# Knotted Web Origami Tessellation

Posted by **Madonna Yoder** on

### Isoarea

If you're already familiar with isoarea tessellations, you might notice something unusual about this one - it uses normal, open hexagon twists!

Most of the time you'll see isoarea patterns in tessellations with all the same kind of twist - squares, equilateral triangles, right triangles, and occasionally rhombi - placed on opposite sides of the paper in equivalent positions.

These uniform tilings are ideal for isoarea tessellations because it's easy to map each tile to another tile of the same type on the other side.

And if the isoarea tessellation isn't on a uniform tiling, it often involves some kind of isoarea twist with half of the edges reversed.

These isoarea twists tend to be difficult to fold, often involving precreasing.

But Knotted Web breaks all of these rules - it has two different kinds of twists (closed triangle and open hexagon) and both twists are in their standard forms.

### Alternation

The trick with Knotted Web is that when you alternate neighboring twists on either side of the paper, you end up with an equal number of hexagons (and triangles) on the front as you have on the back.

This is only possible when the hexagon doesn't occupy a position of 6-fold rotational symmetry.

In fact, each hexagon twist here has only 3-fold rotational symmetry in the broader pattern.

Each open hexagon is neighbored by three open hexagons on the opposite side and three isoarea clusters of closed triangle twists.

In order to get this alternation, the hexagons are next to each other in the tiling and have different twist directions.

In the diagram above, you can see the twist direction in the color - red for right handed and white for left-handed twists.

If no pleats are reversed, then every twist must have neighbors of the opposite twist direction.

For typical hexagons and triangles tessellations, this is achieved by putting six triangles next to each hexagon and three hexagons next to each triangle.

Under this typical regime, all hexagons are the same twist direction and all triangles are the other twist direction and so no isoarea tessellations can be formed without isoarea hexagon twists.

But with this 3-fold arrangement of hexagons and triangles, you'll get an isoarea tessellation every time you alternate the twists front and back as long as each triangle is the same as every other triangle and each hexagon is the same as every other hexagon.

### Balanced

One of the (many) cool thinks about Knotted Web is that I didn't even need a rotated grid for the pattern to perfectly align itself on a hexagon.

This is highly convenient - and fairly rare.

The only reliable way to get this alignment is with mirror symmetries and without those it's a happy accident.

I won't say that this tessellation is easy to fold, but it's not too dense, has lots of tube pleats for references, and has stunning results to pull you through the process.

If you'd like to give it a try, the crease pattern is available for purchase here and you'll be able to access the file immediately after purchase.