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- 1. I. History of Origami II. Terms related to origami III. Origami and Mathematics (Some neat theorms ) IV. Constructing Polygons (Yet another neat theorem) V. Constructing Polyhedra (Modular Origami) VI. Other ways how maths is used in origami
- 2. •In ancient Japanese ori literally translates to folded while gami literally translates to paper. Thus the term origami translates to folded paper 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. Early origami was only performed during ceremonial occasions (i.e. weddings, funerals, etc.).
- 3. FLAT FOLD – An origami which you could place flat on the ground and compress without adding new creases. CREASE PATTERN – The pattern of creases found when an origami is completely unfolded. MOUNTAIN CREASE – A crease which looks like a mountain or a ridge. VALLEY CREASE – A crease which looks like a valley or a trench. VERTEX – A point on the interior of the paper where two or more creases intersect.
- 4. The difference between the number of mountain creases and the number of valley creases intersecting at a particular vertex is always…
- 5. • The all dashed lines represent mountain creases while the dashed/dotted lines represent valley creases. 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
- 6. 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
- 7. 1. Take your piece of paper and fold it into an origami so that the crease pattern has only one vertex. 2.Take the flat origami with the vertex pointing towards the ceiling and fold it about 1½ inches below the vertex. 3.What type of shape is formed when the “altered” origami is opened? polygon 4. As the “altered” origami is closed, what happens to the interior angles of the polygon? 5. Some get smaller – Mountain Creases Some get larger – Valley Creases polygon
- 8. When the “altered” origami is folded up, we have formed a FLAT POLYGON whose interior angles are either: 0° – Mountain or 360° – Valley Creases 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
- 9. s side Shape Angle sum 180 3 5 180(5) – 360 or 540
- 10. What is the sum of the interior angles of any polygon? SIDES n SHAPE ANGLE SUM (180n – 360)° or 180(n – 2)°
- 11. So, we have that the sum of all of the interior angles of any polygon with n sides is: 180(n – 2) 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. Equating both of these expressions we get: 180(n – 2) = 0M + 360V Recall that n = M + V. So, we have: 180(M + V – 2) = 0M + 360V 180M + 180V – 360 = 360V 180M – 180V = 360 M–V=2
- 12. 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 This completes our proof!
- 13. POLYHEDRON – A solid constructed by joining the edges of many different polygons. (Think 3-Dimensional polygon.)
- 14. 0 SONOBE – A flat origami which when pieced together with identical SONOBE units can be used to modularly construct polyhedra.

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