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 .

### 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
Fig.2 Resulting Pentagon
Fig.3 After 5 repetitions

### 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.

```
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.

```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 Fig.23-b:  Paper Strings used #2     click here to enlarge     Open PDF for print

```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

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.

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.

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 Fig. 35  Step 3:  Print and cut-out template          click here to enlarge Fig. 36: Step 4:  Prepare a tracing paper tape          click here to enlarge Fig.37: Step 5:  Cover the template by tape          click here to enlarge Fig.38:  Step 6:  Trace template's pattern          click here to enlarge Fig.39:  Final: Unfold the tracing paper tape          click here to enlarge

```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 .

```

```    --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
```

```     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.

```

## Detailed dicussions about each case

### Pentagon case (N = 5)

(1) How to get a paper strip to construct pentagon ring

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.
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.

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.

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.

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.

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.

#### 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.
```

### More samples of pentagon based pattern

```     There are many variations of pentagon based patterns.
The typical cases are shown below.
```
```   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   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 sample #8          click here to enlarge      click to see backside      click to see the template sample #9          click here to enlarge      click to see backside      click to see the template
```  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 sample #11          click here to enlarge      click to see backside      click to see the template sample #12          click here to enlarge      click to see backside      click to see the template
```  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   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
```

### 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
```

#### 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 Trial #1 Result    enlarge Front    View Back

#### Hexagon Trial #1 Folding Process

 Hexagon #4 End    enlarge Front    View Back Hexagon #5 Front        click here to enlarge Hexagon #5 End    enlarge Front    View Back
 Hexagon #6 Front        click here to enlarge Hexagon #6 End    enlarge Front    View Back
```     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.
```

#### 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 Trial #2 Result    enlarge the image

#### 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.
```

#### 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 Example #1 Result      enlarge the image

#### 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.
```

#### 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.
```

### 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
```

#### 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.
```

#### Heptagonal Fold Trial - shading variation

```	Addition of colors to the previous example makes the heptagon ring
a bit more artistic.
```

#### 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 Paper Strings used #2     click here to enlarge     Open PDF for print

#### Folding Process

```      Basically the folding steps are similar to the 10 pentagon ring case.
Then add #2 Paper strip (Red).

```

#### 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 template #2    larger JPG image    PDF Heptagon template #3    larger JPG image    PDF

### 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

Fig 7-0-b: Template Ex-#2b

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
Open JPG for print

Fig 7-2: Sample image file #2
Open JPG for print

Fig 7-2: Folding diagram
Open JPG for print

Fig 7-3: Cutout Heptagon

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
Open JPG for print

Fig 7-5-1: Cutout Heptagon

Fig 7-5-2: Initial Step

Fig 7-5-3: Cutout Heptagon

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

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

Octagon Knot

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

Octagon_1 Front

Octagon_1 Back

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

Octagon_2 Front

Octagon_2 Back

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,
In the end the reader will end up with a ring of 8 octagons
as shown in the picture.

8 Octagon Start

8 Octagon Start-Back

8 Octagon Final

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

4 Octagon Case

8 Octagon Case

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

8 Octagon - single continuous strip

8 Octagon-single strip

8 Octagon-single strip  Back

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

Octagon template #2

larger JPG image

Octagon template #3

larger JPG image

Octagon template #4

larger JPG image

Octagon template #5

larger JPG image

Octagon template #6

larger JPG image

Octagon template #7

larger JPG image

Examples of Octagon Pattern

#1 example-1

#1 example-2

#1 example-3

#3 example-1

#3 example-2

#1 example-3

#5 example-1

#5 example-2

#7 example-1

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

PDF image

Test #1  Result

Nonagon Test Fold #2

PDF image

Test #2  Result

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

Nonagon template #2

large JPG image

PDF

Nonagon template #3

large JPG image

PDF

Examples of Nonagon Pattern

#1 example-1

#1 example-2

#1 example-3

Number of Nonagon -- 3 and 9

Nonagon Case #1

Nonagon Case #2

6. Decagon case (N = 10)
Number of decagon -- 5 and 10

Decagon Case #1

Decagon Case #2

7. Dodecagon case (N = 12)
(Under construction - to be added soon))
Number of dodecagon -- 3, 4, 6, 12

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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:

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

Internet resources:

Maekawa,Jun:  A Study on Knots of Tapes ,2010

Conley,E.,Meehan,E.,Terry,R.:  Flat Folded Ribbons
Kauffmann,E.:  Minimal Flat Knotted Ribbons

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Last Updated Jul 15 ,2016