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

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

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.

History of Origami

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.

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.

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.

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

History of Origami

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.

Origami & Mathematics: Some neat theorems

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

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

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.

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

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… Maybe reword? 2

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

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

Proof of Maekawa’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

Proof of Maekawa’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.

Proof of Maekawa’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.

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

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

Proof of Maekawa’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

Proof of Maekawa’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

Proof of Maekawa’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 Make note...when polygon is flattened!!!!

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

Proof of Maekawa’s Theorem (Jan Siwanowicz – 1993) What is the sum of the interior angles of any polygon? SIDES SHAPE ANGLE SUM 4 180°(4) – 360° or 360°

Proof of Maekawa’s Theorem (Jan Siwanowicz – 1993) What is the sum of the interior angles of any polygon? SIDES SHAPE ANGLE SUM 5 180°(5) – 360° or 540°

Proof of Maekawa’s Theorem (Jan Siwanowicz – 1993) What is the sum of the interior angles of any polygon? SIDES SHAPE ANGLE SUM 6 180°(6) – 360° or 720°

Proof of Maekawa’s Theorem (Jan Siwanowicz – 1993) What is the sum of the interior angles of any polygon? SIDES SHAPE ANGLE SUM n (180n – 360)° or 180(n – 2)°

Proof of Maekawa’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)

Proof of Maekawa’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.

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

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

Proof of Maekawa’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

Proof of Maekawa’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

Proof of Maekawa’s Theorem (Jan Siwanowicz – 1993) This completes our proof!

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

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

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

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

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 )

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

Polygons and Paper Folding

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.

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.

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

Constructing Polyhedra

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

Terms STELLATED POLYHEDRON – Created by taking the edges of a polyhedron and extending them out into space where they eventually intersect with each other and close in a new polyhedron.

Terms SONOBE – A flat origami which when pieced together with identical SONOBE units can be used to modularly construct polyhedra.

Terms Note: There is no distinct way that a SONOBE must be formed. According to M. Mukhopadhyay only three properties must be met in order for an origami to be considered a SONOBE. The unit must have two points. The unit must have two pockets. A point must be able to fit within a pocket.

Creating a SONOBE Start with the white side up and do a valley fold horizontally in the middle of the paper. Open the paper back up after the fold.

Creating a SONOBE Now take the paper and valley fold horizontally in-between the last valley fold and the ends of the paper.

Creating a SONOBE Mentally label the corners 1, 2, 3, and 4 as I have done. Valley fold corners 1 and 3 as shown below.

Creating a SONOBE Now valley fold the corners once more as shown…think of the type of fold used to make certain paper air-planes.

Creating a SONOBE Now refold the paper along the valley folds done in step (2).

Creating a SONOBE Now valley fold the corners mentally labeled 2 and 4.

Creating a SONOBE You can tuck the corner flap underneath the “air-plane” fold.

Creating a SONOBE Now flip the paper over and valley fold the two points in to make a square.

Creating a SONOBE Finally mountain fold the paper down the diagonal found between the two flaps.

Creating a SONOBE When unfolded the final SONOBE should look like this.

BUILDING A CUBE This requires 6 SONOBE pieces. Fit a point of one piece into a pocket of another piece.

BUILDING A CUBE Fit a point of another piece into the inside pocket of the previously added piece.

BUILDING A CUBE Fold the flap of the original piece over and tuck it into the pocket of the newly added piece. A pyramid should be formed.

BUILDING A CUBE Using two more pieces and one of the existing points, attach another pyramid.

BUILDING A CUBE Fold the pyramids along the common side and tuck in the appropriate points. It should basically resemble a cube with two flaps at the top.

BUILDING A CUBE Now add in the last piece. Again, place points into pockets and pockets over points. Here’s our cube!

BUILDING A STELLATED OCTAHEDRON This requires 12 SONOBE pieces. Do steps 1 to 4 that were done in the construction of the cube.

BUILDING A STELLATED OCTAHEDRON Now add another pyramid by using one of the existing pieces and two new pieces.

BUILDING A STELLATED OCTAHEDRON Using one more piece and two of the existing pieces form another pyramid so that 4 pyramids are clustered around a common apex.

BUILDING A STELLATED OCTAHEDRON Flip the structure over and using one of the existing flaps and two new pieces add yet another pyramid. Do the same for the opposite flap.

BUILDING A STELLATED OCTAHEDRON Now just close up the structure. Match points with flaps so that around every apex you have 4 pyramids.

BUILDING OTHER POLYHEDRA If you cluster 5 pyramids around a common apex then you will create a STELLATED ICOSAHEDRON (made up of 20 pyramids!). How many pieces will you need? Why?

BUILDING OTHER POLYHEDRA You can mix and match the number of pyramids that you cluster around a common apex. By doing this you can create several really strange polyhedra. Here’s a construction consisting of three pieces. I call it “FRANKENSTEIN’S CUBE”.

GO NUTS! Try building a strange polyhedron, or try constructing a stellated icosahedron. It’s fun and very relaxing…I tend to fold while listening to soothing music. If you make something really neat send me a picture and I can put it up on my webpage. Again, my email is jkudrle@cem.uvm.edu.