1 / 19 Algebraic Analysis of Huzita s Origami Origami Operations and their Extensions Fadoua Ghourabi, Asem Kasem, Cezary Kaliszyk University of Tsukuba, Japan. Yarmouk Private University, Syria University of Innsbruck, Austria ADG, 17 September 2012
2 / 19 Outline Background What is Origami Computational Origami Eos System Origami Operations Origami axioms Superposition pairs General Origami Principle Power of Origami Field of Origami constructible numbers Folding with conic sections Conclusion
3 / 19 Outline Background What is Origami Computational Origami Eos System Origami Operations Origami axioms Superposition pairs General Origami Principle Power of Origami Field of Origami constructible numbers Folding with conic sections Conclusion
4 / 19 What is Origami Ancient Japanese art of paper folding Representation of objects using paper folds traditionally no cutting or glue Used in education, sometimes even daily life Tool for geometrical constructions Instead of ruler and compass in Euclidean geometry Start with a square surface Paper folds describe new lines Intersections of lines give rise to new points
5 / 19 Computational Origami Scientific discipline studying mathematical and computational properties of origami. Mathematical theories of paper folding Modelling origami by algebraic and symbolic methods Analyzing origami with computers Simulating origami constructions Proving properties of constructions Correspondence between geometry and algebra Expressed logically Axiom system for origami operations (complete)
6 / 19 Advancements in Origami Making origami more formal Larger community effort More problem specifications System of Operations and their power Computational Geometry Methods
6 / 19 Advancements in Origami Making origami more formal Larger community effort More problem specifications System of Operations and their power Computational Geometry Methods Origami in Education HS Students are shown the basic operations and given simple tasks Dividing a segment into n equal sub-segments Constructing a square, regular hexagon, equilateral triangle Bi-secting an angle Properties of constructions shown on the blackboard Haga theorem (2/3 with one fold) Various methods of angle trisection
7 / 19 Eos System for visualizing origami constructions and proving their properties with the help of Mathematica [Ida et al] Visualizing constructions based on Huzita s axioms Analysing the origami folds algebraically Showing properties of the constructions Algorithmic translation of folds into algebraic properties Usage of Gröbner bases or CAD on whole formulas Flat constructions (every fold followed by unfold) Trisection of an angle Maximum equilateral triangle Regular Heptagon Morley s Triangle Layers and sides Crane WebEos
8 / 19 Outline Background What is Origami Computational Origami Eos System Origami Operations Origami axioms Superposition pairs General Origami Principle Power of Origami Field of Origami constructible numbers Folding with conic sections Conclusion
9 / 19 Origami operations (trad. axioms) (O1) Given two points P and Q on the origami O, we can fold O along the line that passes through P and Q. (O2) Given two points P and Q on the origami O, we can fold O to superpose P and Q. (O3) Given two lines m and n which pass through the origami O, we can fold O to superpose m and n. (O4) Given a point P and a line m passing through the origami O, we can fold O along the line that is perpendicular to m and passes through P. (O5) Given a line m and two points P and Q, which the origami O is passing through, it is decidable whether we can fold O to superpose P and m along the line that passes through Q. (O6) Given two points P and Q and two lines m and n, which the origami O is passing through, it is decidable whether we can fold O to superpose P and m, and Q and n, simultaneously. (O7) Given two lines m and n and a point P, which the origami O is passing through, it is decidable whether we can fold O to superpose P and m along the line perpendicular to n.
10 / 19 Superposition Superposition pair, s-pair, (α, β) Point-point superposition distinct points P and Q (P, Q) defines a unique fold line that superposes P and Q perpendicular bisector (P Q) Line-line superposition When lines m and n are equal, infinite set of fold lines (with m excluded) Otherwise two distinct lines Point-line superposition (P, m) If P m tangent to the parabola with focus P and directrix m. Otherwise any line perpendicular to n or passing by P. Defines the set Γ(P, m)
11 / 19 Superpositions in Huzita s fold principle Table: Superpositions in Huzita s fold principle operation s-pairs degeneracy incidence (1) (P, P), (Q, Q) P = Q (2) (P, Q) P = Q (3) (m, n) m = n (4) (m, m), (P, P) (5) (P, m), (Q, Q) P m (6) (P, m), (Q, n) P = Q m = n P m Q n (7) (P, m), (n, n) P m We can reformulate Huzita s fold principle using the concept of superposition!
12 / 19 General Origami Principle (G) Given two points P and Q and two lines m and n, fold O along a line to superpose P and m, and Q and n. Can do all the rest of fold operations? Yes, but we need to carefully analyze the degenerate and incident cases. We analyze the origami constructible s-pairs. details in the paper
13 / 19 Principle G related to HO incidence degeneracy operation movement P m, Q n m n B(m) (, ) (m n) P = Q I(P) (, ) (m n) P Q (O1) (, ) (m n) P Q (O4) (, ) (m n) P Q (O4) (, ) P m, Q n (O5) (, ) (O7) (, ) P m, Q n (O5) (, ) (O7) (, ) P m, Q n m = n P = Q Γ(P, m) (, ) (m = n P = Q) (O6) (, ) Table: (G) to perform (O1), (O4) - (O7)
14 / 19 Outline Background What is Origami Computational Origami Eos System Origami Operations Origami axioms Superposition pairs General Origami Principle Power of Origami Field of Origami constructible numbers Folding with conic sections Conclusion
15 / 19 Increasing the power of Fold Compass Simplifies constructions But the power is same (equations can be reduced) Multi-fold Arbitrary degree of equations But not feasible by hand Conic sections! Superposition of a point and a conic on the origami
16 / 19 General conic fold operation Abstract from the method used to draw a conic on the origami pins, strings, pencil and straightedge Add general fold operation: Given two points P and Q, a line m and a conic section C, where P is not on C and Q is not on m, fold O along a line to superpose P and m, and Q and C. Analyzing the equation of the fold like we get the result: Then the slope of the fold line satisfies a polynomial equation of degree six over the field of origami constructible numbers
17 / 19 Example Fold lines k 1,, k 6 whose slopes are the six distinct real solutions of the equation 16t 6 78t 5 + 84t 4 + 39t 3 66t 2 + t + 8 = 0 Mathematica gives 6 approximate solutions to the equation By sliding the operation can be performed by hand
18 / 19 Outline Background What is Origami Computational Origami Eos System Origami Operations Origami axioms Superposition pairs General Origami Principle Power of Origami Field of Origami constructible numbers Folding with conic sections Conclusion
19 / 19 Conclusion and Future work Traditional Origami Operations Reformulation using s-pairs Precise degeneracy and incidence conditions General fold operation Conditions for reducing it to the reformulated operations Fold with conic sections Gives rise to 6 possible fold lines
19 / 19 Conclusion and Future work Traditional Origami Operations Reformulation using s-pairs Precise degeneracy and incidence conditions General fold operation Conditions for reducing it to the reformulated operations Fold with conic sections Gives rise to 6 possible fold lines Degenerate and incident cases? What equations of degree 6 can be solved? Formalized origami theory