Origami & Mathematics: Fold tab A to flap B?

1. Origami & Mathematics:Fold tab A to flap B? By Joseph M. Kudrle Department of Mathematics/Statistics University of Vermont

2. Content • Introduction • History of Origami • Origami and Mathematics (Some neat theorems) • Constructing Polygons (Yet another neat theorem) • Constructing Polyhedra (Modular Origami)

3. Objectives: What I want you to get out of this talk. • An appreciation of the art of origami. • An appreciation of the mathematics that are found in the art of origami. • Some “wicked cool” origami models that you can show off to all of your friends.

4. History of Origami

5. History of Origami • Origami – In ancient Japanese ori literally translates to folded while gami literally translates to paper. Thus the term origami translates to folded paper.

6. History of Origami • Origami has roots in several different cultures. The oldest records of origami or paper folding can be traced to the Chinese. The art of origami was brought to the Japanese via Buddhist monks during the 6th century. • The Spanish have also practiced origami for several centuries.

7. History of Origami • Early origami was only performed during ceremonial occasions (i.e. weddings, funerals, etc.). • Traditionally origami was created using both folds and cuts, but modern rules established in the 1950’s and 1960’s state that only folds shall be allowed.

8. History of Origami • Early origami was simple in form and fold. A classic example is the crane.

9. History of Origami

10. History of Origami • Modern origami still utilizes the same ideas found in the traditional models; however, the folds are becoming increasingly more difficult. • Some modern origami model’s folds are highly kept secrets and can take hours and hours for an experienced folder to complete.

11. Origami & Mathematics:Some neat theorems

12. Terms • FLAT FOLD – An origami which you could place flat on the ground and compress without adding new creases.

13. Terms • CREASE PATTERN – The pattern of creases found when an origami is completely unfolded.

14. Terms • MOUNTAIN CREASE – A crease which looks like a mountain or a ridge. • VALLEY CREASE – A crease which looks like a valley or a trench.

15. Terms • VERTEX – A point on the interior of the paper where two or more creases intersect.

16. Maekawa’s Theorem(1980’s) The difference between the number of mountain creases and the number of valley creases intersecting at a particular vertex is always… 2

17. Example of Maekaw’s Theorm • The all dashed lines represent mountain creases while the dashed/dotted lines represent valley creases.

18. Maekawa’s Theorem(1980’s) • Let M be the number of mountain creases at a vertex x. • Let V be the number of valley creases at a vertex x. • Maekawa’s Theorem states that at the vertex x, M – V = 2 or V – M = 2

19. Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) • Note – It is sufficient to just focus on one vertex of an origami. Let n be the total number of creases intersecting at a vertex x. If M is the number of mountain creases and V is the number of valley creases, then n = M + V

20. Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) Take your piece of paper and fold it into an origami so that the crease pattern has only one vertex.

21. Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) Take the flat origami with the vertex pointing towards the ceiling and cut it about 1½ inches below the vertex.

22. Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) What type of shape is formed when the “altered” origami is opened? POLYGON

23. Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) How many sides does it have? : where n is the number of creases intersecting at your vertex. n sides

24. Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) As the “altered” origami is closed, what happens to the interior angles of the polygon? Some get smaller – Mountain Creases Some get larger – Valley Creases

25. Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) When the “altered” origami is folded up, we have formed a FLAT POLYGON whose interior angles are either: 0° – Mountain Creases or 360° – Valley Creases

26. Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) • Recap – Viewing our flat origami we have an n-sided polygon which has interior angles of measure: 0° – M of these 360° – V of these Thus, the sum of all of the interior angles would be: 0M + 360V

27. SIDES SHAPE ANGLE SUM 3 Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) What is the sum of the interior angles of any polygon? 180°

28. Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) What is the sum of the interior angles of any polygon? 180°(4) – 360° or 360°

29. Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) What is the sum of the interior angles of any polygon? 180°(5) – 360° or 540°

30. Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) What is the sum of the interior angles of any polygon? 180°(6) – 360° or 720°

31. Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) What is the sum of the interior angles of any polygon? (180n – 360)° or 180(n – 2)°

32. Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) So, we have that the sum of all of the interior angles of any polygon with n sides is: 180(n – 2)

33. Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) But, we discovered that the sum of the interior angles of each of our FLAT POLYGONS is: 0M + 360V where M is the number of mountain creases and V is the number of valley creases at a vertex x.

34. Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) Equating both of these expressions we get: 180(n – 2) = 0M + 360V Recall that n = M + V.

35. Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) So, we have: 180(M + V – 2) = 0M + 360V 180M + 180V – 360 = 360V 180M – 180V = 360 M – V = 2

36. Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) • Note – If we had directed the vertex of our origami towards the ground when we made our cut, the end result would be: M – V = -2 or V – M = 2

37. Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) Thus, we have shown that given an arbitrary vertex x with M mountain creases and V valley creases, either: M – V = 2 or V – M = 2

38. Proof ofMaekawa’s Theorem(Jan Siwanowicz – 1993) This completes our proof!

39. Corollary to Maekawas Theorem(Unknown Date) The number of creases at a particular vertex on a CREASE PATTERN of a FLAT ORIGAMI must always be: EVEN

40. Proof of Corollary(Thomas Hull – 1990’s) Let M be the number of mountain creases and let V be the number of valley creases at a vertex x. Maekawa’s Theorem states that, M – V = 2 or V – M = 2

41. Proof of Corollary (Thomas Hull – 1990’s) Let n be the total number of creases intersecting at a vertex x. If M is the number of mountain creases and V is the number of valley creases, then n = M + V

42. Proof of Corollary (Thomas Hull – 1990’s) Using some tricky algebra we get: n = M + V n = (M – V )+ 2V

43. Proof of Corollary (Thomas Hull – 1990’s) Now apply Maekawa’s Theorem. n = (2)+ 2V = 2(1 + V ) or n = (-2) + 2V = 2(-1 + V )

44. Proof of Corollary (Thomas Hull – 1990’s) Both 2(1 + V ) and 2(-1 + V ) are even numbers. This completes the proof.

45. Polygons and Paper Folding

46. Polygon Theorem(Author - Date Unknown) Any polygon drawn on a sheet of paper can be extracted from the paper by only one cut, provided the paper is folded into a proper flat origami.

47. Polygon Theorem(Author - Date Unknown) • Regular Polygon – A convex polygon where all sides have equal measures and all interior angles have equal measures. • Test the theorem on your set of regular polygons.

48. Polygon Theorem(Author - Date Unknown) Challenge Question: What is the least number of folds that it takes to extract a regular n sided polygon? Contact me if you get an answer…I will keep working on it myself. jkudrle@cem.uvm.edu

49. Constructing Polyhedra

50. Terms • POLYHEDRON – A solid constructed by joining the edges of many different polygons. (Think 3-Dimensional polygon.)