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Geometrical Principles in 2d Mailling: 101; Regular Tessellation
Article © MAIL User: Karpeth

A ring is just what it sounds like; an approximately perfectly round polygon, which makes it perfect for using it as other shapes.

While rings are round, our weaves are not. The European 4 in 1 rings acts as square tiles. This is the most basic tessellation pattern, as the single ring acts as a square, so does the fivelet and larger patches. As the square is a special case rectangle, and in turn the rectangle a quite special polygon, all ring meshes where rings acts as squares, the mesh and rings can interact as rectangles and vice versa; this comes from that the four sides of a square/rectangle come in pairs; no matter what, the angle keeps at 90 degrees, but the sides can differ in length.

There is only one other way that tessellation can occur, and that is the triangle/hexagon pairing.

If we look at the Japanese weaves, this is quite clear: there are only three Japanese basic sheets, the Japanese 4 in 1, Japanese 3 in 1, and Japanese 6 in 1.

Why's that? well; the answer is simple. In J4-1, the ring and fivelet acts as a square; equidistant connector rings, 90 degrees apart for a sum of 360 degrees. In the fourlet of J3-1, the rings are 120 degrees apart, for a sum of 360 degrees, while they act as a triangle with 60 degree angles. J3-1 acts as a web, as the connector rings connect to the next fourlet centre ring. The effect is that the effective angles used are 120, and the polygon with the inner angles of 120 is the hexagon, leaving the hexagonal openings in the J3-1 web.

The sevenlets of J6-1 are 60 degrees apart, for a sum of 360 degrees. See a pattern here? With an external angle of 120, it's quite simple to add up to 360 degrees; namely, make a triangle with the connector rings.

That's the key to all of euclidian 2d and 1d tessellations; 360 degrees make a full turn, and keeps the sheet flat. 180 degrees makes for a line, and 360 degrees makes a full turn of the cross section.

What does this mean?
When this was submitted, the following two weaves exist in the library:
Moorish Rose
Brej√£o Flowers

Where one is using the square square tessellation, the other is using the hexagonal/triangular tessellation. Using the arguments in this article, there should be one more, and just one more. The search is not long, with only 10 weaves (as of the submission of this article) in the library with both the tags mobius and web. M3 behaves just as predicted; it's a triangular/hexagonal web consisting of interconnected 3-ring Mobius Ball units.

Please remember this when sculpting and creating weaves. This is one of the reasons why I dislike weaves such as Celtic Visions Pentagon Sheet, as it is not a sheet, it's 3d; it's the basis for a dodecahedron.
Original URL: http://www.mailleartisans.org/articles/articledisplay.php?key=756