Selective Fractalization of Chevron-Type Polygons Edges

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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).

Contributed by: Vasil Saroka (January 2018)
Open content licensed under CC BY-NC-SA



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 [1] similar to what has been done for phosphorene quantum dots [2]. The shaded concave chevron-type polygon is a mathematical representation of the unit cell of zigzag-shaped graphene nanoribbon superlattices [1], which are also referred to in the literature as edge-modified zigzag-shaped ribbons [3] or jagged graphene nanoribbons [4].


[1] V. A. Saroka and K. G. Batrakov, "Zigzag-Shaped Superlattices on the Basis of Graphene Nanoribbons: Structure and Electronic Properties," Russian Physics Journal, 59(5), 2016 pp. 633–639. doi:10.1007/s11182-016-0816-6.

[2] 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.

[3] V. A. Saroka, K. G. Batrakov and L. A. Chernozatonskii, "Edge-Modified Zigzag-Shaped Graphene Nanoribbons: Structure and Electronic Properties," Physics of the Solid State, 56(10), 2014 pp. 2135–2145. doi:10.1134/S106378341410028X.

[4] V. A. Saroka, K. G. Batrakov, V. A. Demin and L. A. Chernozatonskii, "Band Gaps in Jagged and Straight Graphene Nanoribbons Tunable by an External Electric Field," Journal of Physics: Condensed Matter, 27(14), 2015 145305. doi:10.1088/0953-8984/27/14/145305.

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