Research Plan

1
JSSST 2006
Logical and Algebraic Formulation of Origami
Axioms
Fadoua Ghourabi Tetsuo Ida, Hidekazu Takahashi,
Mircea Marin and Asem Kasem Symbolic COmputation
REsearch Group University of Tsukuba
2
Outline

• Introduction
• Logical View of Huzitas Axioms
• Algebraic Interpretation
• Applications
• Conclusion

3
Introduction
Origami (???) is the Japanese paper folding art.

• Power of Origami.
• Origami Mathematics.
• Computational Origami.
• Computer assisted construction of geometrical
  objects by means of paper folds.

4
Logical View of Huzitas Axioms

• 1989 definition of origami construction by
  axioms.
• Given points and lines in an origami paper, what
  kind of folds can we make ?

5
Logical View of Huzitas Axioms
Why first-order predicates logic ?

• Abstraction of origami construction.
• Appropriate to model geometric properties.

6
Logical View of Huzitas Axioms

• Syntax
• Language L over a signature (P, F ).
• Terms t p l f(t, ..., t) p are
  points, l are lines, f ? F.
• Formulas prenex normal formula ?.
• F A F ? F F ? F F
• F is quantifier free formula, A atomic formula

7
Logical View of Huzitas Axioms

• Axiom (O1)

Given two points P and Q, we can make a fold
along the fold line that passes through P and Q.
? P, Q ? Point ? k ? Line OnLineP, k ?
OnLineQ, k
8
Logical View of Huzitas Axioms
Axiom (O4)
Given 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.
? P ? Point ? m ? Line ? k ? Line OnLineP, k ?
Perpendiculark, m
9
Logical View of Huzitas Axioms
Axiom (O6)
Given 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.
? P, Q ? Point ? m, n ? Line ? k ?
Line OnLineSymmetricPointP, k,
m? OnLineSymmetricPointQ, k, n
10
Algebraic Interpretation
Logical formulas
Algebraic forms Polynomials equalities

• Why?

Further applications

• Realization of origami construction.
• Proof of the correctness of the origami
  construction.

11
Algebraic Interpretation
Logical formulas
Algebraic forms Polynomials equalities
Transformation rule
F set of formulas R set of polynomials
equalities
A F ? R
F ? F A F p10, , pn 0 Where pi ?
Rx, ?i?1, , n
12

• P (x1, y1)
• m a1 x b1 y c1 0
• n a2 x b2 y c2 0
• A OnLineP, m a1 x1 b1 y1 c10,
  a12b12 -10
• A Perpendicularm, n a1 a2 b1 b2 0
  ,
• a12b12 -1 0, a22b22 -1 0
• A SymmetricPointP, kQ

Algebraic Interpretation
A applied to atomic formulas
13
Algebraic Interpretation
A over origami formulas
1/ Conjunction
A ? i?1,...,n Fi ? i?1,...,n A Fi
Example
A F1 p10 A F2 q10, q20 A
F1 ? F2 p10, q10, q20
14
Algebraic Interpretation
2/ Disjunction
A ? i?1,...,n Fi p1 pn 0
lt p10, , pn0gt? ?
i?1,...,n AFi
Example
A F1 p10 A F2 q10, q20 A
F1 ? F2 p1 q1 0, p1 q20
15
Algebraic Interpretation
3/ Negation
A ?F ? p0?AF (p ?p - 1)
Example
A F1 p10 A ?F1 p1 ? p1 - 1 0

16
Applications

• Origami Construction

Stepwise construction where each step is a fold
operation that satisfies one of Huzitas
axioms. Geometric constraints are defined to
specify origami geometric properties.
formulate origami axioms.
17
Applications
18
Applications
Example trisecting the angle ?FEG
(O1) Construction of the edges of ?FEG c
Constraintk ? Line, ThruQE, G, k s
SolveConstraintc Output k ?Line-1, 1,
-1 (O3) Trisection of ?FEG flines x,
y/.SolveConstraintConstraintx ? Line,
y?Line, yBringLineQEF, x?xBringLineQEG,
y Outputs Three numeric solutions
Second case is trisection of the internal angle
?FEG.
19
Applications

• Origami Theorem Proving

• Collecting the logical constraints of the origami
  construction.
• Translating the logical constraints into
  polynomials and then into algebraic equations
  Premises P.
• Translating the conclusion into algebraic form
  Conclusion C.
• Proving using theorem proving methods. P gt C.

20
Conclusion

• Logical and algebraic formulation of Huzitas
  axioms.
• Constraint solving Theorem proving.
• Future improvements

Generalization of logical formulation.
Optimization of number of variables and
polynomials.
21
Thanks for your attention
22
Algebraic Interpretation
Example Axiom (O2)
A ?P, Q ? Point ?k ? Line SymmetricPointP, k
Q