Shape Computation Lab

Shape Machine

A query of a triangle in Shape Machine under similarity transformations a Query b initial shape c One of the four matches of the identity rule under similarity transformations

A twostep query of a kshape in Shape Machine under similarity transformations a Query b initial shape c One of the eight infinite types of matches of the identity rule under similarity transformations

A query of a triangle in Shape Machine under affinity transformations a Query b initial shape c One of the four matches of the identity rule under affinity transformations

A query of two Lshapes in Shape Machine under affinity transformations a Query b initial shape c One of the eight matches of the identity rule under affinity transformations

A query of an arrangement of an Lshape and a line in Shape Machine under linearity transformations a Query b initial shape c One of the 24 matches of the identity rule under linearity transformations

A query of an arrangement of arcs in Shape Machine under isometry transformations a Query b initial shape c One of the six matches of the identity rule under isometry transformations

A query of a complex arrangement of arcs in Shape Machine under similarity transformations a Query b initial shape c One of the six matches of the identity rule under similarity transformations

A construction in Shape Machine using a single shape rule LHS RHS in the schema x Ty under similarity transformations a Shape Rule b One of the eight possible matches of the RHS shape under similarity transformations c Final design after the parallel execution of all eight matches

A construction in Shape Machine using a single shape rule LHS RHS in the schema x Ty under isometry transformations a Shape Rule b One of the 16 possible matches of the RHS shape under isometry transformations c Final design after the parallel execution of all 16 matches

A construction in Shape Machine using a single shape rule LHS RHS in the schema x Ty under isometry transformations a Shape Rule b One of the 32 possible matches of the RHS shape under isometry transformations c Final design after the parallel execution of all 32 matches

The Celtic knot grammar--a Two shape rules substituting a curve with a loop in two handed versions b One of the eight possible matches of the shape Tx under a similarity transformation T and its corresponding replacement of the shape Ty c a design after the serial execution of four shape rules

A Celtic knot grammar--Modification stage a Two shape rules signifying tying and untying a loop b One of the two possible matches of the untying shape rule c a design after the serial execution of two shape rules

Athanassios Economou and Kurt Hong
2019

Keywords: Shape Grammar Interpreter; Shape Grammars; Computer-Aided Design; Design Automation; Visual Indexing; Algebras of Shape

The Shape Machine is a new computational technology that fundamentally redefines the way shapes are digitally represented, indexed, queried and operated upon. By allowing users to draw shape rules rather than write them in a programming language, it provides a robust and disruptive technology for engineers, computer scientists, designers, students and educators, and in general academics and professionals who use drawings and visual models to develop and communicate their ideas. Significantly, the Shape Machine is the first software that manages to solve the subshape recognition problem in vector graphics and the first one to perform the operations with shapes envisioned by the academic community since the early seventies.

The current working prototype is implemented in Python as a plug-in within Rhinoceros 5 from Robert McNeel and Associates and includes three proprietary modules: shape representation, shape recognition, and shape modification, all for straight lines and arcs under Euclidean transformations. The software allows users to program custom functions by drawing shapes instead of writing code in a programming language.

The intellectual property of the Shape Machine software technology is solely originated in the Shape Computation Lab, led by Athanassios Economou and engineered by Kurt Hong. All technical specifications pertaining to the maximal line representation of shape are protected under a pending patent application at the US Patent and Trademark Office (USPTO) – “Shape Computational Technology”, No. 62/831,919. The provisional application was filed by Troutman Sanders LLP on behalf of the inventors and the Georgia Tech Research Corporation (GTRC) and it was officially received by USPTO on April 10th, 2019.

DrawScript

A DrawScript for Chinese character sorting a A DrawScript with a shape rule specifying the order between the two Chinese characters left and right b one application during the execution of the DrawScript

A DrawScript for an unsigned binary adder a Four of the rules from the DrawScript that process the addition for the first digit b one application during the execution of the DrawScript

A DrawScript for an unsigned binary adder a Four of the rules from the DrawScript that process the addition for the general digits in the second phase b the final result of the execution of the DrawScript

A DrawScript for a Tic-Tac-Toe program a Four of the rules from the DrawScript that place an X to avoid losing the game b one application during the execution of the DrawScript

A DrawScript for a Tic-Tac-Toe program a Four of the rules from the DrawScript that place an X to win the game b one application during the execution of the DrawScript

A DrawScript for a maze solver a Four of the rules from the DrawScript that close the dead ends b one application during the execution of the DrawScript

A DrawScript for a maze solver a Four of the rules from the DrawScript that close the dead ends b the final result of the execution of the DrawScript

A DrawScript for a ladder game solver a Four of the rules from the DrawScript that find the path b one application during the execution of the DrawScript

A DrawScript for a ladder game solver a Four of the rules from the DrawScript that find the path b the final result of the execution of the DrawScript

A DrawScript for a bouncing ball animation a Three rules that define ball movement b one application showing ball movement during the execution of the DrawScript

A DrawScript for a bouncing ball animation a Three rules that define ball movement b one application showing ball bouncing during the execution of the DrawScript

Kurt Hong
2019

Keywords: Programming; Scripting; Shape Grammar Interpreter; Shape Grammars; Computer-Aided Design; Design Automation

DrawScript is a radically new environment to learn and do computation! DrawScript does programming with shapes and it be extended in a complete scripting language where if/then statements, loops, conditionals and so forth can be entirely declared in terms of shapes and visual symbols. In shape machine a line of code is literally a line or an arrangement of geometric lines!

Grape

Stinys iceray checkerboard lattice grammar implemented in GRAPE ab Original rules c New rules substituting the square motifs by new motifs with the same diagonal symmetry

Stinys iceray checkerboard lattice grammar implemented in GRAPE

Some new designs in the checkerboard lattice language implemented in GRAPE

Sample rules from Marchs nucleated and linear distributions implemented in GRAPE a Binary division to 4 squares b Tertiary division to 9 squares

A production in the binary division grammar implemented in GRAPE

Some new designs in alternative sequences of the binary and tertiary division rules implemented in GRAPE

An Escheresque grammar implemented in GRAPE

A production in the Escheresque grammar implemented in GRAPE

$Some fractallike designs in the Escheresque grammar implemented in GRAPE$ The Dirksen grammar James Park implemented in GRAPE Three types of representation of a shape rule--a axonometric projection b orthographic projection c Graph

The Dirksen grammar James Park implemented in GRAPE Three types of representation of a shape rule--a axonometric projection b orthographic projection c Graph

A sample derivation of a Miesian courthouse design from the software implementation of the Dirksen grammar in GRAPE

Sectional models of six Miesian courthouse designs generated in GRAPE All models are satisfying the requirement of having 24 courtrooms and appropriate volume of public--office and supporting spaces

Thomas Grasl and Athanassios Economou
2016

Keywords: Shape Grammar Interpreter; Shape Grammars; Computer-Aided Design; Design Automation; Visual Indexing; Algebras of Shape

The challenges for designing a general-purpose parametric shape grammar application are numerous. Several accounts in the literature focus on technical and/or expressive characteristics of interpreters including underlying computing language, subshape recognition, the dimensionality of shapes, and so forth, or the tasks for programs that implement shape grammars, for example, generation, parsing, and inference tasks and their interactions with CAD modelers. And still other accounts focus on usage in design, including general interpreters versus specific domain applications, schematic design versus design development, industrial strength interpreters versus proof-of-concept applications, and so forth.

The work here presents a new general parametric shape grammar interpreter, code-named GRAPE, that uses graph grammars to carry out computations on graphs that encode maximal representations of shapes. We argue that what distinguishes the application from other applications in the field is that emergent shapes and general parametric rules are both supported – while fully acknowledging that parametric subshape recognition is NP-hard.

Graphs are a popular choice for representing the structure of shape. Typically, graphs are seen as multidimensional data structures that, with an appropriate embedding, can formally structure and visually resemble the shapes they represent. Still, the encoding of shapes in terms of these alternative graph representations is not straightforward. Various approaches to representing shapes as graphs have been attempted in the past but none has managed to provide a viable and useful solution for a shape grammar interpreter. The resulting formalism here uses two parallel representations, a graph-theoretic one that is used for the subshape detection, and a visual one that illustrates the shape and which can be tested for intersections. The implementation presented here also maps the shape rules into graph rules. Once a rule has been selected, matching subgraphs are mapped back to shapes for further user interaction, such as picking a shape to which to apply the rule. The shape grammar library has been implemented in C# and it is code named GRAPE to affectionately recall the parallel representations of “GRAph” and “shAPE.”