**The Arc Approach**

A flat puzzle (tiling) with dozens or hundreds of identical pieces may sound a little dull and predictable. But what is the most interesting shape we can use, to get the most unusual designs and the most variety? To make it more visually interesting, let’s say we want a shape with no straight edges—only curves. The following guidelines should help us get started.

- Let’s use circular arcs, all with the same radius of a unit length. Hereafter we won’t talk about lengths; just about angles. These are angles of the arcs and of the corner angles. For good tiling these angles need to be divisors of 360° such as multiples of 12° or 15°: “agreeable” angles.
- Since the arcs must fit together there must be as much concave arc as convex arc.
- We’ll look at shapes that at least tile periodically—that is, by repeating it in simple translation—but are looking for tiles that fit together after rotation, with the more options the better.
- Let’s say we are free to use the reflection or mirror image of the shape. This might not seem important at first with symmetrical shapes but will be important later with more complex shapes and tilings.

**Circling the Square**

It is simplest to start with a square, since we can just replace the sides with two concave and two convex arcs, and get tiling based on adjacent squares as shown below. We can start with 90° arcs, which could encircle the square. The bottom shape below will show up again. It has been used for centuries. Since it is could be viewed as a stylized horseshoe crab, we’ll call it the Crab.

The arc angle can be any value up to 180°, as shown below.

Similar results can be achieved when starting with a rhombus, but with a more distorted view:

**Trying Triangles**

It we want to start with a triangle, it becomes more difficult due to the three sides. We can’t just replace the three sides of an equilateral triangle with identical arcs, since we won’t be able to get the same amount of convex and concave arc.

A 45° right triangle can be easily converted by putting a 180° arc on the hypotenuse, and 90° arcs on the two smaller sides. This gives us the Crab again.

Any right triangle can be converted to a tiling shape, by putting a convex 180° arc on the hypotenuse, and same-radius concave arcs on each of the smaller sides. This is because any right triangle can be inscribed in a half circle.

This shape can tile periodically, and some special cases–such as conversions from 45° and 30°/60° right triangles—result in shapes with agreeable angles that can also tile with rotations. But with all other right triangles we can’t easily get the final agreeable angles we want.

**Coming Full Circle**

If we start with a whole circle, we will want to replace half the circumference with concave arcs. We could start with creating two 90° concave arcs, either opposite each other or next to each other–and get the two same shapes we got initially using squares.

We can also use three 60° concave arcs. This can be done in the three arrangements shown below.

These shapes can also be made using a hexagon as the starting point.

The shape on the right above–with the three adjacent concave cutouts—can be modified with other sizes or concave arcs. If we stay symmetrical we can use various combinations of concave arcs totaling 180° as shown below. These will all tile in the same periodic manner. If the bottom middle cutout is reduced to nothing, we will have just two 90° concave arcs: the Crab again.

This approach with three concave cutouts in the lower half can also be used based on a lens shape. The lens is created by taking one arc (up to 180 degrees) and mirroring it about its endpoints. This is the more general case of the circle. As we did with the circle, we can make three concave cutouts bounded by one of the arcs, with similar periodic tiling.

All the shapes above primarily tile predictably and periodically, albeit with a wide range of possible arc angles and corner angles. Some of them can fit together in more complex ways, with rotation and more choices for tiling. How can we get the most flexibility from a single shape; or better, from a family of shapes?

**Trifocal Lenses**

The shape family with the most overall flexibility has three sides. But is not constructed from a triangle; rather it starts with the desired corner angles or arcs in the framework of a lens shape.

Let’s say we want a triangle-like shape with the usable corner angles of 30° and 60°. These will also be the angles of the two concave arcs. We could start construction with these, but it’s easier to start with the large-arc lens which will be the sum of these, or 90°. So we make a 90° arc and mirror it to make a lens shape. Then mark two smaller arcs—where they meet on the mirrored arc—and mirror each of them about their endpoints.

The resulting shape allows surprising flexibility for tiling.

The big advantage with this approach is that we choose the corner angles first, and the rest follows. If we want to build tiling around 5-pointed stars or flowers, we can choose small angles of, say, 36° and 72°.

Assuming we use reasonable angles, this construction and tiling works for any large angle up to 180°, and any proportioning of the two smaller arcs. The corner angle opposite the large convex arc is always the supplement (difference from 180°) of the large arc. And the smaller corner angles are always the same as the concave arcs.

**Conclusion**

The above approach lets us make a wide range of shapes, with complex and varied tilings that are radial/polar, periodic, or non-periodic, or some combination of these. This new family of shapes we can call tricurves.

Please try this out, explore the possibilities, and share what you find!

For more information on tricurves, see National Curve Bank entry and article and more images.