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Polygonal string - paper strip origami

Polygonal string ( make rings of regular polygon using strings of paper)

Note from the author to 2016 OrigamiUSA attendants:
Chapter 1-a: How to fill the cnter piece
&
Chapter 3-a: Seven pointed star in the center
are added to help the class attendants .

Table of Content

        Basic Idea
    1. Pentagon (N = 5) case
    1-a. How to fill the center piece
    2. Hexagon (N = 6) case
    3. Heptagon (N = 7) case
    3-a. Seven pointed star in the heptagon center
    4. Octagon (N = 8) case
    5. Nonagon (N = 9) case
    6. Decagon (N = 10) case
    7. Dodecagon (N = 12) case
    8. Tools required
    9. References

Basic Idea - Polygonal Knot

It is well known that if a strip of paper is knotted once (Fig. 1) and carefuly pressed flat , the folds
will form a regular pentagon (Fig. 2) . And this is known as "Polygonal knots".(ref 1 )
But so far the author has not found any publication which reports what happens if the process
is repeated on the same paper strip.
The author found that the result is a ring made up of either 5 or 10 unit pentagons.
Five unit pentagons case was shown in Fig. 3.
A example shown below is done by using 3/8 inch "quilling paper" strip .
     Fig.1 A simplest knot
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     Fig.2 Resulting Pentagon
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     Fig.3 After 5 repetitions
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1. Pentagon Case (N = 5)

Step by step instructions of making the simplest pentagon ring

A simple hand-on exercise will help the readers to understand the concept of this new "paper" pastime.
Objective: create a ring of 5 polygons (Fig. 5) using paper strips drawing in Fig. 4.

           Fig.4:  Paper Strings used
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         Fig. 5:  Resulting ring
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    Fig. 6   Step 2:
Inside boundary lines on the back
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         Fig. 7  Step 3:
      Cutting out paper strips
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         Fig. 8: 
     Connecting strips #1 & #2
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           Fig.9: 
     All strips connected
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         Fig.10:  Step 6a:
       Make crease lines on #1
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         Fig.11:  Step 6b:
      Make crease lines on #2
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           Fig.12: 
     After step 6a & 6b
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         Fig.13: 
     After processing #5 strip
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         Fig.14: 
     Finished look
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Step 1  Print out paper strip image shown in Fig. 4.
Step 2  There are two types of broken lines,on outside boundarties and inside with numbers.
       On the back of the printed paper, mark the location of the inner broken lines by 
       a pencil.( Fig. 6 ) A light box is a handy tool if available, but a window pane does  
       the job too.
Step 3  Using a very sharp blade,cut out 5 pieces of strips with the same width. (Fig. 7 )   
Step 4  Take strip #1 and #2 ,lay the left back side of #2 strip on the right edge of 
       #1 strip. Make sure that the dotted lines coincide.(Fig. 8 )
Step 5  Repeat this "glueing front to back" process for Strips #3,4,& 5.
        The result is shown in Fig. 9 .
Step 6a  Press an old ball point pen cartridge along the 3 solid lines on Tape #1. 
        (Fig. 10) Then apply "Yama Ori" for the creased lines. 
        The third folding must go through under two layers of strip. The result is shown 
        in Fig. 10.
Step 6b  Repeat the same Process on Tape #2 (Fig. 11). The result is shown in Fig. 12.
Step 7  Repeat the same process on tape #3 , 4 & 5. The result is shown in Fig. 13.
       You will see both the ends of #1 and #5 are sticking out. 
       Using the friction between your fingers and the paper strip, push them 
       into each other slowly. Cutting the corners of edges will make this process easier.
        Despite no glue usage during the folding stage ,the final product looks
       very stable. Fig. 14

Step by step instructions of making a ring of 10 pentagons (#1)

Objective: create a ring of 10 polygons (Fig. 16) using paper strips drawing in Fig. 15.

           Fig.15:  Paper Strings used
   click here to enlarge     open PDF for print
ten_pentagon_strip-8.5x11-2004
         Fig. 16:  Resulting ring
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ten_pent_final

    Fig. 17   Step 1-2:
    Fold the right hand side
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ten_pent_step1-2
         Fig. 18  Step 3:
Left tape through the pentagon
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ten_pent_step-3
         Fig. 19: Step 4:
     Make the left pentagon
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ten_pent_step-4
           Fig.20: Step 5:
     Connect two units
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ten_pent_step-5
         Fig.21:  Step 6:
       Connect the 5-th unit
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ten_pent_step-5b
         Fig.22:  Final:
      Final look
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ten_pent_final

Step 1  Print out paper strip image shown in Fig. 15.
       Mark the crease line by a ball point,then fold along the lines only locally. 
       "center" line is "Yama" ori ,and "dotted" line is "Tani" ori.
Step 2  Fold the right hand side. See Fig. 17.
Step 3  Put the LHS tape through the RHS pentagon's slot and pull it while holding right 
       hand side. (Fig. 18 ) 
Step 4  Make the left pentagon. Two pentagons unit is complete. (Fig. 19 ) 
Step 5  Repeat the process and make one more unit .(Fig. 20 )
       Connect these 2 units by inserting into each other. 
Step 6  Insert the final unit into 4 connected units . (Fig. 21)

Step by step instructions of making a ring of 10 pentagons (#2)

Objective: create a ring of 10 polygons (Fig. 28) using paper strips drawing in Fig. 23-a & b.

           Fig.23-a:  Paper Strings used #1
    click here to enlarge     Open PDF for print
ten_pent_unit-2004
           Fig.23-b:  Paper Strings used #2
    click here to enlarge     Open PDF for print
ten_pent_unit-a-2004

       Fig. 24   Step 1:
  Make the "green"pentagon.
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pent2_step-1
         Fig. 25  Step 2:
   Prepare the second strip.
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pent2_step-2
         Fig. 26: Step 3-a:
     Insert short end strip
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pent2_step-3a
           Fig.27: Step 3-b:
     Insert longer strip
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pent2_step-3b
         Fig.28:  End:
       Final Look
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pent2_final

Step 1  After printing out Fig.23-a, connect 5 strips as it is done in the simple 
       pentagon case.The difference is that there is a gap between pentagons.
       Then construct a pentagon ring as shown in Fig. 24. All the fold is "Yama-Ori".    
       When properly done, the same letters ,either "R" or "L"  must show up 
       between pentagons. The side of "R" is temporarily called the "front" face.
Step 2  After printing out Fig.23-b, connect 5 strips as it is done in step 1. 
       See Fig. 25.
Step 3-a  Make crease lines on #1 section of the pink strip, and do Yama-ori.
         This will cover the space between "green" pentagons. Short end goes down 
         the slot in the back face.(Fig. 26 )
Step 3-b  The longer strip will go upward through slots on the front face (Fig. 27).  
Step 4  Repeat the step-3b process for the rest of the strip. 
       The final look is as shown in Fig. 28.

Multiple ways to cover a regular polygon by constant width tape

                    Fig.29:  Regular Heptagon Case
                           click here to enlarge
regular_heptagon

 Let us consider how to wrap around general regular polygon by a constant width tape.
As an example, a regular heptagon as shown in Fig. 29 is chosen.
Each apex can be connected to other 3 apexes. Then observing the resulting network of lines,
it is easy to figure out that there are three different modes of tape wrapping.
Three tapes of width h1, h2 and h3 are used for this operation. The shaded areas in the center are
the areas which is not covered by these tapes.
In general,
The number of modes for a given polygon 2x(N+1) , or 2xN+1 is N.

For example, a pentagon (N=2) , hexagon(N=2), heptagon(n=3),octagon(N=3), nonagon(N=4), etc.
mode-1: made up of unit isosceles triangle including two sides of the polygon
mode-2: made up of unit trapezoid ,the top of which is the side of the polygon.
It is interesting to note that the simple exercise explained on the top of this article can be constructed
using mode-1 of the regular pentagon using the strip shown in Fig. 30.

           Fig.30:  Mode-1 String
   click here to enlarge     Open PDF to print
pent_mode1_string
         Fig. 31:  Resulting ring
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fig-31

The final look of the pentagon ring is shown in Fig. 31. It is clear from this picture that
it is not easy to make the boundary of lines to line up to form a neat overall pattern.
The reason is obvious.
Basically what we are trying to do is squeezing one pentagon inside the same sized pentagon.
And it is not possible how thin the paper is.
Another sring pattern is tried using the string shown in Fig 32. and its result , Fig. 33.
It was slightly easier to work on this, and the result looks a little better , but not much.

           Fig.32:  Mode-1 String
   click here to enlarge     Open PDF to print
pent_mode1_string
         Fig. 33:  Resulting ring
         click here to enlarge
fig-33


So the author came to the conclusion that " mode-1 " is not suitable to creating a nice pre-designed
overall pattern. But this seemingly a futile effort gave a very interesting hint for a new approach
to polygonal knots.

A general approach of polygon making using constant width tape

1. Motive
So far we have tried to create a pentagon ring with one particular pattern by using paper strips
so designed . But we know that it will be more difficult to design the paper strips as the number
of sides of polygon increases. This will restrict the extension of this idea to polygons of
more than 5 sides ( 6,7,8,9 ,.. ) .
Now let us go back to see how the string is made. The process will give us a hint of more general scheme
of making a variety of polygonal knots.

    Fig. 34   Step 1-2:
    Template drawing
    Enlarge      Print
pent_mode1_final
         Fig. 35  Step 3:
  Print and cut-out template
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fig-35
         Fig. 36: Step 4:
 Prepare a tracing paper tape
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fig-36
           Fig.37: Step 5:
 Cover the template by tape
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fig-37
         Fig.38:  Step 6:
  Trace template's pattern
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fig-38
         Fig.39:  Final:
 Unfold the tracing paper tape
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fig-39

1. Printout Fig. 34 with hatching and solid border lines with the scale factor 1.5.
   In the drawing the strip width is set to a unit value,so the strip width is
   1.5 inch when printed out. 
   Paste this onto any paper of higher weight index (around 60 ~ 100 lbs.).  
   Cut out the pentagon. Now we have a thick pentagon template. (Fig. 35)
2. Cut out strips of 1.5 inch wide using "tracing paper" and connect them into a strip
   long enough to cover this pentagon 2 times around by MODE-1. (Fig. 36)  
3. Using the tape, wrap around the template.  At the start and end of this process,
   "double sided tape" is used because the tape must be peeled off at the later step.
   Fig. 37
4. Now draw lines on the tracing paper strips. 
   For the hatching,  use colored pencils. (Fig. 38)
5. When boundary lines and hatching are marked, peel off string. (Fig. 39)
6. Using this tape as a reference, make folding strip drawing using your favorite CAD 
   software  The author used AutoCAD R-2013 , but any software package is fine.
   Drawing of Fig. 30 is created in this manner, and Fig. 32 is a modification based
   on Fig. 30.
2. Trials:
    mode 1

    Let us replace the tracing paper in step 4 by a synthetic decorative adhesive tape, and
   custom made tape with repeating patterns,width of which are the same 1.5 inch.
   They are shown in Fig. 40, and the results , Fig. 41 , 42 below .

           Fig.40:
     Strips used
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fig-40
         Fig.41:
       Adhesive tape case
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fig-41
         Fig.42:
      Paper tape case
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fig-42

    --Mode 2--
    Preparation of a template and its matching tape
    If we watch carefully at Fig.4 & 5, and reverse the process from Fig. 4 to Fig. 5 
   backward,we can make a pentagon ring.
       The process goes this way.
   step 1:  print the pentagon ring drawing (Fig. 43), and paste it onto a thick piece 
           of paper.The paper width in the drawing is set to one unit, so if the scale   
           factor is 1.5,this ring require a strip of 1.5 inch wide.
   step 2:  Cut along the inner & outer boundary lines, and make a template. (Fig. 44)
   step 3:  Print out the paper tape pattern drawing . The width of the strip is set to 
           1.0 , so print out with scale factor of 1.5. (Fig. 45)
   Note: The total length of tape = 39.9 X tape width       

           Fig.43:
     template used
   click here to enlarge     Print PDF
fig-43
         Fig.44:
       Cut out template
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fig-44
         Fig.45:
      Image pattern used
click here to enlargeopen PDF to print fig-45
         Fig.46:
       Strip used
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fig-46

     Folding Process
      step-4  Before the folding begins, a few tricks must be applied. 
             Triangle areas between corner pentagons must be covered by short pieces 
             of the strip. Front and rear views are shown in Fig. 47 & 48. 
             There is a reason for this operation.
             For details, refer to  Tape Weaving Section.

         Fig.47:
       Template-Front View
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fig-47
         Fig.48:
       Template-Rear View
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fig-48
         Fig.49:
       Start Folding
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fig-49
         Fig.50:
   Folding at half way point
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fig-50
         Fig.51:
  Handling of the end piece
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fig-51
         Fig.52:
       End of Folding
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fig-52

Detailed dicussions about each case

Pentagon case (N = 5)


(1) How to get a paper strip to construct pentagon ring
         Start
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         Step 1
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         Step 2
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         Step 3
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Refer to figures above .
Start
 First draw a pentagon, and add A-D, and 0-4 to vertices and edges as shown. Then connect all vertices by lines.
This will divide the pentagon into four trapezoids. ( ABCD, BCDE, CDEA, and EABC )
Extend lines CB, DA CD & BE to the outside of this pentagon.
At point "A", draw a line parallel to CD, and this line intersects the "CB" extension at point "A2".
From point "A" drop a line perpendicular to BA2, and name this point "A1".
Do similar operations on the right hand side, and mark points "E1", and "E2".
 We can view the pentagon as made up of these 4 trapezoids layered on top of the others.
(i.e. ABCD is on top of CDAE, which is on top of EABC, which is on top of BCDE)
It is also interesting to observe that
(1) Line segments AB and ED forms an angle of 36 degrees(= 360 / 10).
(2) Line segments AA1 and EE1 forms an angle of 72 degrees (= 360 / 5).
Step 1
 Let us try to peel these four trapezoids one by one startig with ABCD.
In doing so, triangle ABA2 is also added to this trapezoid.
The result is shown as Step 1 in the figure.
Step 2
A similar porcess is applied to CDEA. The result is Step 2 picture.
Step 3
Then finally Step 3 shows how it looks when all trapezoids are peeled off from the original pentagon shape.
(2). Geometry of unit strips for rings of 5 and 10 pentagons
 From the discussions above, readers will understand that pentagon case #1 and #2 can be
constructed using two kinds of strings shown in the figure left below.
         Unit strips
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         Geometry of basic element
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 Basic dimension of the trapezoidal element is shown the the figure right above.
The numbers are based on basic pentagon radius r.
  edge = 2r sin(36),   base = 2r sin(36){1 + 2sin(18)},   height = 2rsin(36)cos(18)
When a tape with given height (e.g. 1", 3/4" ,etc) is used, it is handy if these figures
are given based on the height (h).
  r = (0.8946)h,   edge = (1.0515)h,   base = (1.7013)h
(3) Pentagon string examples
    Simple geometric shapes
In the case when the pattern is of simple geometric nature, it is easy to create the pentagon string pattern
using a standard CAD software.(case #1 through #9).
The reason is that the pattern is limited within the range of one fifth segment,thus allowing the five repetitions of the same pattern.
           Pentagon Case #1
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     click here to see the paper strip
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         Pentagon Case #2
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     click here to see the paper strip
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         Pentagon Case #3
         click here to enlarge
     click here to see the paper strip
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           Pentagon Case #4
         click here to enlarge
     click here to see the paper strip
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         Pentagon Case #5
         click here to enlarge
     click here to see the paper strip
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         Pentagon Case #6
         click here to enlarge
     click here to see the paper strip
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           Pentagon Case #7
         click here to enlarge
     click here to see the paper strip
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           Pentagon Case #8
         click here to enlarge
     click here to see the paper strip
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         Pentagon Case #9
         click here to enlarge
     click here to see the paper strip
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    simple but slightly complex pattern
But when basic pattern overflows its basic pentagon region (case #10 & #11), the string pattern creation bocomes slightly complex.
The figure below illustrates how such a pattern is created into a unit string.
               How to make a pentagon string
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           Pentagon Case #10
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     click here to see the paper strip
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           Pentagon Case #11
         click here to enlarge
     click here to see the paper strip
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    more complex pattern- Fractal pattern
As long as the whole pattern is 5 axis-symmetric, any complex figure can be converted to a set of 5 same strings.
           Pentagon Case #12
         click here to enlarge
     click here to see the paper strip
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           Pentagon Case #13
         click here to enlarge
     click here to see the paper strip
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         Pentagon Case #14
         click here to enlarge
     click here to see the paper strip
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    Using commercially available printed paper
The author came across printed papers of beautiful pattern in the scrap book supply section of a craft shop.
There are hundreds of patterns and paper weight and textures, and its size is 12 inch square.
The author printed basic string line pattern on the back of the paper. Of course there are pros and cons.
Nice thing about this is the the cutting lines do not show up in the output, but the output will lose symmetrical beauty.
           Pentagon Case #15
         click here to enlarge
     click here to see the paper strip
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           Pentagon Case #16
         click here to enlarge
     click here to see the paper strip
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         Pentagon Case #17
         click here to enlarge
     click here to see the paper strip
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    Using commercially available "DUCT TAPE" or "DECORATIVE TAPE"
The author was really surprised to find out that there are many colorful "DUCT TAPE" sold at the hardware stores.
The same sized tape ,but of different material is sold as "decorative tape' at the craft shop.
The output #19 is made using the "DUCT TAPE" and #18, by "decorative tape".
The width is 1 7/8 ". Later the author found that the duct tape of 1.42 " wide is also available .
As expected making creases is more difficult than working on paper, but the result is worth the efforts.
It is sturdy, water propellant, so it may be used as a decorative coaster.
           Pentagon Case #18
         click here to enlarge
     click here to see the paper strip
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           Pentagon Case #19
         click here to enlarge
     click here to see the paper strip
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Usage of pentagon pattern templates

     The introduction of pentagon patterned templates, which are made up of 
    equilateral triangle grids, opens up a wide open area of beautiful patterns.
    Here are several examples ranging from basic to their variations.

     Pentagon template #1
         click here to enlarge
pent_temp1-200.jpg
     Pentagon template #2
         click here to enlarge
pent_temp2-200.jpg
     Pentagon template #3
     click here to enlarge     
pent_temp3-200.jpg
     Pentagon template #4
         click here to enlarge
pent_temp4-200.jpg
     Pentagon template #5
         click here to enlarge
pent_temp5-200.jpg
     Pentagon template #6
     click here to enlarge     
pent_temp6-200.jpg

More samples of pentagon based pattern

     There are many variations of pentagon based patterns. 
    The typical cases are shown below.
           sample #1
         click here to enlarge
     click to see the template
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           sample #2
         click here to enlarge
     click to see the template
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         sample #3
         click here to enlarge
     click to see the template
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   All the tape widths are the same. This is a very rare case.
   Weaving pattern:
    	Outer most is mode-1 of pentagon.
    	Middle is mode-2 of decagon.
    	Inner most is mode-4 of decagon.
    sample #1	All tape width = 3/4 inch	 synthetic decorative
    sample #2	All tape width = 15 mm		 Washi decorative	
    sample #3	All tape width = 10 mm		 Washi decorative
           sample #4
         click here to enlarge
     click to see the template
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           sample #5
         click here to enlarge
     click to see the template
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         sample #6
         click here to enlarge
     click to see the template
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  sample #4   1 inch wide custom made paper strips  see tape 1  see tape 2
sample #5 43 mm wide strip cutout from a border wall paper see the border wall paper Both top and bottom pieces will be used later for octagon cases. Advantage of using a border wall paper is the availability of long, seamless (60 ~ 80 inches) strips . sample #6 3/4 inch wide synthetic adhesive tapes. Cut the template along the solid lines to get outer & inner pieces. The inner piece cut line is offset about 1/100 inch toward the center so that both pieces will fit after taping.
           sample #7
         click here to enlarge
     click to see backside
     click to see the template
i
           sample #8
         click here to enlarge
     click to see backside
     click to see the template
i
         sample #9
         click here to enlarge
     click to see backside
     click to see the template
i
  sample #7   The template is similar to that of sample #6 except its interior pattern.
  sample #8   Mode 2 for basic pentagon with 3/4 inch silver & red adhesive tapes
  sample #9   The template is the same as sample #6 except its width (1 inch).
	      16 mm Washi tape (yellow) is added onto the interior blue tape.


           sample #10
         click here to enlarge
     click to see backside
     click to see the template
i
           sample #11
         click here to enlarge
     click to see backside
     click to see the template
i
         sample #12
         click here to enlarge
     click to see backside
     click to see the template
i
  sample #10   Similar to sample #6
  sample #11   Basic pentagon mode-2   3/4 inch adhesive tape
  sample #12   Basic pentagon mode-2   3/4 inch adhesive tape

           sample #13
         click here to enlarge
click to see the template
i
           sample #14
         click here to enlarge
click to see the template
i
         sample #15
         click here to enlarge
click to see the template
i
  sample #13   Similar to sample #6
  sample #14   Basic pentagon mode-2   3/4 inch adhesive tape
  sample #15   Basic pentagon mode-2   3/4 inch adhesive tape

1-a. How to fill the center pentagon
Most of the examples shown for the pentagon case above are without the pentagon
 in the center. When it becomes necessary to fill up that void space to make
 a complete pentagon, we have to do the following tricks. The precedure is explained 
using a sample pattern used for the 2016 OrigamiUSA convention class.


             Paper Strings used
    click here to enlarge     open JPG for print
i
           strip and center piece
   click here to enlarge     open JPG for print
i
             Finished Model : Front & Rear View
    click here to enlarge     open JPG for print
i
             Folding lines for center piece
    click here to enlarge     open JPG for print
i

             Outside pentagon ring
    click here to enlarge    
i
             Option-1 result
    click here to enlarge    
i
             Option-2 result
    click here to enlarge    
i
             Final step of option-2
    click here to enlarge    
i

Steps to make the filled up pentagon.
 step 1: cut out 5 pieces of strips. Follow the "Step by step instructions of making
     the simplest pentagon ring . This will result in the configuration as shown
     in Fig 5. There is a vacant pentagon space in the center.
 step 2:There are 2 options to make a pentagon which will fit the center.
    option 1: using a single paper strip in the figure above "strip and center piece"
            Notice the pattern differs slightly from the pentagon strip in the left above.
    option 2: apply "twist fold" to the pentagon figure in   "strip and center piece".
In order to make the folding lines clear, a separate figure is shown above.
 "mouintain fold" is shown by red color, and "valley fold", by blue.   
The resulting center pieces after folding are also shown for both cases.
One more step is required for option 2 to make the back side match the 
surrounding pattern.   Mark pencil lines along the folded edges( shown by arrows),and
make scissor cuts along the lines. Open all 5 flaps as shown in the figure.
Glue these flaps and the option-2 process is finished.        		

2. Hexagon case (N = 6)

The basic idea of hexagon knot is shown below.
      Figure A   Knot pattern by two pipe cleaners.
      Figure B   Knot by 3/8 inch quilling tape
      Figure C   quilling tapes flattened 

      Figure A
      click here to enlarge i
      Figure B
      click here to enlarge i
      Figure C
      click here to enlarge i

Hexagonal Fold trial #1

Let us try to create a hexagon ring made up of 6 hexagon units.
     The first step is to print out the drawing below. Then using the old ballpoint
    cartridge, trace the red lines. This process makes "mountain fold" easier .
    Using a very sharp pointed knife, cut out 6 pieces of strips. This ring requires
    6 hexagon units and we start the folding process following the figures from
    "Hexagon #1" through "Hexagon #6".
            Trial #1 Drawing
         click here to enlarge & print
i
       Trial #1 Result
   enlarge Front    View Back
i

Hexagon Trial #1 Folding Process

     Hexagon #1 Front
         click here to enlarge
i
     Hexagon #1 Back
         click here to enlarge
i
     Hexagon #1 End
     Front      Back
i
     Hexagon #2 Front
         click here to enlarge
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     Hexagon #2 Back
         click here to enlarge
i
     Hexagon #2 End Front
   enlarge Front    View Back
i
     Hexagon #3 Front
       click here to enlarge
i
     Hexagon #3 End
   enlarge Front    View Back
i
     Hexagon #4 Front
       click here to enlarge
i
     Hexagon #4 End
   enlarge Front    View Back
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     Hexagon #5 Front
       click here to enlarge
i
     Hexagon #5 End
   enlarge Front    View Back
i
     Hexagon #6 Front
       click here to enlarge
i
     Hexagon #6 End
   enlarge Front    View Back
i
     When printer papers of 6 different colors are used instead of a plain white color, 
    and the same process is applied , the result will be as shown in the following figure.
    Can you notice the difference ?
     One of the problem of this folding scheme is that the unit hexagon is not a complete
    hexagon which is missing just one triangular segment. The following "Trial fold #2"
    addresses this issue. It also gives a hint on how to make the predetermined hexagonal
    pattern.
         Variation of #1
         click here to enlarge
i

Hexagonal Fold trial #2

    Comparing two drawings of Trial #1 & #2 , the readers will notice that the only difference
    are the extra triangles at the right and left edges marked by the characters "B" and "E".
     And you will see these two characters filling the missing triangular spaces in Trial #2 result.
            Trial #2 Drawing
         click here to enlarge & print
i
       Trial #2 Result
   enlarge the image   
i

Hexagon Trial #2 Folding Process

      Basically the folding steps are similar to #1 case except the edges extend out 
     one more triangle units. And these extras will be folded and tucked into the
     hexagon unit. The following figures illustrate the first hexagon making process.

       step 1: Cross 2 strips just like #1 case.
       step 2: Intersect these strips to form a hexagon.
               Notice extra triangle pieces. 
       step 3: Fold back these extra pieces into the hexagon.

     Hex #2 step-1
         click here to enlarge
mod_hex_1.jpg
     Hex #2 step-2
         click here to enlarge
mod_hex_2.jpg
     Hex #2 step-3
     click here to enlarge     
mod_hex_3.jpg

Examples of Modified Hexagon Folding

      Now we know how to repeat the same hexagon pattern, a few examples will be shown.
            Example #1 Drawing
         click here to enlarge & print
i
       Example #1 Result
     enlarge the image   
i

Number of hexagon -- 3, and 12 cases

     As shown in the photos below, it is also possible to create rings made up 
    of 3 and 12 hexagon units. The paper strings required are  repetitions of 
    equilateral triangle pattern.
           Hexagon Case #1
     click here to enlarge
i
           Hexagon Case #3
         click here to enlarge
i

Usage of hexagon pattern templates

     The introduction of hexagon patterned templates, which are made up of 
    equilateral triangle grids, opens up a wide open area of beautiful patterns.
    Here are several examples ranging from basic to their variations.

     Hexagon template #1
         click here to enlarge
mod_hex_1.jpg
     Hexagon template #2
         click here to enlarge
mod_hex_2.jpg
     Hexagon template #3
     click here to enlarge     
mod_hex_3.jpg

     Hexagon template #4
         click here to enlarge
mod_hex_1.jpg
     Hexagon template #5
         click here to enlarge
mod_hex_2.jpg
     Hexagon template #6
     click here to enlarge     
mod_hex_3.jpg

     Hexagon template #7
         click here to enlarge
mod_hex_1.jpg
     Hexagon template #8
         click here to enlarge
hexagon_template-82.jpg
     Hexagon template #9
     click here to enlarge     
hexagon_template-9.jpg

     Hexagon template #10
         click here to enlarge
hexagon_template-10.jpg
     Hexagon template #11
         click here to enlarge
hexagon_template-11.jpg
     Hexagon template #12
     click here to enlarge     
hexagon_template-12.jpg

     Hexagon template #13
         click here to enlarge
hexagon_template-13.jpg
     Hexagon template #14
         click here to enlarge
hexagon_template-14.jpg
     Hexagon template #15
         click here to enlarge
hexagon_template-15.jpg

Examples of Hexagon Patterns

     #1 example-a
         click here to enlarge
mod_hex_1.jpg
     #1 example-b
         click here to enlarge
mod_hex_2.jpg
     #1 example-c
     click here to enlarge     
mod_hex_3.jpg

     Template Ex-#2a
         click here to enlarge
mod_hex_1.jpg
     Template Ex-#2b
         click here to enlarge
mod_hex_2.jpg
     Template Ex-#2c
     click here to enlarge     
mod_hex_3.jpg

     Template Ex-#3a
         click here to enlarge
hex_temp_3a.jpg
     Template Ex-#3b
         click here to enlarge
hex_temp_3b.jpg
     Template Ex-#2c
     click here to enlarge     
mod_hex_3.jpg

     Template Ex-#4a
         click here to enlarge
hex_temp_4a.jpg
     Template Ex-#4b
         click here to enlarge
hex_temp_4b.jpg
     Template Ex-#4c
     click here to enlarge     
hex_temp_4c.jpg

     Template Ex-#5a
         click here to enlarge
hex_temp_5a.jpg
     Template Ex-#5b
         click here to enlarge
hex_temp_5b.jpg
     Template Ex-#5c
     click here to enlarge     
hex_temp_5c.jpg

     Template Ex-#6a
         click here to enlarge
hex_temp_6a.jpg
     Template Ex-#6b
         click here to enlarge
hex_temp_6b.jpg
     Template Ex-#6c
     click here to enlarge     
hex_temp_6c.jpg

     Template Ex-#7a
   enlarge front enlarge back
hex_temp_7af.jpg
     Template Ex-#7b
         click here to enlarge
hex_temp_7b.jpg
     Template Ex-#7c
     click here to enlarge     
hex_temp_7c.jpg

     Template Ex-#8a
         click here to enlarge
hex_temp_3a.jpg
     Template Ex-#8b
         click here to enlarge
hex_temp_3b.jpg
     Template Ex-#8c
     click here to enlarge     
hex_temp_8c.jpg

     Template Ex-#9a
         click here to enlarge
hex_temp_9a.jpg
     Template Ex-#9b
         click here to enlarge
hex_temp_3b.jpg
     Template Ex-#9c
     click here to enlarge     
hex_temp_9c.jpg

     Template Ex-#10a
         click here to enlarge
hex_temp_3a.jpg
     Template Ex-#10b
         click here to enlarge
hex_temp_3b.jpg
     Template Ex-#10c
     click here to enlarge     
mod_hex_3.jpg

     Template Ex-#11a
         click here to enlarge
hex_temp_3a.jpg
     Template Ex-#11b
         click here to enlarge
hex_temp_3b.jpg
     Template Ex-#11c
     click here to enlarge     
mod_hex_3.jpg

     Template Ex-#12a
         click here to enlarge
hex_temp_12a.jpg
     Template Ex-#11b
         click here to enlarge
hex_temp_12b.jpg
     Template Ex-#11c
     click here to enlarge     
mod_hex_3.jpg

     Template Ex-#13a
         click here to enlarge
hex_temp_13a.jpg
     Template Ex-#13b
         click here to enlarge
hex_temp_13b.jpg
     Template Ex-#11c
     click here to enlarge     
mod_hex_3.jpg

3. Heptagon case (N = 7)

The basic idea of heptagon knot is shown below.
      Figure A   Knot pattern by a pipe cleaner.
      Figure B   Knot by 3/4 inch quilling tape with double pattern
      Figure C   quilling tapes flattened 

      Figure A
      click here to enlarge hept_knot-200.jpg
      Figure B
      click here to enlarge i
      Figure C
      click here to enlarge i

Heptagonal Fold Trial

Let us try to create a heptagon ring made up of 7 heptagon units.
     The first step is to print out the drawing below. Then as is done in "Pentagon Case"
    mark two dotted lines in the back of the printed sheet to identify the boundaries
    to connect paper strips. Before cutting 7 pieces of paper strips, it is recommended
    to trace the "mountain fold" lines using the unusable cartridge of any ball point pen .
    The reason is that it is much easier to trace lines with some pressure before all pieces 
    are still connected in a sheet of paper instead of separate pieces.
     This process makes "mountain fold" easier during the folding process.
    Using a very sharp pointed knife, cut out 7 pieces of strips. This ring requires
    7 hexagon units and theoretically we can make it from a single strip made up of 7 unit
    strips.
     But practically it is easier to start with a strip with two unit strips connected,
    and after making one heptagon unit, add a next strip , and so on. The result will look
    like the one shown in the photo image below.
            Trial Drawing
         click here to enlarge & print
hept_string_basic-250.jpg
       Trial Result
     click here to enlarge   
example_heptagon-250.jpg

Heptagonal Fold Trial - shading variation

	Addition of colors to the previous example makes the heptagon ring 
       a bit more artistic.
            Variation Drawing
         click here to enlarge & print
hept_string_basic-a-250.jpg
       Variation Result
     click here to enlarge   
example_heptagon-a-250.jpg

How to make a ring of 14 heptagons

	Just like the pentagon case, it is also possible to make a ring made up of
       14 heptagon units.

             Paper Strings used #1
    click here to enlarge     Open PDF for print
hepta_string_14a-250
             Paper Strings used #2
    click here to enlarge     Open PDF for print
hepta_string_14b-250


Folding Process

      Basically the folding steps are similar to the 10 pentagon ring case.
     Start with one ring made up of #1 Paper Strip (Green)shown above. 
     Then add #2 Paper strip (Red). 
     

     Green Heptagon Ring
         click here to enlarge
hepta_2x_1.jpg
     Start Red Heptagon
         click here to enlarge
hepta_2x_2.jpg
     Final Look
     click here to enlarge     
hepta_2x_3.jpg

Usage of heptagon pattern templates

     The introduction of heptagon patterned templates, which are made up of 
    regular heptagon grids, opens up a wide open area of beautiful patterns.
    Here are several examples ranging from basic to their variations.

     Heptagon template #1
   larger JPG image    PDF
heptagon_temp-1-200.jpg
     Heptagon template #2
   larger JPG image    PDF
heptagon_temp-2-200.jpg
     Heptagon template #3
   larger JPG image    PDF
heptagon_temp-3-200.jpg

Examples of Heptagon Patterns

     #1 example-a
         click here to enlarge
hepta_temp1_ex1.jpg
     #1 example-b
         click here to enlarge
hepta_temp1_ex2.jpg
     #1 example-c
     click here to enlarge     
hepta_temp1_ex3.jpg

     Template Ex-#2a
         click here to enlarge
hepta_temp1_ex4.jpg
     Template Ex-#2b
         click here to enlarge
hepta_temp1_ex5.jpg
     Template Ex-#2c
     click here to enlarge     
hepta_temp2_ex1.jpg

     Template Ex-#3a
         click here to enlarge
hepta_temp3_ex1.jpg
     Template Ex-#3b
         click here to enlarge
hepta_temp3_ex2.jpg
     Template Ex-#2c
     click here to enlarge     
hepta_temp3_ex3.jpg

     Template Ex-#3a
         click here to enlarge
hepta_temp3_ex4.jpg
     Template Ex-#4a
         click here to enlarge
hept_temp4_ex1.jpg

3-a: Seven pointed star in the heptagon center

 We have shown that a ring of 7 heptagons can be created from a continuous strip of constant
width. Suppose we now have a ring maed iup of 7 regular heptagons as shown in two figures below.
They both are created usng 1 inch wide strip. Fig 7-0-a & b.

Next question is how we can fill up the vacant center area.  Or if the readers are only interested
in making a seven pointed star .One  answer is to use the Origami's "twist anf fold" techinique.

Here is the procedure.
Fig 7-0-a: Variation Result click here to enlarge example_heptagon-a-250.jpg Fig 7-0-b: Template Ex-#2b click here to enlarge hepta_temp1_ex5.jpg
Step1: Print one of the image files (Fig 7-1 & 2), and after printing on one side, print the heptagon pattern
	shown in Fig 7-2 making sure that the center of the pictures is as close as possible
Step 2: Cut out the printed paper along the periphral lines of the heptagon. Fig 7-3
Fig 7-1: Sample image file #1 click here to enlarge Open JPG for print hepta_string_14a-250 Fig 7-2: Sample image file #2 click here to enlarge Open JPG for print hepta_string_14b-250
Fig 7-2: Folding diagram click here to enlarge Open JPG for print hepta_string_14a-250 Fig 7-3: Cutout Heptagon click here to enlarge hepta_string_14b-250


Step 3: Score all the red, blue and black broken lines.
Step 4: Only for red and blue lines going outward from the inner heptagon, apply the
	following.  Mountain fold for the "red" , and valley fold  for the blue, while keeping the
	inner heptagon flat. Fig 7-4
Fig 7-4: Initial Step click here to enlarge Open JPG for print hepta_string_14a-250 Fig 7-5-1: Cutout Heptagon click here to enlarge hepta_string_14b-250
Fig 7-5-2: Initial Step click here to enlarge hepta_string_14a-250 Fig 7-5-3: Cutout Heptagon click here to enlarge hepta_string_14b-250
Step 5: Open the flap on the small heptagon side, then apply mountain fold to the "black" line.
	Fig 7-5-1 and 7-5-2.  When you let fingers go,  a star shape will form automatically bending
	at the red line on the outside . Fig 7-5-3 and fig 7-5-4.
	Repeat the same 6 more times for other edges.
	Result will look like Fig 7-5-5. 
	 If "mountain" and "valley" folding are switched, the final pattern looks like Fig 7-5-6.
Final Look:  Fig 7-6: Heptagon Ring with 7 pointed star inside
Fig 7-6: Heptagon Ring with 7 pointed star inside click here to enlarge hepta_string_14a-250

4. Octagon case (N = 8)

Octagon Knot using "Pipe-Cleaner"

	In the regular octagon drawing below, the numbers (1- 8) shows the sequence
       of how the strip goes around the paper plane. Solid lines cover the front , 
       and dotted lines,the back side of the plane. The right hand side photo shows  
       how this is reperesented by the pipe cleaner.      
           Regular Octagon
         click here to enlarge
octagon_knot_base-200
         Octagon Knot
         click here to enlarge
octagon_knot-200

First Trial- A ring of 4 Octagons

	      When the "Octagon_1" drawing below is printed out, and strips 
              are glued together by the triangular shaped areas on both edges,
              (though there are 8 strips in the drawing, only 4 pieces are used)
              the result will be a square shaped ring made up of 4 octagons.
             (Surprise ! surprise !)  Both front and back views are shown.
              8 different colors are used here to make the folding sequence
              easier to be see.              

           Octagon_1 Strip
         click here to enlarge
octagon_1_strip-200
         Octagon_1 Front
         click here to enlarge
octagon_1-200
         Octagon_1 Back
         click here to enlarge
octagon_1_back-200

Second Trial- A ring of 4 Octagons-preparation for 8 Octagons

	      When the "Octagon_2" drawing below is printed out, and strips 
              are glued together by the rectangular shaped areas on both edges,
              the result will be a square shaped ring made up of 4 octagons.
             (Again ?)  Both front and back views are shown.
              But this gives us a hint on how to make a ring of 8 octagons
              using exactly the same strings.              

           Octagon_2 Strip
         click here to enlarge
octagon_2_strip-200
         Octagon_2 Front
         click here to enlarge
octagon_2-200
         Octagon_2 Back
         click here to enlarge
octagon_2_back-200

Third Trial- A ring of 8 Octagons

	      First, prepare 4 strips cut from octagon_2 drawing shown above.
             Then connect two pieces together by glueing the end rectangular 
             area. Using this strip make an octagon in the vacant area between
             two octagons (as shown in the first picture below).
             Use the rear view as reference.
	     After the first step is done, connect the next strip,
             and follow the similar process.
             In the end the reader will end up with a ring of 8 octagons
             as shown in the picture.              

      8 Octagon Start
         click here to enlarge
octagon_3_1-200
      8 Octagon Start-Back
         click here to enlarge
octagon_3_1_back-200
      8 Octagon Final
         click here to enlarge
octagon_3_final-200

Fourth Trial- A ring of 8 Octagons with modified string

	     Now that we know how to make a ring of 8 ocatgons,
            let us modify the string pattern in order to appreciate the
            octagonal symmetry of this ring.
            The drawing below allows the experimenter to make a ring
            of either 4 or 8 octagons.
            4-octagon ring requires only 4 pieces of strip.
            The instruction is shown in the drawing.
            The resulting rings are shown. 

       Note: two separate continuous strips are used to make this ring.
            Is it possible to use only one contionuous tape to make a
            complete ocatgon ring made up of 8 octagons ?
            The answer is Yes, but with some bend of rule.
            See the Fifth Trial section.                         

      8 Octagon Start
         click here to enlarge
octagon_3_strip-200
      4 Octagon Case
         click here to enlarge
four_octagon-200
      8 Octagon Case
         click here to enlarge
eight_octagon-200

Fifth Trial- A ring of 8 Octagons with one continuous paper strip

	    The first drawing below illustrates the idea of making a single
           continuous paper strip to make a ring of 8 octagons.
           The sequence number 9 and 10 are extras, and they are revisiting
           the points 1 and 2. 
             The bext drawing has 4 pieces of short strips, and
            8 of longer ones.
            By connecting longer strip to shorter one, repeating 4 times
            the result will be a long one continuous paper strip. 
           But this method is more difficult than two pieces approach.                             

      8 Octagon with a single strip
         click here to enlarge
octagon_5-200
      8 Octagon - single continuous strip
         click here to enlarge
octagon_4_strip-200
   8 Octagon-single strip
         click here to enlarge
octagon_4-200
   8 Octagon-single strip Back
         click here to enlarge
octagon_4_back-200

Usage of octagon pattern templates

     The introduction of octagon patterned templates, which are made up of 
    regular octagon grids, opens up a wide open area of beautiful patterns.
    Here are several examples ranging from basic to their variations.

     Octagon template #1
   larger JPG image
oct_temp_1-200.jpg
     Octagon template #2
   larger JPG image
oct_temp_2-200.jpg
     Octagon template #3
   larger JPG image
oct_temp_3-200.jpg
     Octagon template #4
   larger JPG image
oct_temp_4-200.jpg
     Octagon template #5
   larger JPG image
oct_temp_5-200.jpg
     Octagon template #6
   larger JPG image
oct_temp_6-200.jpg
     Octagon template #7
   larger JPG image
oct_temp_7-200.jpg

Examples of Octagon Pattern

     #1 example-1
         click here to enlarge
oct_temp1_1-200.jpg
     #1 example-2
         click here to enlarge
oct_temp1_2-200.jpg
     #1 example-3
     click here to enlarge     
oct_temp1_3-200.jpg
     #3 example-1
         click here to enlarge
oct_temp3_1-200.jpg
     #3 example-2
         click here to enlarge
oct_temp3_2-200.jpg
     #1 example-3
     click here to enlarge     
oct_temp1_3-200.jpg
     #5 example-1
         click here to enlarge
oct_temp5_1-200.jpg
     #5 example-2
         click here to enlarge
oct_temp5_2-200.jpg
     #7 example-1
     click here to enlarge     
oct_temp7_1-200.jpg

5. Nonagon case (N = 9)


Nonagon (N=9) Fold Trial

	Thw following drawings can be used to weave a beautiful nonagon ring.  
      
            Nonagon Test Fold #1
    click here to enlarge & print     PDF image
nona_string_3_plus-200.jpg
       Test #1 Result
     click here to enlarge   
nonagon_test_1-200.jpg
            Nonagon Test Fold #2
    click here to enlarge & print     PDF image
nona_string_3_plus-200.jpg
       Test #2 Result
     click here to enlarge   
nonagon_test_2-200.jpg

Usage of nonagon pattern templates

     The introduction of nonagon patterned templates, which are made up of 
    regular nonagon grids, opens up a wide open area of beautiful patterns.
    Here are several examples ranging from basic to their variations.

     Nonagon template #1
large JPG image     PDF
nona_temp1-200.jpg
     Nonagon template #2
large JPG image     PDF
nona_temp2-200.jpg
     Nonagon template #3
large JPG image     PDF
nona_temp3-200.jpg

Examples of Nonagon Pattern

     #1 example-1
         click here to enlarge
nona_temp1_ex1-200.jpg
     #1 example-2
         click here to enlarge
nona_temp1_ex2-200.jpg
     #1 example-3
     click here to enlarge     
oct_temp1_3-200.jpg
Number of Nonagon -- 3 and 9
           Nonagon Case #1
         click here to enlarge
i
         Nonagon Case #2
         click here to enlarge
i

6. Decagon case (N = 10)

Number of decagon -- 5 and 10
           Decagon Case #1
         click here to enlarge
deca-5-900
         Decagon Case #2
         click here to enlarge
deca-10-900

7. Dodecagon case (N = 12)

(Under construction - to be added soon)) Number of dodecagon -- 3, 4, 6, 12
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

What is needed to enjoy this paper pastime

      These are absolutely necessasry.
    (1) Printer, which can print various size of papers. Direct feed type works better.
        It can handle heavy weight (up to around 65 lb.) paper.
    (2) Sharp paper cutter (roller or knife edge type), exact knife and replacement blades
    (3) Self healing vinyl mat (minimum 12 x 17 inches)
    (4) Paper glue, scissors (a good quality & very sharp), ruler (longer than 12 inches )

      The following are optional , but very handy if you have .
    (5) Light Box (used to trace on the back of printed paper in DUCT TAPE case.)
    (6) CAD software to draw your own pattern and make paper strings for printout.

Historical references

  (1) Ref. 4 & 5 offer another way of making a polygon using a ribbon.
  (2) In Ref. 2, A.R.Pargeter of Southampton gave a full count of a most interesting method of constructing
polyhedra by simply plaiting flat strips together. This method was discovered by a nineteenth-century doctor
named John Gorham, of Tonbridge, Kent, Englland, and he published a book "Plaited Crystal Models" in 1888.
So the topic discussed here may be called "Plaited Polygonal Models".

References

Books:

  1. Cundy,H.M. and Rollett,A.P.: Mathematical Models, ISBN4-89491-065-9,Oxford University Press,Oxford,1951
  2. Pargeter,A.R.: Plaited Polyhedra, Mathematical Gazette, Vol. 43, No. 344, pp.88 - 101, 1959
  3. Walser, Hans: The Golden Section, Translated from the original German, ISBN-0-88385-534-8,MAA,2001
  4. Hilton,P. & Pedersen,J.: Build your own POLYHEDRA, ISBN-0-201-49096-X,Addison-Wesley,1988
  5. Hilton,P., Pedersen,J.,Donmoyer,S.: A Mathematical Tapestry, ISBN-978-0-521-12821-6,Cambridge Univ. Press,2010
  6. Patterson,J.L: Create your own "printable" Scrapbook Papers , ISBN-0-486-99171-7,Dover,2011
  7. Farris, Frank A.: Creating Symmetry:the artful mathematics of wallpaper pattern, ISBN-0-691-16173-9,Princeton Univ. Press,2015

Internet resources:

  1. Maekawa,Jun: A Study on Knots of Tapes ,2010
      For math savvy readers:
  2. Conley,E.,Meehan,E.,Terry,R.: Flat Folded Ribbons
  3. Kauffmann,E.: Minimal Flat Knotted Ribbons

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Last Updated Jul 15 ,2016

Copyright 2006 Takaya Iwamoto   All rights reserved.