Computational Construction of a Maximum Equilateral Triangle Inscribed in an Origami

Transcription

1 omputational onstruction of a Maximum quilateral Triangle Inscribed in an Origami Tetsuo Ida, Hidekazu Takahashi, Mircea Marin, adoua hourabi, and sem Kasem epartment of omputer Science University of Tsukuba, Tsukuba , Japan {ida, hidekazu, mmarin, ghourabi, kasem}@score.cs.tsukuba.ac.jp bstract. We present an origami construction of a maximum equilateral triangle inscribed in an origami, and an automated proof of the correctness of the construction. The construction and the correctness proof are achieved by a computational origami system called os (-origami system). In the construction we apply the techniques of geometrical constraint solving, and in the automated proof we apply röbner bases theory and the cylindrical algebraic decomposition method. The cylindrical algebraic decomposition is indispensable to the automated proof of the maximality since the specification of this property involves the notion of inequalities. The interplay of construction and proof by röbner bases method and the cylindrical algebraic decomposition supported by os is the feature of our work. 1 Introduction Origami is a traditional Japanese art of paper folding, which is now enjoyed by many people all over the world. omputational origami is a scientific discipline to study computational aspects of origami, and aims to complement and even goes beyond traditional paper folding by hands [4]. We are aiming at overcoming the limitations of traditional origami construction, and, with the use of computers, at providing capabilities that support geometrical reasoning about origami. We study the problem of computational origami construction of an equilateral triangle that is maximally inscribed in an origami square. lthough the problem is easy, we use it as an example to appeal the importance of computational origami and of the effectiveness of our approach, since it reveals the essence of computational origami construction and of the automated correctness proof of the construction thereafter. In the construction we apply the techniques of geometrical constraint solving, and in the automated proof we apply röbner bases theory [] and the cylindrical algebraic decomposition method [1]. We reported earlier some examples of computational origami constructions [9, 8], such as the construction of Morley s triangle by origami with correctness proof. What is new in this paper is the application of cylindrical algebraic decomposition (cad) to prove the correctness of our construction. The use of cad

2 has lead to further improvement of our computational origami system called os. The paper is organized as follows. In section, we briefly explain our computational origami environment as we will use it for the study of origami construction of a maximum equilateral triangle. In section 3, we will present a method for the construction using os. In section 4, we will give the proof of the correctness of the construction. We will show two methods of proving the correctness. In section 5, we draw some conclusion and point out future research. -origami system os efore we describe the construction of the equilateral triangle, we briefly explain the main features of os. os is a collection of Mathematica programs, specialized in origami processing, that have capabilities of basic geometrical computing tailored to origami, of constraint solving both by numeric and symbolic methods, of theorem proving, of visualizations of origami and of web interfaces. It performs origami construction that a human would do by hand with a piece of paper. Moreover, it performs automated proving of geometrical properties of the constructed origami, which would require much geometric intuition in addition to complicated and tedious algebraic manipulation..1 Origami construction by os os implements Huzita s origami axioms [7], which describe the basic folding operations. Huzita s axiom set is more powerful than the straight-edge and compass method in uclidean plane geometry [5, 6]. Using Huzita s axiom set, by origami we can construct a trisector of an angle, whereas by the straight-edge and compass we can not. Huzita s axiom set is described by the following statements about the fundamental origami folding operations. (O1) iven two points, we can make a fold along the fold line that passes through them. (O) iven two points, we can make a fold to bring one of the points onto the other. (O3) iven two lines, we can make a fold to superpose the two lines. (O4) iven a point P and a line m, we can make a fold along the fold line that is perpendicular to m and passes through P. (O5) iven two points P and Q and a line m, we can make a fold to superpose P and m along the fold line that passes through Q. (O6) iven two points P and Q and two lines m and n, we can make a fold to superpose P and m, and Q and n, simultaneously. These axioms are formalized in a properly chosen sub-language of first-order logic, with function symbols for the geometric constructs and predicate symbols for the geometric properties mentioned there. or example, axiom (O5) is formalized as follows: {k Line} OnLine[Q, k] OnLine[SymmetricPoint[P, k], m].

3 The atomic formula OnLine[Q, k] specifies that point Q is on line k, and the term SymmetricPoint[P, k] denotes the symmetric point of P with respect to line k. The algebraic meaning of each predicate is specified by a set of equations. or example, the meaning of OnLine[Q, k] is given by a x 1 + b y 1 + c = 0, where Q is positioned at (x 1, y 1 ) and k is defined by the equation a x + b y + c = 0. Origami constructions proceed stepwise, where each step indicates a fold operation that satisfies one of axioms (O1) (O6). os provides a function old to realize the fold steps. In addition, every call of old has two effects: (1) to visualize the result of the fold step by computing the fold line and performing the fold, and () to record the geometric constraints that characterize the fold step. In this way, os provides interactive access to (1) a view of the origami constructed so far, and () the collection of geometric constraints that describe the origami constructed so far.. Theorem proving of origami construction fter origami construction with os, we can proceed to prove geometrical properties of the construction in the following steps. Premise generation: xtract the geometrical properties of the construction and transform them into polynomial equalities and inequalities which form the premise of the theorem to be proved. onclusion formulation: Represent the conclusion in polynomial equalities and/or inequalities. Proof generation: ive the polynomial equalities and inequalities to a theorem prover such as Theorema when the theorem is described in polynomial equalities, or to cylindrical algebraic decomposition programs when the theorem is described partly in inequalities, and obtain the proof result. The most important step is the formalization of the geometrical properties to be proved. Note that although the conclusion formulation step is in the third computation step, we have to think about the conclusion at first. lso note that the steps except for the conclusion formulation step are automated. 3 onstruction of an equilateral triangle In this section we illustrate origami construction of an equilateral triangle by os. straightforward method to construct an equilateral triangle is to use one of the sides of the origami as a side of an equilateral triangle. The following snapshot of the origami construction using the os website illustrates the construction. However, we immediately recognize that this is not a maximum triangle inscribed in the origami. In the following, we will show another origami construction that creates a maximum equilateral triangle. Since we want to make

4 ig. 1. webpage of WebOrigami the side of a triangle as long as possible, it is natural to take one of the corners of the triangle to be also a corner of the origami. We expect that the other two vertices are also on the sides of the origami. We will illustrate a step-wise construction of a maximum equilateral triangle via a Mathematica notebook interface rather than os web interface, since in the notebook we can use all the capabilities of os and Mathematica. 3.1 onstruction We start our origami construction by declaring the origami to be used. In the following example, we will use a unit square paper with four vertices marked as,, and. eginorigami[{1, MarkPoints {,,, }, igureaption Step }] Step 1

5 The calls of function old and others with various parameters will be selfexplanatory. We make a fold to bring point onto (application of (O)), and then make a fold to bring point onto (application of (O)). unction Unfoldll completely unfolds the folded origami into the original origami. These steps are illustrated below. old[,]; old[,];unfoldll[]; H H Step Step 3 I Step 4 fter these preparatory steps, we make a fold to superpose and line HI along the fold line that passes through (application of (O5)) 1. In this case we have two possible folds. It can be shown that to find the fold line in the use of axiom (O5) is a constraint solving problem formulated with a set of algebraic equalities of the second order. Hence, we have at most two choices. os will interact with the user to ask for the choice, as seen below. old[, HI, Through, Markrease {}]; Specify the fold line. H 1 I Step 5 There are two ways to proceed. One is to specify the line number shown in the origami, and the other is to give the constraint that uniquely determines the choice. In the above case, both responses will do. However, there is a difference when it comes to proving later. y specifying the constraint, os not only selects one of the numeric solutions using the constraint but also saves the constraint symbolically, which can be used to prove geometrical properties later. Since we are interested in geometric theorem proving we specify the constraint as follows. 1 XY denotes a segment between point X and point Y and XY denotes the line that extends segment XY, i.e the line that pass through X and Y.

6 old[, HI, Through,Markrease {}, onstraint OnSegment[,HI]]; H I K Step 6 In the above we specify the constraint that point is on the segment HI. This constraint determines the choice of the fold line 1. Similarly we will construct point L and, finally, apply (O1) to fold the origami along line KL. old[,, Through, onstraint OnSegment[, ]]; old[, long KL]; L H L H I K Step 8 I K Step 9 The triangle LK appears to be equilateral, and at this point we conjecture that this is the case. The following complete development of the origami, which is obtained by the call of Unfoldll, will be helpful for further study of the geometrical properties. L H I K Step 10 ig.. omplete development of the folded origami

7 The figure is called ori-zu. It shows the creases made by both valley folds and mountain folds, and the points, created in the whole construction. 4 Proof of the correctness of the construction We will now prove the following theorem: Theorem 1 (Maximum quilateral Triangle Origami Theorem). iven an origami, the origami construction in subsection 3.1 has the following properties: (a) LK is equilateral, and furthermore (b) LK is a maximum equilateral triangle inscribed in the origami. The automated proof of this theorem is given in the following two subsections. We will follow the proof procedure as outlined in subsection., interacting with os. 4.1 Proof that LK is equilateral utomated proof by os interacting with Theorema. or proving, we switch the context from construction to proof by calling function eginproof. eginproof[ ]; The ori-zu shown below contains more information than the complete development of the folded origami of ig.. In the figure, X n indicates that point X is on the origami whose id is n. Origami id is not shown in the figures, but can be displayed by setting the switch on. H L I 5 K 6 1 We first consider part (a) of the theorem. The proof is based on the röbner bases method []. ori means fold and zu means figure.

8 Premise generation. or theorem proving, we transform the symbolic constraints accumulated during the construction of origami into algebraic form. This is achieved by deciding the coordinate system and then translating the symbolic representation of the geometrical properties into polynomials. The generated premise sometimes needs to be strengthened in order to eliminate degenerate cases and other unwanted geometrical configurations. In our case, we need the condition that K 6, 1 and L 7 are not collinear. This is related to the fact that at steps 6 and 8 we have two folding possibilities. xiom (O5) describes a geometric constraint that is expressed in second degree polynomials. This gives rise to sets of solutions. lthough at steps 6 and 8, we have chosen one of the two possibilities, in this premise generation phase we do not specify these committed choices in order to allow possibly more general statements about the intended geometrical properties. In our case, we would expect that in four possible cases we could construct equilateral triangles. However, this is not the case as shown in the following figure. ig. 3. Possible cases of the construction s seen in the figure, only the first and third cases make the equilateral triangles. The second and the fourth cases can be eliminated by specifying that ollinear[k 6, 1, L 7 ]. We can translate the constraints related to points K 6, 1 and L 7 from symbolic to algebraic form as follows. premisepoly = Tolgebraic[{K 6, 1, L 7 }, InitialShape {Point[0, 0], 1}, onstraints { ollinear[k 6, 1, L 7 ]}] 1+a +b, c+ax+by, 1+a3 +b3, a3+b3+c3, 1+a4 +b4, a4+ b4 + c4, 1 + a1 + b1, c1 + a1x3 + b1y3, y4, c3 + a3x4 + b3y4, x5, c4 + a4x5 + b4y5, a1, b1 + c1, b1( 1 + x1) + ( 1)a1y1, c1 + 1 b1y1 a1(1 + x1) +, b3( 1 + x) + ( 1)a3y, c3 + 1 b3y ax1 a3(1 + x) +, bx1 + a( 1 + y1), c b(1 + y1), b4x3 + a4(1 + ( 1)y3), c4 + a4x3 + 1 b4(1 + y3), 1 + (x5 + ( 1)y5 + x4( 1 + y5) + (1 + ( 1)x5)y4)ξ1 In our representation of a line a x + b y + c = 0, we need to make sure that both a and b are not equal to zero simultaneously, namely a + b 0. nd without loss of generality, we can add the constraint that a + b = 1.

9 Note also that ollinear[k 6, 1 L 7 ] is translated to equalities using Rabinowich trick. This can be further transformed to the equivalent logic form (output not shown). premise = ToLogic[premisePoly]; onclusion formation. The next step is to transform the conclusion that LK is an equilateral triangle into algebraic form. n equilateral triangle is characterized by the property that all the three sides are equal in length. Hence we have the following specification. conclusion = ToLogic[{ istance [K 6, L 7 ] istance [ 1, L 7 ], istance [K 6, L 7 ] istance [ 1, K 6 ] }] (1 x4) + (x4 x5) (1 y4) + (y4 y5) == 0 (x4 x5) + (y4 y5) (1 x5) (1 y5) == 0 Note that we use two kinds of equality symbols; == as the output of Mathematica, and = as in the ordinary mathematics. Proof generation. os can prove geometric properties of origamis by accessing to the theorem prover Theorema [3]. Since both os and Theorema are implemented in Mathematica, accessing to Theorema is straightforward. irst we call Theoremaormula["quilateral Triangle Th", Variables[premisePoly], premise conclusion, "(1)"]; This method call encodes the proposition premise conclusion into a Theorema formula, labeled "quilateral Triangle Th". The variables occurring in the formula are universally quantified. Next, we invoke Theorema to prove it: Prove[ormula["quilateral Triangle Th"], using {}, by roebnerasesprover]//timing {3.03 Second, ProofObject } This call will yield a human-readable proof with proof text structured as nested cells of Mathematica. Proof by os using the cylindrical algebraic decomposition. The success of our proof by röbner bases method relies on the crucial observation that we must add the condition ollenear[k 6, 1, L 7 ]. In the following we will show an alternative proof based on the cylindrical algebraic decomposition. We recall that in order to specify one of the two choices of the fold lines we specified the constraint that OnSegment[, HI] at step 6 and OnSegment[, ] at step 8. These additional constraints are not included in the previous proof since they give rise to inequalities, which are not handled by röbner bases method. We now add these constraints by setting the option ddonstraint to True in the call of function Tolgebraic.

10 premise = ToLogic[Tolgebraic[{K 6, 1, L 7 }, InitialSahpe {Point[0, 0], 1}, ddonstraint True ]]; The outcome is that premise contains the following additional inequalities: ( x3 + x4)( x3 + x5) 0 (y4 + ( 1)y3)(y5 + ( 1)y3) 0 (x6 + ( 1)x)(x7 + ( 1)x) 0 (y6 + ( 1)y)(y7 + ( 1)y) 0 We next perform the cylindrical algebraic decomposition [1] of the above formula by calling the Mathematica function ylindricalecomposition. This will transform the above formula to a fully solved form. spremise = ylindricalecomposition[premise,variables[premise]]; Now it is very easy to prove the conclusion by substituting those values in the conclusion formula. We can also use the previous code and call Theorema to prove the theorem as follows. proposition = spremise conclusion Theoremaormula["Maximum quilateral Triangle Th", vars, proposition, "()"]; 3 Prove[ormula["quilateral Triangle Th"], using {}, by roebnerasesprover]//timing {3.094 Second, ProofObject } We should note in passing the following properties. Let = = = = 1. { Let h = H and x = K at step 7. Then we have the set of equations 1 eqns = 4 + h == 1, (1 h) + ( ) x 1 == (1 x) }, whose solutions are {{ given by x 1 3, h } { 3, x 1 + 3, h }} 3. The first solution is irrelevant because x = K has to be non-negative. rom the second solution we learn that point K is at ( 3-1, 0). Therefore, the area of the triangle is This property is used to establish the proof of the part (b) of Theorem Proof that LK is the maximum equilateral triangle inscribed in the origami To prove part (b) of Theorem 1, we proceed as follows. We consider an equilateral triangle XYZ as depicted below, where we have (0,0), (1,1), X(x, y), Y(0, b), Z(c, 0) with b, c (0, 1). 3 vars is the set of variables occurring in proposition.

11 Y X Z XYZ is equilateral, and therefore YZ = YX = XZ. This geometric constraint is expressed algebraically as: (y b) + x = b + c = (x c) + y. We can solve these algebraic constraints by calling Solve [{( (y b) + x ) ==b + c == (x c) + y }, {x, y} ] {{ y b 3c, x 3b+c }, { y b+ 3c, x 3b+c This call computes x and y as functions of b and c. We know that the area of an equilateral triangle with edge of length l is 3l /4. Since YZ = a + b, we learn that the area of XYZ is S = 3(b + c )/4. Our problem is to compute the values of b and c for which S is maximal. Let S 0 be the area of the equilateral triangle LK constructed with os. Since S 0 = 3 + 3, we know that the maximum value of S is greater than or equal to S 0. We can compute the values of b, c for which S S 0 by using : }} 1. If y = (b + 3c)/ and x = ( 3b + c)/ then the call ylindricalecomposition[{s ( ) S( 0, 0 x )} 1, 0 y 1, ] 0 < b < 1, 0 < c < 1, y== 1 b + 3c, x== 1 3b + c, {x, y, b, c} yields the cylindrical decomposition y = 1 x = 1 c = b = If y = (b 3c)/ and x = ( 3b + c)/ then the call ylindricalecomposition[{s ( ) S( 0, 0 x )} 1, 0 y 1, ] 0 < b < 1, 0 < c < 1, y== 1 b 3c, x== 1 3b + c, {x, y, b, c} yields alse. We conclude that the maximum area of XYZ is reached only when x = 1, c = and b = In this case, we have X=, Y=L and Z=K. Therefore, we can conclude that Theorem 1.(b) holds. 5 onclusion We have shown the origami construction of an equilateral triangle and the automated proof of the property that the constructed triangle is equilateral, maximally inscribed in the origami by the computational origami environment os. It not only simulates origami folds, but also computes and proves geometric

12 properties of the construction. os keeps track of the geometrical properties of all points and lines during the construction. rom those properties, polynomial algebraic constraints are generated, which then are supplied to theorem provers. Recently, we added several improvements in the proof part. y specifying additional constraints using equalities and inequalities during the construction, the system can perform more automated deduction. Inequality constraints require more powerful solving methods such as cylindrical algebraic decomposition. Our experience with os shows that the number of polynomials grows very rapidly as the number of origami construction steps grows. ven a small number of fold steps (for example, 30 steps) may generate over 100 polynomials. Handling constraints of this size is a challenging task for most of the geometric solvers and provers. We can easily generate sets of constraints which make a röbner bases prover run out of computing resources. These limitations can be overcome by adopting a computational framework with access to networked resources. cknowledgements This research is supported in part by the JSPS rantsin-id for Scientific Research No and No , and by MXT rant-in-id for xploratory Research NO References 1.. S. rnon,.. ollins, and S. Mcallum. ylindrical algebraic decomposition I: The basic algorithm. SIM Journal on omputing, 13(4): , uchberger. in algorithmisches Kriterium für die Lösbarkeit eines algebraischen leichungssystems. equationes mathematicae, 4(3): , uchberger,. upre, T. Jebelean,. Kriftner, K. Nakagawa,. Văsaru, and W. Windsteiger. The Theorema Project: Progress Report. In M. Kerber and M. Kohlhase, editors, Symbolic omputation and utomated Reasoning: The alculemus-000 Symposium, pages , St. ndrews, Scotland, ugust emaine and M. L.emaine. Recent results in computational origami. In Thomas Hull, editor, Origami 3 : Third International Meeting of Origami Science, Mathematics and ducation, pages 3 16, Natick, Massachusetts, 00. K Peters, Ltd. 5. R. eretschläger. eometric onstructions in Origami. Morikita Publishing o., 00. In Japanese, translation by Hidetoshi ukagawa. 6. T. Hull. Origami and geometric constructions thull/omfiles/geoconst.html. 7. H. Huzita. xiomatic evelopment of Origami eometry. In H. Huzita ed, editor, Proceedings of the irst International Meeting of Origami Science and Technology, pages T. Ida,. Ţepeneu,. uchberger, and J. Robu. Proving and onstraint Solving in omputational Origami. In Proceeding of the 7th International Symposium on rtificial Intelligence and Symbolic omputation (IS 004), volume 349 of Lecture Notes in rtificial Intelligence, pages 13 14, T. Ida, H. Takahashi,. Ţepeneu, and M. Marin. Morley s Theorem Revisited through omputational Origami. In Proceedings of the 7th International Mathematica Symposium (IMS 005), 005.