Papers by Fadoua Ghourabi
Artificial Intelligence and Symbolic Computation, 2018
Origami geometry is based on a set of 7 fundamental folding operations. By applying a well-chosen... more Origami geometry is based on a set of 7 fundamental folding operations. By applying a well-chosen sequence of the operations, we are able to solve a variety of geometric problems including those impossible by using Euclidean tools. In this paper, we examine these operations from spatial qualitative point of view, i.e. a common-sense knowledge of the space and the relations between its objects. The qualitative spatial representation of the origami folds is suitable for human cognition when practicing origami by hand. We analyze the spatial relations between the parameters of the folding operations using some existing spatial calculus. We attempt to divide the set of possible values of the parameters into disjoint spatial configurations that correspond to a specific number of fold lines. Our analyses and proofs use the power of a computer algebra system and in particular the Gröbner basis algorithm.
We present a method of superposing rectangles. The superposition is under the condition that some... more We present a method of superposing rectangles. The superposition is under the condition that some of the regions should be visible. We first define a qualitative spatial representation of the rectangles. In particular, direction relations are used to express the positions of the must-be-visible regions. The representation is extendable to accommodate higher degree of granularity, and therefore to cover any arrangement of regions. Properties of success and effectiveness are defined to evaluate the superposition.
We present a method of superposing rectangles. The superposition is under the condition that some... more We present a method of superposing rectangles. The superposition is under the condition that some of the regions should be visible. We first define a qualitative spatial representation of the rectangles. In particular, direction relations are used to express the positions of the must-be-visible regions. The representation is extendable to accommodate higher degree of granularity, and therefore to cover any arrangement of regions. Properties of success and effectiveness are defined to evaluate the superposition. Qualitative spatial representation emerged as an area of knowledge representation. The foundation in qualitative spatial representation is to treat objects of the space qualitatively, i.e. what matters is how objects are related. Positions of objects in the space is one of the relevant problems that is addressed by the field of qualitative spatial representation. The direction relations describe where an object is positioned w.r.t. a reference. We distinguish two categories o...
Abstract. Computational origami is the computer assisted study of origami as a branch of science ... more Abstract. Computational origami is the computer assisted study of origami as a branch of science of shapes. The origami construction is a countably finite sequence of fold steps, each consisting in folding along a line. In this paper, we formalize origami construction. We model origami paper by a set of faces over which we specify relations of overlay and adjacency. A fold line is determined by a specific fold method. After folding along the fold line, the structure of origami is transformed; some faces are divided and moved, new faces are created and therefore the relations over the faces change. We give a formal method to construct the model origami. The model furthermore induces a graph of layers of faces. We give two origami examples as the application of our model. They exhibit non-trivial aspects of origami which are revealed only by formal modeling. The model is the abstraction of the implemented core of the system of computational origami called Eos (E-origami system). 1
Proceedings of the International Conference on Agents and Artificial Intelligence, 2015
We formalize and verify the superposition of rectangles in Isabelle/HOL. The superposition is ass... more We formalize and verify the superposition of rectangles in Isabelle/HOL. The superposition is associated with the arrangement of rectangular software windows while keeping some regions visible and other hidden. We adopt a qualitative spatial reasoning approach to represent these rectangles and the relations between their regions. The properties of the model are formally proved and show some characteristics of superposition operation. Although, this work is limited to 29 structures of rectangles, the superpositions produce hundreds of cases that are tedious to tackle in Isabelle/HOL. We also explain our strategy to optimize the proofs.
2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, 2013
ABSTRACT We present computer-assisted construction of reg-ular polygons by knot paper fold. The c... more ABSTRACT We present computer-assisted construction of reg-ular polygons by knot paper fold. The construction is completed with an automated proof based on algebraic methods. Given a rectangular origami or a finite tape, both of an adequate length, we can construct the simplest knot by making three folds. The shape of the knot is made to be a regular pentagon if we fasten the tape tightly without destroying the tape. We performed the analysis of the knot fold further formally towards the computer assisted construction and verification. Our study yielded more rigor and in-depth results about the subject.
Lecture Notes in Computer Science, 2013
We investigate the basic fold operations, often referred to as Huzita's axioms, which represent t... more We investigate the basic fold operations, often referred to as Huzita's axioms, which represent the standard seven operations used commonly in computational origami. We reformulate the operations by giving them precise conditions that eliminate the degenerate and incident cases. We prove that the reformulated ones yield a finite number of fold lines. Furthermore, we show how the incident cases reduce certain operations to simpler ones. We present an alternative single operation based on one of the operations without side conditions. We show how each of the reformulated operations can be realized by the alternative one. It is known that cubic equations can be solved using origami folding. We study the extension of origami by introducing fold operations that involve conic sections. We show that the new extended set of fold operations generates polynomial equations of degree up to six.
Lecture Notes in Computer Science, 2011
A proof document for origami theorem proving is a record of entire process of reasoning about ori... more A proof document for origami theorem proving is a record of entire process of reasoning about origami construction and theorem proving. It is produced at the completion of origami theorem proving as a kind of proof certificate. It describes in detail how the whole process of an origami construction and the subsequent theorem proving are carried out in our computational origami system. In particular, it describes logical and algebraic transformations of the prescription of origami construction into mathematical models that in turn become amenable to computation and verification. The structure of the proof document is detailed using an illustrative examples, which reveal the importance of such a document in the analysis of reasoning of origami theorem proving.
Journal of Symbolic Computation, 2015
We present computer-assisted construction of regular polygonal knots by origami. The construction... more We present computer-assisted construction of regular polygonal knots by origami. The construction is completed with an automated proof based on algebraic methods. Given a rectangular origami or a finite tape, of an adequate length, we can construct the simplest knot by three folds. The shape of the knot is made to be a regular pentagon if we fasten the knot tightly without distorting the tape. We perform the analysis of the knot fold further formally towards the automated construction and verification. In particular, we show the construction and proof of regular pentagonal and heptagonal knots. We employ a software tool called Eos (e-origami system), which incorporates the extension of Huzita's basic fold operations for construction, and Gröbner basis computation for proving. Our study yields more mathematical rigor and in-depth results about the polygonal knots.
Electronic Notes in Theoretical Computer Science, 2008
Computational origami is the computer assisted study of mathematical and computational aspects of... more Computational origami is the computer assisted study of mathematical and computational aspects of origami. An origami is constructed by a finite sequence of fold steps, each consisting in folding along a fold line. We base the fold methods on Huzita's axiomatization, and show how folding an origami can be formulated by a conditional rewrite system. A rewriting sequence of origami structures is viewed as an abstraction of origami construction. We also explain how the basic concepts of constraint and functional and logic programming are related to this computational construction. Our approach is not only useful for computational construction of an origami, but it leads us to automated theorem proving of the correctness of the origami construction.
… Koen Ronbunshu (CD …, 2006
We describe Huzita's origami axioms in logical and algebraic point of view. Observing that H... more We describe Huzita's origami axioms in logical and algebraic point of view. Observing that Huzita's axioms are statements about the existence of certain origami constructions, we can generate basic origami constructions from those axioms. We give the logical specification of ...
EPiC Series in Computing
Making a knot on a rectangular origami or more generally on a tape of a finite length gives rise ... more Making a knot on a rectangular origami or more generally on a tape of a finite length gives rise to a regular polygon. We present an automated algebraic proof that making two knots leads to a regular heptagon. Knot fold is regarded as a double fold operation coupled with Huzita's fold operations. We specify the construction by describing the geometrical constraints on the fold lines to be used for the construction of a knot. The algebraic interpretation of the logical formulas allows us to solve the problem of how to find the fold operations, i.e. to find concrete fold lines. The logical and algebraic framework incorporated in a system called Eos (e-origami system) is used to simulate the knot construction as well as to prove the correctness of the construction based on algebraic proof methods.
We prove the 169 compositions of time interval relations. The proof is first-order and inferred f... more We prove the 169 compositions of time interval relations. The proof is first-order and inferred from an axiomatic system on time intervals. We show a general proof template that can alleviate the manual proof with Isar.
Computational Origami is a branch of the science of shapes, where we study computational and math... more Computational Origami is a branch of the science of shapes, where we study computational and mathematical aspects of origami. One of the foundational studies of the computational origami is the axiomatic definition of origami foldability by Huzita in 1989. We describe Huzita's origami axioms from the logical and algebraic points of view. Observing that Huzita's axioms are statements about the existence of certain origami constructions, we can generate basic origami constructions from those axioms. Origami construction is performed by repeated application of Huzita's axioms. We give the logical specification of Huzita's axioms as constraints among geometric objects of origami in the language of the firstorder predicate logic. For this purpose, we define a many-sorted language L to give a logical formalization to Huzita's axioms and specify other geometric properties of origami elements, as well. The logical specification is then translated algebraic forms, i.e. polynomial equalities, disequalities and inequalities. If inequalities are not involved, by constraint solving we obtain solutions that satisfy the logical specification of the origami construction problem. The solutions include fold lines along which origami paper has to be folded. The obtained solutions both in numeric and symbolic forms make origami computationally tractable for further treatments, such as (1) simulation of the origami construction parallel with the graphic visualization of construction steps and (2) automated theorem proving of the correctness of the origami construction. To simulate origami construction, we model origami paper by a set of faces over which we specify relations of overlay and adjacency. After folding along the fold line, the structure of origami is transformed. Some faces are divided and moved, new faces are created and therefore the relations over the faces change. We give a formal method to construct the model origami. Using the algebraic theorem prover Gröbner basis, we prove some properties of the final origami object.
… of the 2011 ACM Symposium on …, Jan 1, 2011
Journal of symbolic …, Jan 1, 2010
Morley's theorem states that for any triangle, the intersections of its adjacent angle trisectors... more Morley's theorem states that for any triangle, the intersections of its adjacent angle trisectors form an equilateral triangle. The construction of Morley's triangle by the straightedge and compass is impossible because of the well-known impossibility result of the angle trisection. However, by origami, the construction of an angle trisector is possible, and hence of Morley's triangle. In this paper we present a computational origami construction of Morley's triangle and automated correctness proof of the generalized Morley's theorem.
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Papers by Fadoua Ghourabi