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 partiallytransparent patterns, such as window
screens or chainlink 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.
Moirélike 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 moirélike 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.
