Composite Thue-Morse zone plates

Bifractal focusing and imaging properties of Thue-Morse Zone Plates.

We present a new family of Zone Plates (ZPs) designed using the Thue-Morse sequence. The focusing and imaging properties of these aperiodic diffractive lenses coined Thue-Morse Zone Plates (TMZPs) are examined. It is demonstrated that TMZPs produce a pair of self-similar and equally intense foci along the optical axis. As a consequence of this property, under broadband illumination, a TMZP prod...

When Thue-morse Meets Koch

In this paper, we reveal a remarkable connection between the Thue-Morse sequence and the Koch snowflake. Using turtle geometry and polygon maps, we realize the Thue-Morse sequence as the limit of polygonal curves in the plane. We then prove that a sequence of such curves converges to the Koch snowflake in the Hausdorff metric. In the final section we consider generalized Thue-Morse sequences an...

Rarified Sums of the Thue-morse Sequence

Let q be an odd number and Sq,0(n) the difference between the number of k < n, k ≡ 0 mod q, with an even binary digit sum and the corresponding number of k < n, k ≡ 0 mod q, with an odd binary digit sum. A remarkable theorem of Newman says that S3,0(n) > 0 for all n. In this paper it is proved that the same assertion holds if q is divisible by 3 or q = 4N + 1. On the other hand, it is shown tha...

An Overview of the Thue-Morse Sequence

Nonoverlap Property of the Thue-Morse Sequence

In this note, we provide a new proof for the nonoverlap property of the ThueMorse sequence using a Boolean functions approach and investigate other patterns that occur in a generalization of the Thue-Morse sequence.