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Growth in crystals can be { usually } described by field equations such asthe Kardar-Parisi-Zhang (KPZ) equation. While the crystalline structure can becharacterized by Euclidean geometry with its peculiar symmetries, the growthdynamics creates a fractal structure at the interface of a crystal and itsgrowth medium, which in turn determines the growth. Recent work (The KPZexponents for the 2+ 1 dimensions, MS Gomes-Filho, ALA Penna, FA Oliveira;\textit{Results in Physics}, 104435 (2021)) associated the fractal dimension ofthe interface with the growth exponents for KPZ, and provides explicit valuesfor them. In this work we discuss how the fluctuations and the responses to itare associated with this fractal geometry and the new hidden symmetryassociated with the universality of the exponents.

The fractal geometry of growth: fluctuation-dissipation theorem and hidden symmetry