Layered Growth of 3D Snowflake Subject to Membrane Effect and More than One Nucleation Center by Means of Cellular Automata

César Renán Acosta (), Irma Martín and Gabriela Rivadeneyra
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César Renán Acosta: Applied Physics Department, Faculty of Engineering, Autonomous University of Yucatan, Mérida 97310, Yucatán, Mexico
Irma Martín: Applied Physics Department, Faculty of Engineering, Autonomous University of Yucatan, Mérida 97310, Yucatán, Mexico
Gabriela Rivadeneyra: Applied Physics Department, Faculty of Engineering, Autonomous University of Yucatan, Mérida 97310, Yucatán, Mexico

Mathematics, 2025, vol. 13, issue 3, 1-12

Abstract: In this work, it is taken into account that in nature, due to pressure and temperature, water drops in general are either spherical or ellipsoidal. Thus, starting from a more general structure, a 3D elliptical surface (oblate spheroid) is constructed, which, by means of parameters, can be turned into a spherical shape. Hexagons are built on a rectangular horizontal plane, then this plane is passed through an elliptical surface at height h , which is determined by a parameter θ . As a result of the cutting of these surfaces, a curve and a plane are obtained, both horizontal ellipsoidal; if these hexagons are within the perimeter of the horizontal ellipse obtained as a function of θ , they are marked with an N , and if they are outside the perimeter, they are marked with an E . Several frozen nucleation centers are established, either in the same layer or in different planes, marking them with an F and their first eight neighbors with a B . The calculations based on a modified snowflake model are carried out tile by tile and layer by layer, governed by the thermodynamic factors α , β , and γ , leading to results that depend on the position of the nucleator, which can be symmetrical or asymmetrical for a snowflake with more than one nucleation center and an external surface formed by water vapor that functions as a membrane.

Keywords: cellular automaton; snowflake; hexagonal lattice; discrete Laplacian (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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