This Demonstration shows how the ideal Koch curve replacing edges of various regular polygons changes upon different types of randomization.

Snapshot 1: "ideal"—original Koch curves

Snapshot 2: "plus random"—randomized Koch curves with only outward notches allowed

Snapshot 3: "minus random"—randomized Koch curves with only inward notches allowed

Snapshot 4: "full random"—randomized Koch curves with outward and inward notches allowed

Flat nanostructures such as jagged graphene nanoribbons [1, 2, 3] or silicene quantum dots [4] can be defined by a bounding polygon, which when overlaid with a 2D crystal of graphene or silicene isolates a group of atoms forming the unit cell of the ribbon or quantum dot. The edges of this bounding polygon can be replaced by the Koch curve with randomly generated parameters as demonstrated above to model fluctuations in the structure synthesis conditions.

[1] 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.

[2] 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.

[3] 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.

[4] H. Abdelsalam, M. H. Talaat, I. Lukyanchuk, M. E. Portnoi and V. A. Saroka, "Electro-absorption of Silicene and Bilayer Graphene Quantum Dots,"

*Journal of Applied Physics*,

**120**(1), 2016 014304.

doi:10.1063/1.4955222.