Selective Fractalization of Chevron-Type Polygons Edges
This Demonstration explores the fractalization of arbitrarily chosen edges of arbitrary polygons. A chevron-type concave polygon is used as a representative geometrical figure. The particular edges of this chevron-type polygon are fractalized with Koch curves. Then the copies of the resulting polygons are tiled and concatenated via straight non-fractalized edges so that they form elongated structures. Two types of fractalization are considered for comparison: randomized Koch curve (red, left) and regular Koch curve (blue, right).
The chevron-type concave polygon is described by three vectors. Two-arm vectors are defined as
where is the apex angle.
The third width vector is
Snapshot 1: tiled chevron structure with symmetric arms and symmetric width vector
Snapshot 2: tiled chevron structure with asymmetric arms and symmetric width vector
Snapshot 3: tiled chevron structure with symmetric arms and asymmetric width vector
This model of the edge randomization by Koch curves can be applied to study the edge disorder in zigzag-shaped graphene nanoribbons  similar to what has been done for phosphorene quantum dots . The shaded concave chevron-type polygon is a mathematical representation of the unit cell of zigzag-shaped graphene nanoribbon superlattices , which are also referred to in the literature as edge-modified zigzag-shaped ribbons  or jagged graphene nanoribbons .
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 V. A. Saroka, I. Lukyanchuk, M. E. Portnoi and H. Abdelsalam, "Electro-optical Properties of Phosphorene Quantum Dots," Physical Review B, 96(8), 2017 085436. doi:10.1103/PhysRevB.96.085436.
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