economou and Thomas Grasl
2006

Keywords: Enumeration; Symmetry; Configuration; Polya’s Theorem of Counting; Recursion

A class of recursive patterns that satisfy given area coverage constraints and given patterns of distribution is specified here. The elementary models of these patterns involve four properties: a) topology, b) area, c) scale, and d) configuration. A pattern may appear to be nuclear or linear with respect to the basic topology of the repeated element in the configuration. Similarly, these same patterns may be given in the complimentary form where the figure-ground relationship between shape and background is reversed. The same topological patterns may be used for different amounts of coverages. Similarly, the same medium-dispersed nuclear patterns may be used in their corresponding linear form for different area coverages. A pattern may appear to be concentrated or dispersed with respect to the number of parts that are taken to constitute the pattern. The patterns are assumed to be continuous and isotropic and there are three geometric arrangements – triangular, rectangular, or rhombic, and hexagonal that can satisfy these constraints.

Innercube

Lattice of the symmetry group of the cube

Catalogue of the subsymmetry groups 1-15 out of the 98 subsymmetry groups of the cube

Catalogue of the subsymmetry groups 16-30 out of the 98 subsymmetry groups of the cube

Catalogue of the subsymmetry groups 31-45 out of the 98 subsymmetry groups of the cube

Catalogue of the subsymmetry groups 46-60 out of the 98 subsymmetry groups of the cube

Catalogue of the subsymmetry groups 61-75 out of the 98 subsymmetry groups of the cube

Athanassios Economou
2006

Keywords: Symmetry; Point groups; Kindergarten method

The structure of the Froebel gifts can be explored in terms of the isometries that leave them invariant. This classification is similar to the classifications employed for the partition , say, of the Euclidean group, E, consisting of all isometries, to the ten groups Eij, or the partition of the three dimensional point group , E03, consisting of all rotations, reflections and rotor reflections about a point 0, in the seven polyhedral groups and the seven infinite sets of point symmetry ; the difference is that while those classifications are built upon classes of equivalent isometries in the Euclidean space, i.e. all reflections or all n-fold rotations are of the same type respectively, this classification distinguishes between these symmetries, so far considered typologically the same, in respects to the effect that they impose on each individual spatial structure . In these corresponding spaces, neither all reflections are the same, nor all two-fold rotations are of the same type. In the cube, for example, a reflection in a face plane interchanges all three pairs of faces while a reflection in an edge plane interchanges two faces and leaves two faces still. Both isometries permute the faces in a different way and thus they are classified as different. Informally, two isometries are equivalent or conjugate when they impose the same kind of rearrangement in the structure. Formally, given elements x, y of a group G, x is conjugate toy if gxg-1 = y for some g E G. The set of all conjugates of x, {gxg-1 19 E G} is called the conjugacy class of x in G and all the elements in the same class have the same order.

Within this context the forty-eight elements of the symmetry group of the cube naturally split in ten conjugacy classes (Yale, 1988). The conjugacy classes of the symmetries of the other three Froebel blocks are subsets of these ten classes. The sixteen elements of the pillar are partitioned in eight classes, the eight elements of the oblong in four classes and the four elements of the half-cube in three classes. The explicit descriptions of these types of symmetries and their order for the four Froebel blocks are given in Table 3. These conjugate classes of symmetries combine to create conjugate groups within the symmetry groups of each individual solid. The conjugate subgroups show essentially the different types of symmetries within each block. There are thirty-three conjugate classes in the octahedral group and it can easily deduce that the symmetry groups of the pillar, the oblong and the half-cube can be further partitioned in twenty-three, eight, and four conjugate classes respectively. The symmetry groups can be even further scrutinized in terms of all the distinct subgroups that are formed out of all possible combinations of the elements of the group.

The structure of the cube, the octahedral group, has ninety-eight subgroups which all together form the thirty-three conjugacy classes of the cube. All the subgroups of the other Froebel gifts are contained within these ninety-eight groups. It can be easily checked that the symmetry group of the pillar contains thirty-five subgroups, the symmetry group of the oblong contains sixteen subgroups, and the symmetry group of the half-cube contains four subgroups. The complete number of all conjugate subgroups for the cube is given here for a spatial relation between a cube and an oblong.

Cube Orbits

The 48 cycles of permutations of the vertices of the cube induced by symmetries of the cube 1-12

The 48 cycles of permutations of the vertices of the cube induced by symmetries of the cube 13-24

The 48 cycles of permutations of the vertices of the cube induced by symmetries of the cube 25-36

The 48 cycles of permutations of the vertices of the cube induced by symmetries of the cube 37-48

The 48 cycles of permutations of the faces of the cube induced by symmetries of the cube 1-12

The 48 cycles of permutations of the faces of the cube induced by symmetries of the cube 13-24

The 48 cycles of permutations of the faces of the cube induced by symmetries of the cube 25-36

The 48 cycles of permutations of the faces of the cube induced by symmetries of the cube 37-48

The 48 cycles of permutations of the edges of the cube induced by symmetries of the cube 1-12

The 48 cycles of permutations of the edges of the cube induced by symmetries of the cube 13-24

The 48 cycles of permutations of the edges of the cube induced by symmetries of the cube 25-36

The 48 cycles of permutations of the edges of the cube induced by symmetries of the cube 37-48

Jerry Hsu and Athanassios Economou
2005

President’s Undergraduate Research Award (PURA) Award, Georgia Tech

Keywords: Cube; Hexahedral group; Permutation groups; Cycle index; Symmetry; Configuration

The cube has always been one of the most ubiquitous shapes in three-dimensional spatial design. Aspects of the mathematical structure of the cube have been explored with different tools drawn from Euclidean geometry, descriptive geometry, group theory, permutations and combinatorics. Here one of the most interesting visual representations of the structure of the cube is given in terms of the cycle index of the permutation groups of the vertices, edges and faces of the shape under the symmetry group of the cube. All permutations are given in a consistent projection depicting the symmetry transformation that induces them as the geometrical loci of points that remain invariant under the transformation (typically axes of rotation and rotor reflection, and planes of reflections) and the cycles of permutations of the vertices, edges and faces as closed rings circumscribing them.

Four Eights

Four shapes with equal order of symmetry a Shaved octagonal truncated pyramid b Isosceles tetrahedron c Shaved square prism d Rectangular prism

The octagonal prism as the underlying geometric structure of the four shapes of the previous figure a Shaved octagonal truncated pyramid b Isosceles tetrahedron c Shaved square prism d Rectangular prism

Pictorial representations of the symmetry elements of four shapes a Shaved octagonal truncated pyramid b Isosceles tetrahedron c Shaved square prism d Rectangular prism

Diagrammatic representations of the four algebraic structures of order 8 a Shaved octagonal truncated pyramid b Isosceles tetrahedron c Shaved square prism d Rectangular prism The corresponding 7 symmetry classes of order 8 are shown as well

Lattices of subgroups of four shapes with symmetry 8 a Shaved octagonal truncated pyramid b Isosceles tetrahedron c Shaved square prism d Rectangular prism

Some shapes rules instantiating 3d shapes to substitute the labels in the partial order lattices of order 8

Partial order set of parts used for the generation of a design in one of the lattices of order 8

Axonometric views of a design in the language

Athanassios Economou
2002

Keywords: Symmetry; Prismatic groups; Configurations; Shape grammars

A constructive program for the generation of three-dimensional languages of designs based on nested group structures is outlined using the four possible algebraic infinite symmetry structures of 3-dimensional space, that is, the cyclic and dihedral groups and their direct product groups, and their corresponding seven prismatic groups in Euclidean space. The pedagogical material and the studies themselves focus on a specific order, here the order 8, to showcase the expressiveness of these structures in design.