Polygonal Medallion

Class-1 -- Pentagonal Medal - I --

 ```In this class we make a pentagon medal with an internal star at the center. To help the attendant to understand the folding diagram, a rectangular vellum sheet is used in a practice folding exercise. Result shown in Fig 1-1. The final target is to make a nice art piece as shown in Fig 1-2 using a paper with folding lines and five axis symmetric pattern printed. ```
 Fig.1-1 Vellum Practice Fold result          click to enlarge & print Fig.1-2 Final Fold appearance          click to enlarge & print

Explanation of folding process using a Vellum paper

Step-1 : Cut out a pentagon from square or rectangular paper

 ```The first step is to cut out a pentagon from a square or rectangular paper. Here a commercially available light weight vellum paper is used. The procedure is shown in the following web site. "How to Make a Decagon" URL: http://origami.oschene.com/cp/DEcagon%20SCP.pdf by Philip Chapman-Bell . The reason I like this method is that most of the lines created in the process is used effectively when this model is folded. At the end of the original PDF file, a small drawing is added to show the discrepancy of this method against the exact pentagon geometry. ```
 Fig. 1-3   Make pentagon Steps          click here to enlarge

Step-2 : Fold the vellum paper according to the instructions

 ```For clarity purposes, the front side of the paper is shown in white, and back side in black. The whole process is described in 16 steps on 2 pages as shown in Fig 1-4A & Fig 1-4B. In the drawing "red" color denotes "mountain (Yama) fold", and "blue", "valley (Tani)". ```
 Fig. 1-4A   Page-1          click here to see PDF Fig. 1-4B   Page-2          click here to see PDF

Fold the paper with the printed diagram

 ```First fix the printed paper atop a heavy weight paper using tape. Then score all the red and blue lines using a ruler and a ball point pen. After scoring is done, peel off the scored paper. Using a scissor, cut along the outer pentagon boundary lines. Precrease mountain fold lines, then valley fold lines. Construction is similar to the process used in the vellum exercise session. ```
 Fig. 1-5A   Spiral          click here to enlarge Fig. 1-5B   Folding Diagram          click here to enlarge

Note: How this paper is prepared

 ```If you observe the two pictures above (Fig 1-5A & 5B), it is clear that Fig 1-5A is a composite of one symmetric image and Fig 1-5B. This suggests that anyone can create his own folding diagram if Fig 1-5B is printed over any image available. ```

Class-2 -- Pentagonal Medal - II --

 ```In this class we make a pentagon medal with an opening at the center. To help the attendant to understand the folding diagram, a rectangular vellum sheet is used in a practice folding exercise. Result shown in Fig 2-1. The final target is to make a nice art piece as shown in Fig 2-2 using a paper with folding lines and five axis symmetric pattern printed. ```
 Fig.2-1 Vellum Practice Fold result          click to enlarge & print Fig.2-2 Final Fold appearance          click to enlarge & print

Explanation of folding process using a Vellum paper

Step-1 : Cut out a pentagon from square or rectangular paper

 ```The first step is to cut out a pentagon from a square or rectangular paper. Here a commercially available light weight vellum paper is used. The procedure is shown in the following web site. "How to Make a Decagon" URL: http://origami.oschene.com/cp/DEcagon%20SCP.pdf by Philip Chapman-Bell . The reason I like this method is that most of the lines created in the process is used effectively when this model is folded. At the end of the original PDF file, a small drawing is added to show the discrepancy of this method against the exact pentagon geometry. ```

 Fig. 2-3   Make pentagon Steps          click here to enlarge

Step-2 : Fold the vellum paper according to the instructions

 ```For clarity purposes, the front side of the paper is shown in white, and back side in gray. The whole process is described in 16 steps on 2 pages as shown in Fig 2-4A & Fig 2-4B. In the drawing "red" color denotes "mountain (Yama) fold", and "blue", "valley (Tani)". ```
 Fig. 2-4A   Page-1          click here to see PDF Fig. 2-4B   Page-2          click here to see PDF

Fold the paper with the printed diagram

 ```The paper prepared for this model has both front and back printed. The pattern on one side (Fig 2-5A) is a spiral pattern with multiple concentric circles and this is the outside pattern in the finished model. The other side (Fig 2-5B) has folding lines and the colored sections ,which is the interior pattern of the model. First fix the printed paper atop a heavy weight paper using tape. Then score all the red and blue lines using a ruler and a ball point pen. After scoring is done, peel off the scored paper. Using a scissor, cut along the outer pentagon boundary lines. Precrease mountain fold lines, then valley fold lines. Construction is similar to the process used in the vellum exercise session. ```
 Fig. 2-5A   Spiral Pattern          click here to enlarge Fig. 2-5B   Inside Pattern & Diagram          click here to enlarge

Note: How this paper is prepared

 ```It is very important that the centers of the two images above must be located very close. If not, the symmetry of the folded model is lost and the overall looks is not satisfactory to the eyes. ```

Class - 3 -- Gleaner's Joy - Finding a Jewel --

 There are 2 basic ways to accomplish "Twist Fold" though there are many variations based on them.  Here, as an exmple, a regular pentagon is chosen for the purpose of discussion, but the same idea   can be applied to any "N"-gons.  Two pentagons, outer and inner, are arranged so that the line connecting the center point   and each apex of the inner pentagon passes through ,either the apex of the outer pentagon (Type-I) or   the mid-point of the outer pentagon's side (Type-II).  We also observe that in both cases angles are divided into 4 equal parts, at each mid-point of the side (Type-II)  or at each apex (Type-I).  So the each angle is    Type-I       108/4 = 27 degrees    Type-II      180/4 = 45 degrees   Typical diagrams and their dimensions are shown in the following figures.
 Fig. 3-1A Type -I diagram          click to enlarge & print Fig. 3-1B Type-I appearance          click to enlarge & print
 Fig. 3-2A Type-II diagram          click to enlarge & print Fig. 3-2B Type-II appearance          click to enlarge & print
 ```Here we will discuss how to apply some variations to these base model and obtain the aesthetically satisfying models . A few examples are shown in Fig 3-3A & 3B. Fig 3-3A: note that radii of the star in front and pentagon in the back are the same length. Fig 3-3B: note two overlapping pentagons at the center area. ```
 Fig. 3-3A Model (I)          click to enlarge & print Fig. 3-3B Model (II)          click to enlarge & print
 ```Here we will discuss how to apply some variations to these base model and obtain the aesthetically satisfying models . A few examples are shown in Fig 3-3A & 3B. Fig 3-3A: note that radii of the star in front and pentagon in the back are the same length. Fig 3-3B: note two overlapping pentagons at the center area. ```
 Fig.3-4A  Type-I Dimension          click to enlarge & print Fig. 3-4B  Type-II Dimension          click to enlarge & print

Using the dimensional drawing above, it is possible to create folding diagrams for multiple values of radii
of circles circumscribing the inner pentagons. The results are shown below.

Type-I Twist

 Fig. 3-5A  Type-I Group-Front Face          click to enlarge & print Fig. 3-5B  Type-I Group-Back Face          click to enlarge & print

Diagrams for Type-I Group

 Fig. 3-6A  Type-I "B" diagram          click to enlarge & print Fig. 3-6B  Type-I "C" diagram          click to enlarge & print
 Fig. 3-6C  Type-I "D" diagram          click to enlarge & print Fig. 3-6D  Type-I "E" diagram          click to enlarge & print

Type-II Twist

 Fig. 3-7A  Type-II Group-Front Face          click to enlarge & print Fig. 3-7B  Type-II Group-Back Face          click to enlarge & print

Diagrams for Type-II Group

 Fig. 3-8A  Type-II "B" diagram          click to enlarge & print Fig. 3-8B  Type-II "C" diagram          click to enlarge & print
 Fig. 3-8C  Type-II "D" diagram          click to enlarge & print Fig. 3-8D  Type-II "E" diagram          click to enlarge & print
 Fig. 3-8E  Type-II "F" diagram          click to enlarge & print Fig. 3-8F  Type-II "G" diagram          click to enlarge & print
 ``` Observing these 2 figures,we notice that 1. In Type -I, somewhere between "C" and "D" ,the radii of the base pentagon and five-pointed-star become identical. 2. In Type - II, between "F" and "G", the red colored tip will be partially hidden under the center pentagon. How can we find the exact location ? ```

Model - I

 Fig. 3-9A  Dimension for "B" & "D"          click to enlarge & print Fig. 3-9B  Parameter Graph          click to enlarge & print
 ``` The diagrams for case "B" and "D" are drawn on the same drawing. In the drawing, "B" "D" radius of the Star : 0.702 2.351 radius of the base pentagon : 2.038 1.019 This set of data are plotted as a graph , Star Radius ( red ) and Base Radius ( blue ). Two lines intersects at radius = 1.527, this is the radius of the diagram to be used. A new diagram is created and this is pasted over the separately prepared image file. Both are shown in the figures below. ```
 Fig. 3-10A  Model-I base diagram          click to enlarge & print Fig. 3-10A  Model Folded          click to enlarge & print Fig. 3-10B  Example Diagram          click to enlarge & print

Model - II

 Fig. 3-11A  Dimension for "F" & "G"          click to enlarge & print Fig. 3-11B  Parameter Gra[h          click to enlarge & print
 ``` The diagrams for case "F" and "G" are drawn on the same drawing. In the drawing, "F" "G" length of "X" : 2.2546 2.2301 length of "Y" : 1.6832 1.2074 diameter of the base pentagon 1.0938 0.7846 ( X - D) 1.1608 1.4455 ( X - D) - Y -0.5224 0.2381 This set of data are plotted in a graph , parameters shown in different colors. Note the values of ( X - D) - Y. They are the distance of overlapping we are looking for. The point where RED colored line intersects with ordinate value = 0 is the point where the tip of the star touches the base pentagon. For the actual choice of base pentagon dimension, this point is moved slightly to the left to be safe. Its value is 0.958. The diagram using this diameter is shown below. ```
 Fig. 3-12A  Model-II base          click to enlarge & print Fig. 3-12B  Folded Model          click to enlarge & print
 ``` The final diagram for Model-II is created by inserting this diagram on the separately prepared five axis symmetric image drawing. For this exercise another pattern is printed on the back side. Notice that the pattern on the back side is the mirror image of the same pattern with respect to the horizontal axis through the center. ```
 Fig. 3-13A   Pattern Diagram - Front          click here to enlarge Fig. 3-13B  Pattern Diagram - Back          click here to enlarge

References

1. Montroll, John.: Galaxy of Origami Stars,ISBN-10:1480103047,John Montroll , 2012 .
2. Project F.: Origami - Nejiri Ori (in Japanese). Seibundo-Shinkosha,Tokyo,ISBN978-4-416-31200-1,2012 .
Note: Project F: group of three persons-Satoko Saito, Taiko Niwa, Tomoko Fuse