Mathematics in the Modern World - Patterns in Nature and the Fibonacci Sequence

Vocabulary practice flashcards covering patterns in nature (symmetries, fractals, spirals, tessellations, foams, stripes, cracks) and the Fibonacci sequence from Mathematics in the Modern World.

Patterns in Nature

Visible regular forms found in the natural world that can sometimes be modeled mathematically, including symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks, and stripes.

Symmetry

A condition in which different sides of an object are alike, presenting as mirror images with two sides, identical on several sides, or symmetrical on all sides.

Bilateral Symmetry (Mirror Symmetry)

Symmetry with respect to reflection, wherein a figure does not change upon undergoing a reflection.

Rotational Symmetry (Radial Symmetry)

Symmetry found at different scales among non-living and living things where features repeat around an axis, such as a drop forming a crown splash in a pond, or the spheroidal shape and rings of Saturn.

Fivefold Symmetry (Penta-radiate Symmetry)

A form of radial symmetry found in echinoderms—including starfish, sea urchins, and sea lilies—characterized by five repeating sections radiating from a center.

Six-Fold Symmetry

A type of symmetry observed in snowflakes, where each flake's structure forms a record of crystallization conditions with nearly the identical pattern of growth on each of its six arms.

True Crystal Symmetry Limitation

The property stating that true crystals can have a variety of symmetries and habits (such as cubic or octahedral), but cannot possess fivefold symmetry, unlike quasicrystals.

Fractals

Infinitely self-similar, iterated mathematical constructs having fractal dimension; in nature, infinite iteration is impossible, so all natural fractal patterns are only approximate.

Spirals

A curve that emanates from a central point, moving progressively farther away as it revolves around that point.

Tessellations

Patterns formed by repeating tiles all over a flat surface without gaps or overlaps, classified into $17$ wallpaper groups of tilings.

Bubbles / Foams

Physical structures where a film forms a sphere or minimal surface area—the smallest possible surface area for the volume enclosed.

Stripes

Natural visual patterns on animals with an evolutionary explanation: functions that increase the chances that the offspring of the patterned animal survive to reproduce.

Cracks

Linear openings that form in materials to relieve stress, with their pattern indicating whether the material is elastic or not.

Sequence

An ordered list of numbers, called terms, that may have repeated values and whose arrangement is governed by a definite rule.

Fibonacci Sequence

The numerical sequence $0, 1, 1, 2, 3, 5, 8, 13, 21, 34, \text{...}$ where each term after the first two is found by adding up the two preceding numbers.

Fibonacci Recursive Rule

The mathematical formula expressed as $X_n = X_{n-1} + X_{n-2}$, where $X_n$ is term number $n$, $X_{n-1}$ is the preceding term, and $X_{n-2}$ is the term before that.

Fibonacci Divisibility Pattern

The rule stating that every $n\text{th}$ number in the Fibonacci sequence is a multiple of $X_n$ (e.g., every 3rd number is a multiple of $2$, every 4th is a multiple of $3$, and every 5th is a multiple of $5$).