Polygonal Medallion

## Class-1 -- Floral Medallion --

In this class , we make three floral medallions using the papers provided. The final results and the diagrams used are shown below.
 Fig.1-1 Final appearance          click to enlarge & print

### Basic Twist Diagram

The diagrams are created by inserting the following twist fold pattern drawings on top of the flower images.
Notice that the paper to be used has a circular boundary.

## Class - 2 -- Polygonal Medallion -Spirals --

In this class , we make two spiral medallions ,pentagon and hexagon , as shown below.
 Fig. 2-1A  Pentagonal Spiral          click to enlarge & print Fig. 2-1B  Hexagonal Spiral          click to enlarge & print

### Pentagon spiral Model Pattern

The pentagonal spiral is made up of the multiple pieces shown below.
 ``` The final bare minimum folding diagram and its dimensions are shown in the following figures. In the figure with dimensions, there are two center circles. Outer circle is the circle for the pentagon resulted from folding , and the inner one is for the inscribing pentagon . The ratio is the reduction scale factor. The ratio is 0.8416 (1.683 / 2.0) for the pentagon case. Then applying this ratio repeatedly, we will have folding diagrams with constant reduction factor. ```

### Pentagon Basic Pattern & Dimension

 Fig. 2-3A  Pentagon Basic Pattern          click to enlarge & print Fig. 2-3B  Pentagon Basic Pattern Dimension          click to enlarge & print
The diagrams are as follows:

### Hexagon Spiral Model Pattern

The hexagonal spiral is made up of the multiple pieces shown below.
 ``` The final bare minimum folding diagram and its dimensions are shown in the following figures. In the figure with dimensions, there are two center circles. Outer circle is the circle for the hexagon resulted from folding , and the inner one is for the inscribing hexagon . The ratio is the reduction scale factor. The ratio is 0.73205 (1.464 / 2.0) for the hexgon case. Then applying this ratio repeatedly , we will have folding diagrams with constant reduction factor. ```
 Fig. 2-6A  Hexagon Twist Base          click to enlarge & print Fig. 2-6B  Hexagon Twist Basic Dimension          click to enlarge & print
The diagrams are as follows:

## Class - 3 -- CAD Usage for Twist Fold --

 Fig. 3-1A Model (I)          click to enlarge & print Fig. 3-1B Model (II)          click to enlarge & print
 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-II) or   the mid-point of the outer pentagon's side (Type-I).  We also observe that in both cases angles are divided into 4 equal parts, at each mid-point of the side (Type-I)  or at each apex (Type-II).  So the each angle is    Type-I       180/4 = 45 degrees    Type-II      108/4 = 27 degrees   Typical diagrams and their dimensions are shown in the following figures.
 Fig.3-2A  Type-I Dimension          click to enlarge & print Fig. 3-2B  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. 3-3A  Type-I Group-Front Face          click to enlarge & print Fig. 3-3B  Type-I Group-Back Face          click to enlarge & print

### Diagrams for Type-I Group

 Fig. 3-4A  Type-I "B" diagram          click to enlarge & print Fig. 3-4B  Type-I "C" diagram          click to enlarge & print
 Fig. 3-4C  Type-I "D" diagram          click to enlarge & print Fig. 3-4D  Type-I "E" diagram          click to enlarge & print

### Type-II Twist

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

### Diagrams for Type-II Group

 Fig. 3-6A  Type-II "B" diagram          click to enlarge & print Fig. 3-6B  Type-II "C" diagram          click to enlarge & print
 Fig. 3-6C  Type-II "D" diagram          click to enlarge & print Fig. 3-6D  Type-II "E" diagram          click to enlarge & print
 Fig. 3-6E  Type-II "F" diagram          click to enlarge & print Fig. 3-6F  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. 3-7A  Dimension for "B" & "D"          click to enlarge & print Fig. 3-7B  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. 3-8A  Model-I base diagram          click to enlarge & print Fig. 3-8B  Model-I Diagram          click to enlarge & print

### Model - II

 Fig. 3-9A  Dimension for "F" & "G"          click to enlarge & print Fig. 3-9B  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. 3-10  Model-II base          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. ```