### My Foray into Moiré: Exploring the Mathematics of Visual Overlap Patterns

#### Ariana Ray, Hastings High School Marty Cohen and John Noé, Stony Brook Laser Teaching Center

##### Simons Symposium Abstract: August, 2012

My interest in moiré effects dates from a group lunch at the Simons Center Cafe, where my mentor pointed out some strange, wavy lines on the screens covering a heater. The patterns shifted dramatically with small changes in the viewing angle, creating an ethereal appearance. I soon learned that those lines are moiré patterns. Moiré patterns are frequently observed when two objects with partially-transparent patterns, such as window screens or chain-link fences, overlap. The surreal hazy lines that appear result from the differing amounts of visual overlap (transparency or opacity) of the two patterns when one is tilted with respect to the other or has a different period. Moiré-like effects have been known since the Middle Ages, when fabrics such as watered silk were revered for the rippled "watermarks" created by their unique weave. The many modern applications include collimation testing, navigation, counterfeit prevention, and steganography. But moiré-like effects also have undesirable consequences, such as the creation of artifacts ("aliasing") in digital images or sound recordings.

In this work we have explored a variety of moiré effects and ways to create them. In particular, we have experimented with gratings that are not linear or that are not strictly periodic (aperiodic or quasiperiodic). Perhaps the most famous example of a quasiperiodic structure is the Fibonacci sequence; the fruitlets of a pineapple, the scales of a pine cone, and the florets of a sunflower all follow the Fibonacci sequence (1, 1, 2, 3, 5, 8, ...). The experiments consisted of creating overlapping grating patterns in Mathematica. Periodic gratings employed include lines, concentric ellipses, concentric circles, and arrays of dots. A quasiperiodic grating was also studied. It consisted of an array of dots arranged like florets on a sunflower and was generated using H. Vogel's formula (2004). All models were animated using the manipulate tool to allow for user interaction. The resulting interactive demonstrations are good educational tools for teaching mathematical concepts.

Creating these patterns by computer simulation is not the same as having a mathematical expression for them. In the future, we would like to better understand and extend previous work of this type, such as that of Oster et al. (1964) and Creath and Wyant (1992). We are especially interested in quasiperiodic gratings, such as the sunflower floret pattern we modeled.

This work was supported by the Simons Foundation and the Laser Teaching Center.