Notes on Patterns and Numbers in Nature

Mathematics

The science of patterns, structures, relationships, and logical reasoning, enabling analysis, prediction, problem-solving, and precise description of the world.

Mathema

The Greek word from which 'mathematics' is derived, meaning 'knowledge' or 'learning'.

Symmetry

An exact correspondence of form and constituent configuration on opposite sides of a dividing line, plane, or about a center/axis.

Bilateral or Reflection Symmetry

A type of symmetry where a figure can be divided by a line into two parts that are mirror images of each other.

Spiral (Pattern)

A curved pattern that focuses on a point and consists of a series of circular shapes revolving around it.

Fractal

A pattern built from simple repeated shapes that are reduced in size when repeated, creating intricate details at different scales.

Tessellations

Patterns created with identical shapes that fit together with no gaps or overlaps, tiling a plane.

Rotational Symmetry / Angle of Rotation

The smallest measure of angle that a figure can be rotated while preserving its original position.

Order of Rotation

The number of times a figure maps onto itself during a full rotation of $360^{\circ}$.

Leonardo Fibonacci

An Italian mathematician, born about 1170, known for the Fibonacci sequence. His nickname 'Fibonacci' means 'Son of Bonacci'.

Fibonacci Sequence

A sequence of numbers where each number is the sum of the two preceding ones, typically starting with 0 and 1. The sequence is $0,\ 1,\ 1,\ 2,\ 3,\ 5,\ 8,\ 13,\ \dots$

Recursive Definition of Fibonacci Sequence

The formula $F\_n = F\_{n-1} + F\_{n-2}$ for $n \ge 3$, with initial conditions $F\_1 = 1$ and $F\_2 = 1$.

Golden Ratio ($\phi$)

A mathematical proportion approximately equal to $1.618\dots$, given by the formula $\phi = \frac{1+\sqrt{5}}{2}$, found