Mathematics in Our World: Patterns, Fibonacci Sequence, and the Golden Ratio

A set of Question-and-Answer flashcards covering patterns in nature, the Fibonacci sequence, the golden ratio, and the indispensable role of mathematics as presented in the lecture notes.

What is a pattern?

A regular, repeated form or design observed in nature, human-made designs, or abstract ideas; patterns can be visual or abstract like numerical sequences.

Name three types of patterns in nature described in the notes.

Symmetry, fractals, and spirals.

What is bilateral symmetry?

Also known as line symmetry; an imaginary plane divides the body into left and right mirror images.

What is a fractal?

A never-ending pattern whose parts have the same statistical character as the whole; exhibits self-similarity at different scales (e.g., fern, branching tree, lightning).

What is a logarithmic (growth) spiral?

A self-similar spiral curve that focuses on a center point; appears in nature (pine cones, pineapples, hurricanes); described by Descartes and Bernoulli.

What is the Fibonacci sequence?

A sequence where each term is the sum of the two previous ones; commonly 0,1,1,2,3,5,8,13,… with Fn = Fn-1 + Fn-2.

How is the Fibonacci sequence connected to nature?

It appears in natural patterns such as nautilus shells and sunflower seeds; the Fibonacci spiral can be seen in squares, and ratios of consecutive terms approach the golden ratio.

What is the golden ratio (phi)?

Approximately 1.618; the limiting ratio of consecutive Fibonacci numbers; used to create aesthetically pleasing proportions in art, architecture, and design; also called the Divine Proportion.

Who is associated with the history of the Golden Ratio?

Leonardo da Vinci and Luca Pacioli; early 1500s; De Divina Proportione; also linked to the term section aurea (Golden Section).

What is the practical use of the golden ratio in design?

To achieve balance, harmony, and aesthetically pleasing proportions in art, architecture, and graphic design.

Why is mathematics indispensable?

Because it underpins daily life, critical thinking, science and technology, economics and finance, medicine and biology, serves as a universal language, and opens career opportunities.

Name two daily life applications of mathematics mentioned in the notes.

Managing finances (budgeting, saving, investing) and planning/organizing schedules.

What does mathematics develop in terms of thinking?

Critical thinking and problem-solving: analyzing situations, identifying patterns, and developing solutions.

Give an example of mathematics as the backbone of science and technology.

Physics uses math to explain laws; Engineering uses math to design and analyze structures; Computer Science relies on algorithms and data structures.

What does 'universal language' mean in mathematics?

Its principles are consistent and understood globally, enabling collaboration and communication across cultures.

What is 'Mathematics for prediction' about?

Applying probability to estimate the chance of events (e.g., typhoons, meteor showers).

What is 'Mathematics for organization' about?

Using mathematical methods to organize tasks, resources, and activities.

What is 'Mathematics for control' about?

Using mathematics to exert control over oneself and the effects of nature.

Which packing arrangement is more space-efficient for coins, square or hexagonal?

Hexagonal (honeycomb) arrangement; it covers more area and minimizes exposed gaps because interior coins have six neighbors.