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Bifurcation theory studies the changes in the qualitative or topological structure of a given family of solutions to a system as a parameter is varied. This is evident in the branching patterns of trees and rivers.

Fractal patterns appear in nature with self-similar structures at different scales such as coastlines, mountain ranges, and snowflakes. Mathematical fractals are typically described by recursive or iterative processes.

Fourier transforms decompose functions into their constituent frequencies, a principle used in sound analysis and signal processing. This relates to how musical instruments produce sound and how we perceive it.

Benford's Law states that in many naturally occurring collections of numbers, the leading significant digit is likely to be small. It is often observed in sets of economic, demographic, and scientific data.

Chaos theory explains the behavior of systems that are highly sensitive to initial conditions, often leading to seemingly random outcomes like weather patterns. The Lorenz attractor is one example of a chaotic system.

In biology, cellular automata model the growth patterns of organisms and other complex systems through simple, discrete, and deterministic rules applied on a grid.

Network theory provides a framework for understanding the structure and behavior of complex systems such as neural networks, social systems, and ecological food webs.

Dendritic crystal growth, such as seen in snowflakes or mineral crystal formations, is governed by diffusion-limited aggregation, which is a process that forms fractal patterns.

The formation of zebra stripes can be explained by activation-inhibition models, such as reaction-diffusion systems proposed by Turing, which account for the emergence of patterns in nature.

The logistic map is a model of population growth; it exhibits chaotic behavior when the population rate is above a certain threshold.

The optimal foraging theory posits that animals forage in a manner that maximizes their net energy intake per unit time. This can often be modeled using decision trees and game theory.

Wave phenomena can be mathematical described by differential equations, observing patterns in water, sound, or light. The wave equation is typically used to describe these systems.

Symmetry is present in living organisms and crystal structures, often described by mathematical groups that capture the essence of these symmetrical properties.

Reaction-diffusion systems describe patterns in space and time, generated through the interplay of transport mechanisms and local chemical reactions. They explain patterns like spots and stripes on animals.

The meandering of rivers can be modeled using fluid dynamics and sediment transport. Mathematical models can predict the shape and evolution of river bends over time.

Plateau's laws describe the structure of soap films formed in foam, providing insight into the minimal surface problem. These surfaces can be described by minimal surface equations.

The hexagonal crystal structure of snowflakes can be explained by molecular bonding patterns and the minimization of energy, showing the principles of crystallography and solid-state physics.

Tessellations are patterns of shapes that fit together without any gaps or overlaps. Examples include honeycomb structures made by bees and the geometric tiling found in nature such as basalt columns.

Allometric scaling laws in biology describe the relationship between the size of an animal and various physiological traits, often revealing predictable patterns across species.

Packing problems, such as how spheres can be densely packed in space, are seen in crystallography and the arrangement of cells. The hexagonal close packing and face-centered cubic are solutions to sphere packing in 3D.

Observed in various natural phenomena such as the arrangement of leaves on a stem, the bracts of a pine cone, or the fruitlets of a pineapple. The sequence follows the pattern where each number is the sum of the two preceding ones, often starting with 0 and 1.

Many natural phenomena exhibit the golden ratio ($\varphi$), which is approximately 1.618. It can be found in the proportion of various elements of living organisms and is often associated with aesthetically pleasing compositions.

Voronoi diagrams partition spaces based on the distance to a specified set of objects or seeds, which can explain the structure of animal skins, leaves, and forest canopies.

The Earth's elliptical orbit around the Sun can be explained by Kepler's laws of planetary motion, which are based on gravitational forces as described by Newtonian physics.