Nature of Mathematics: Patterns and Numbers in Nature and World (Fibonacci Sequence & Golden Ratio)

Vocabulary-style flashcards covering key concepts from the notes on mathematics, patterns, and natural patterns including Fibonacci sequence and the golden ratio.

Mathematics

A formal system of thought for recognizing, classifying, and exploiting patterns; it uses numbers, symbols, notations, operations, equations, and functions, and is applicable to real-world processes and phenomena.

Patterns

Regular, repeated or recurring forms or designs that help observers anticipate what might happen next.

Ishango bone

An early mathematical artifact used to count cycles (e.g., lunar phases) and explore patterns, illustrating primitive mathematical activity.

Pythagorean Theorem

A fundamental relation in right triangles (a^2 + b^2 = c^2) that exemplified early mathematical invention and discovery with wide architectural impact.

Fibonacci sequence

An integer sequence starting 1, 1, 2, 3, 5, 8, 13, … used to model growth patterns in nature and studied in fields from astronomy to botany and finance.

Golden ratio (Divine Proportion)

The irrational number φ ≈ 1.618…, the ratio of consecutive Fibonacci numbers that appears in natural growth patterns and art; linked to the tendency of growth to follow proportional ideals.

Honeycomb

Bees’ wax hive with hexagonal cells, illustrating a hexagonal tessellation pattern found in nature.

Snowflake

An ice crystal with hexagonal symmetry; its crystalline structure exemplifies repeating hexagonal patterns that aggregate into larger flakes.

Tiger stripes

Camouflage pattern where evenly spaced stripes are generally perpendicular to the spine, aiding concealment in the environment.

Hyena’s spots

Spotted pattern distributed for camouflage, illustrating irregular but patterned animal markings.

Sunflower patterns

Sunflower seed arrangements often form families of spirals, reflecting ordered, natural spirals found in nature.

Snail’s shell

A spiral-shaped shell whose radius increases as the mollusk grows, illustrating geometric spirals in nature.

Flower petals (Fibonacci pattern)

The number of petals often follows Fibonacci numbers (1, 2, 3, 5, 8, 13, …), linking botany to the Fibonacci sequence.

Weather patterns

Cyclic patterns in climate and weather that recur over days, weeks, or months, forming recognizable seasonal sequences.

Human populations (demography)

Distribution of people in a population; often pyramid-shaped in less developed regions and thinning bases as nations develop economically.

Human productivity (80:20 rule)

Observation that about 20% of members often account for roughly 80% of an organization’s output, illustrating unequal contribution patterns.

Symmetry

Invariance under transformations (reflection, rotation, scaling) that gives objects a sense of harmony and balance.

Bilateral symmetry

Left and right halves of an organism (or object) are approximate mirror images along a midline.

Radial symmetry

Symmetry around a central point, as in starfish or jellyfish, with patterns that are identical around the center.

Fractals

Infinitely complex patterns that are self-similar across scales, produced by repeating a simple process; used to model coastlines, snowflakes, and similar natural phenomena.

Tessellations

Repeated shapes covering a plane without gaps or overlaps, seen in natural examples like turtle shells, honeycombs, and pineapples.

Spirals

Logarithmic or growth spirals that are self-similar across scales; common in nature (sunflowers, shells, galaxies) and various natural phenomena.