Mathematics in the Modern World – Patterns, Symmetry & Growth

Flashcards covering definitions, examples, and formulas on patterns, symmetry, tessellation, Fibonacci sequence, and exponential population growth from the lecture notes.

What is the definition of a pattern in mathematics?

A pattern is a regular, repeated, or recurring form or design in which all the members are related to each other by a specific rule.

Why do we study patterns?

Studying patterns helps identify relationships, form generalizations, and make logical predictions.

Name six common types of patterns introduced in the lecture.

Number patterns, text patterns, alphanumeric patterns, skipping patterns, geometric patterns, and cyclic patterns.

What rule generates the sequence 2, 5, 8, 11…?

Add 3 to each term.

Give the next three terms of 256, 128, 64, 32…

16, 8, 4 (rule: divide by 2).

State the rule for the sequence 3, 6, 12, 24…

Multiply by 2 each time.

Which famous integer sequence begins 0, 1, 1, 2, 3, 5, 8…?

The Fibonacci sequence.

Complete the sequence 45, 48, 51, 54, 57, 60, __.

63 (rule: add 3).

What is bilateral symmetry?

A symmetry where a line (line of symmetry) divides a figure into two mirror-image halves.

Give an example of an object with bilateral symmetry mentioned in the slides.

A butterfly.

What is rotational symmetry?

A figure has rotational symmetry if it can be rotated less than 360° about a point and coincide with itself.

How is the angle of rotation calculated for rotational symmetry?

Angle of rotation = 360° divided by the order of rotation (n).

What is the angle of rotation of a regular hexagon?

60° (order of rotation = 6).

Define order of rotation.

The number of times a figure fits onto itself during one 360° rotation.

What natural objects were cited as showing five-fold rotational symmetry?

Echinoderms such as starfish and certain flowers.

What non-living object famously shows six-fold symmetry?

A snowflake.

What is translational symmetry?

A symmetry where a pattern can be slid (translated) along a direction and match exactly with itself.

Describe a logarithmic spiral example given in animals.

The nautilus shell, where each chamber is a scaled copy arranged in a spiral.

What term describes leaf and seed head spirals in plants such as sunflowers?

Phyllotaxis (or parastichy patterns).

Define tessellation.

A pattern that covers a plane completely with no overlaps or gaps using repeated tiles.

State two properties necessary for polygons to tessellate regularly.

No gaps/overlaps; the interior angles meeting at a point must sum to 360°.

Which three regular polygons can form a regular tessellation?

Equilateral triangle, square, and regular hexagon.

Why can’t a regular pentagon tessellate a plane on its own?

Its interior angle (108°) does not divide 360° evenly, leaving gaps.

How many semi-regular (Archimedean) tessellations exist?

Eight.

What is a fractal?

A shape that displays self-similarity; parts of it resemble the whole at different scales.

Name two natural fractal examples cited.

A fern leaf and Romanesco broccoli.

What is the Fibonacci sequence rule expressed algebraically?

Fₙ = Fₙ₋₁ + Fₙ₋₂ with F₀ = 0, F₁ = 1.

Who introduced the Fibonacci sequence to Europe?

Leonardo Pisano Bigollo, nicknamed Fibonacci, in his book Liber Abaci (1202).

What was Fibonacci’s original rabbit problem about?

Modeling rabbit pair growth under ideal reproduction conditions.

Write the first ten Fibonacci numbers.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34.

What constant do successive Fibonacci ratios approximate?

The golden ratio (≈1.618).

State Binet’s formula for the nth Fibonacci number.

Fₙ = (φⁿ – (1–φ)ⁿ) / √5 where φ = (1+√5)/2.

Using the exponential model A = 32,000e^{0.02t}, what is the population at t = 0?

32,000 (the starting population in 2014).

With the same model, estimate the population after 7 years.

≈36,809 (in the year 2021).

In A = 30e^{0.02t}, what is the population one year after 1995?

≈30.606 thousand.

Define meander as a natural pattern.

A sinuous bend in a river that grows as flowing water erodes and deposits material.

What pattern explains zebra stripes evolutionarily?

Stripes provide survival advantages such as camouflage or deterrence of predators/insects.

What is foam in a natural context?

A mass of bubbles whose boundaries often form hexagonal patterns resembling Plateau foam.

Explain the rule for constructing the next number in a sequence generated by adding a constant.

Add the same fixed number to the previous term each time.

Explain skipping patterns briefly.

Sequences that alternate between two or more interlaced sub-sequences (e.g., 10, 8, 11, 9, 12…).

What does cyclic pattern mean in sequences?

A repeating block of terms, such as 1,2,3,1,2,3… repeating indefinitely.

What is the smallest positive angle that returns a figure to itself called?

The angle of rotation.

How many lines of symmetry does a regular pentagon have?

Five.

What does the term ‘self-similarity’ mean in fractals?

Parts of the object are miniature copies (scaled versions) of the whole.

Give a formula to compute the angle of rotation for rotational symmetry.

Angle = 360° / n, where n is the order of rotation.

Why is mathematics called a universal language according to the opening slide of perceptions?

Because mathematical truths and symbols are consistent and understood globally, independent of natural languages.

State two key characteristics of patterns emphasized in learning.

They involve relationships and predictability.

What is the importance of ‘critical thinking’ in studying mathematics as per students’ perceptions?

It enables solving problems, understanding structures, and making logical deductions.

Define ‘order of rotation’ in one sentence.

The number of times a figure maps onto itself during one full 360° rotation.

What key idea links spirals, phyllotaxis, and Fibonacci numbers in plants?

Plant growth often follows Fibonacci ratios, producing spirals that optimally pack leaves or seeds.