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The Tessellation Theory of Japanese Weaves
Article © MAIL User: Drax
The Tessellation Theory of Japanese Weaves
by Drax
This article will give a brief summary of the geometric idea known as "tessellations" and show their ability to describe Japanese weaves. This method is capable of describing all current Japanese weaves and will also serve as a guide for creating endless new types of Japanese weaves.
"To tessellate" means to form a mosaic pattern using geometric shapes. An example of this is shown below; it uses a square to form what we all know as a checkerboard pattern. This is a tessellation. It is also the blueprint for Japanese 4in1 (more on this later).
Another example is given below; this one uses triangles and as we'll see it is the basis for Japanese 6in1.
These are quite simple tessellation patterns; we can make much more complicated ones. This in turn will lead to new and complex chainmaille weaves in the Japanese style.
The way to do this is quite easy. Take your desired tessellated pattern and draw a circle at every place where lines intersect (i.e. at every vertex). Check out the picture below for how to do this with the previously shown grids of squares and triangles.
Now they look like Japanese 4 in 1 and Japanese 6 in 1, don't they! It's as easy as that, but you can make much more complicated things. As a side note, you can determine the level of linkages (3in1, 4in1, etc) by how many lines intersect at each point (I'll point this out later with an example). In Japanese weaves, the minimum is 2in1 and the maximum is 6in1 (I’ll insert a proof on this later).
Patterns derived via this method will look best if the connecting rings (the ones that only link 2 rings) are smaller than the main rings. This makes the pattern clearer to the eye. However, one can of course use rings all of the same size and the resulting pattern will look slightly different. It all comes down to a matter of taste and also the objective of the weave. Other possible variations include doubling the main rings, or even doubling the connector rings; there is a lot of room for variation here as well.
I took a look at the current list of Japanese weaves in the M.A.I.L. weave section, and now I would like to show how those derive from tessellations.
I’ve already shown the origin of 4in1 and 6in1, so let's take a look at the one listed as a Japanese 6 in 1 Variant. This comes from a pattern of hexagons and triangles, as shown below:
As mentioned in the weave library, this is technically a version of 4in1. We can see this by looking at the grid and noticing that there are FOUR lines that intersect at each point. Thus 4in1. Easy!
Next comes Japanese 3 in 1. This draws from a simple grid of hexagons, as shown below (the picture from the library uses the same sized rings for all links -- just picture the 2in1 links as smaller).
Finally there's Sakredchao's Japanese 3 in 1 Octagon / Square Tessellation. This comes from a classic pattern of octagons and squares:
The only one missing now is Japanese 5 in 1, and here’s a possibility:
I say “a” possibility, because here’s another one, grid only shown:
Of course, there are many patterns possible that aren't completely flat. For example the alternating hexagon-pentagon pattern of a soccer ball will work, but it won't be flat. For this tessellation method, the shapes you use should be regular. That is, use polygons of equal sides and equal internal angles; also use the same length sides for different polygons (e.g. the length of a square side should be the same as a triangle side). To know if a pattern will be flat, the angles that meet at a point should all add up to 360 degrees. Below, I've shown an example of a pattern that won't quite lay flat (it will buckle slightly) because the angles at each point add up to 366, not 360.
I would also like to explain the derivation of the "hizashi" subfamily that I recently discovered. But this article is already very long, so if you're still interested, check that out by clicking here (I’ll add the link once this isn’t in the test section anymore).
The possibilities of variations are truly endless. Another thing that is fun to do is vary the coloring schemes of the links! You can create beautiful mosaic-like weaves out of color variations, just remember to color the vertices (where the lines meet) and not fill in the shapes (that won't work).
The patterns you create can also have open geometric spaces to create “webs”, or net-like weaves. I have seen many designs like these that create beautiful overlays (Sakredchao’s Japanese 3 in 1 Octagon / Square Tessellation is an example of this).
If you're interested about tessellations and their geometric rules, there are plenty of sites on the internet, many with very complicated patterns already available. Many also include information on perhaps the most famous tessellator, M.C. Escher! My favorite site for information can be found here at http://library.thinkquest.org/16661/index2.html. Thinkquest makes you go through an ad-gateway first, so just look for a link that takes you to the actual page.
I hope you found this information useful and that you can create some wonderful pieces!
by Drax
This article will give a brief summary of the geometric idea known as "tessellations" and show their ability to describe Japanese weaves. This method is capable of describing all current Japanese weaves and will also serve as a guide for creating endless new types of Japanese weaves.
"To tessellate" means to form a mosaic pattern using geometric shapes. An example of this is shown below; it uses a square to form what we all know as a checkerboard pattern. This is a tessellation. It is also the blueprint for Japanese 4in1 (more on this later).
Another example is given below; this one uses triangles and as we'll see it is the basis for Japanese 6in1.
These are quite simple tessellation patterns; we can make much more complicated ones. This in turn will lead to new and complex chainmaille weaves in the Japanese style.
The way to do this is quite easy. Take your desired tessellated pattern and draw a circle at every place where lines intersect (i.e. at every vertex). Check out the picture below for how to do this with the previously shown grids of squares and triangles.
Now they look like Japanese 4 in 1 and Japanese 6 in 1, don't they! It's as easy as that, but you can make much more complicated things. As a side note, you can determine the level of linkages (3in1, 4in1, etc) by how many lines intersect at each point (I'll point this out later with an example). In Japanese weaves, the minimum is 2in1 and the maximum is 6in1 (I’ll insert a proof on this later).
Patterns derived via this method will look best if the connecting rings (the ones that only link 2 rings) are smaller than the main rings. This makes the pattern clearer to the eye. However, one can of course use rings all of the same size and the resulting pattern will look slightly different. It all comes down to a matter of taste and also the objective of the weave. Other possible variations include doubling the main rings, or even doubling the connector rings; there is a lot of room for variation here as well.
I took a look at the current list of Japanese weaves in the M.A.I.L. weave section, and now I would like to show how those derive from tessellations.
I’ve already shown the origin of 4in1 and 6in1, so let's take a look at the one listed as a Japanese 6 in 1 Variant. This comes from a pattern of hexagons and triangles, as shown below:
As mentioned in the weave library, this is technically a version of 4in1. We can see this by looking at the grid and noticing that there are FOUR lines that intersect at each point. Thus 4in1. Easy!
Next comes Japanese 3 in 1. This draws from a simple grid of hexagons, as shown below (the picture from the library uses the same sized rings for all links -- just picture the 2in1 links as smaller).
Finally there's Sakredchao's Japanese 3 in 1 Octagon / Square Tessellation. This comes from a classic pattern of octagons and squares:
The only one missing now is Japanese 5 in 1, and here’s a possibility:
I say “a” possibility, because here’s another one, grid only shown:
Of course, there are many patterns possible that aren't completely flat. For example the alternating hexagon-pentagon pattern of a soccer ball will work, but it won't be flat. For this tessellation method, the shapes you use should be regular. That is, use polygons of equal sides and equal internal angles; also use the same length sides for different polygons (e.g. the length of a square side should be the same as a triangle side). To know if a pattern will be flat, the angles that meet at a point should all add up to 360 degrees. Below, I've shown an example of a pattern that won't quite lay flat (it will buckle slightly) because the angles at each point add up to 366, not 360.
I would also like to explain the derivation of the "hizashi" subfamily that I recently discovered. But this article is already very long, so if you're still interested, check that out by clicking here (I’ll add the link once this isn’t in the test section anymore).
The possibilities of variations are truly endless. Another thing that is fun to do is vary the coloring schemes of the links! You can create beautiful mosaic-like weaves out of color variations, just remember to color the vertices (where the lines meet) and not fill in the shapes (that won't work).
The patterns you create can also have open geometric spaces to create “webs”, or net-like weaves. I have seen many designs like these that create beautiful overlays (Sakredchao’s Japanese 3 in 1 Octagon / Square Tessellation is an example of this).
If you're interested about tessellations and their geometric rules, there are plenty of sites on the internet, many with very complicated patterns already available. Many also include information on perhaps the most famous tessellator, M.C. Escher! My favorite site for information can be found here at http://library.thinkquest.org/16661/index2.html. Thinkquest makes you go through an ad-gateway first, so just look for a link that takes you to the actual page.
I hope you found this information useful and that you can create some wonderful pieces!
Original URL: http://www.mailleartisans.org/articles/articledisplay.php?key=3
