Recognizing Patterns in Nature

Types of Patterns

• Symmetric patterns occur when an object or shape is identical on both sides when divided along a central axis.
  • Examples: butterfly wings, starfish, snowflakes, and human faces.
• Tessellation patterns are repeated geometric shapes that cover a surface without gaps or overlaps. They must fit together on a flat surface
• Fractal patterns are complex, self-repeating patterns that look similar at different scales, often generated through mathematical equations.
  • Examples: lightning, broccoli, snowflakes, and fern leaves.
• Fibonacci Sequence and Golden Ratio: The Fibonacci Sequence (1, 1, 2, 3, 5, 8…) appears in natural growth patterns and the Golden Ratio (1.618) is linked to aesthetically pleasing proportions.
  • Examples: sunflower seeds, galaxy spirals, and pinecones.
• Wave patterns are repeating patterns formed by smooth, regular curves that go up and down.
  • Examples: sound waves, tide movements, light waves, and seismic waves.
• Spiral patterns often follow the Fibonacci Sequence or logarithmic growth patterns and must originate from a central point.
  • Examples: snail shells, hurricanes, tornadoes, and whirlpools.
• Spots and Stripes (Morphogenesis) result from biological and chemical processes.
  • Examples: Dalmatian spots, zebra stripes, tiger fur, leopard spots, and cheetah spots.
• Cracks patterns are networks of irregular lines or fractures on a surface, usually due to stress, pressure, or temperature changes.
  • Examples: ice cracks, mud cracks, rock or concrete cracks, and glass cracks.
• Polygonal patterns arise due to energy minimization or geometric constraints.
  • Examples: honeycomb hexagons, soap bubbles, and basalt columns.
• Foam patterns are natural, irregular patterns formed by clusters of bubbles or cells packed closely together, typically with polygonal shapes and curved edges.
  • Example: Soap Foam, Foamed Plastic or Packaging

Types of Symmetry

• Lateral Symmetry (Bilateral Symmetry) occurs when a shape can be divided into two equal mirror-image halves along a vertical line.
  • Examples: dragonfly, heart, humans, animals, and leaves.
• Rotational Symmetry is when a shape looks the same after being rotated less than a full circle.
  • Examples: clock, LCD fan, and wheel designs.
• Radial Symmetry is symmetry around a central axis.
  • Examples: orange, starfruit, kiwi, and pizza slices.
• Repeating patterns are series of identical motifs or elements repeated at regular intervals, showing translational symmetry.
  • Examples: honeycomb, floor tiles, and pineapple skin.
• Reflection Symmetry (Mirror Symmetry) is when one half of a figure is a mirror image of the other half.
  • Examples: Human body structure, Logos

Pattern Examples from Quiz

• 10, 20, 30, 40, 50, …
  • Answer: 60, 70
  • Pattern: Arithmetic sequence, adding 10 to the previous term.
• A, C, E, G, I, …
  • Answer: K, M
  • Pattern: Sequence of odd-numbered letters in the alphabet.
• 100, 78, 54, 28, 0, …
  • Answer: -28
  • Pattern: Arithmetic sequence, subtracting a decreasing amount (-22, -24, -26, -28), with the difference decreasing by 2 each time.
• 30, 35, 45, 60, 80, …
  • Answer: 105, 135
  • Pattern: The differences between consecutive terms are 5, 10, 15, 20. Each term is obtained by adding an increasing multiple of 5 to the previous term
• 151, 149, 145, 137, 121, …
  • Answer: 97, 65
  • Pattern: The differences between consecutive terms are -2, -4, -8, -16. This is a geometric sequence of differences where the differences are multiplied by 2 each time.
• 36, 43.5, 51, 58.5, 66, …
  • Answer: 73.5, 81
  • Pattern: Arithmetic sequence, adding 7.5 to the previous term.
• 13, 21, 34, 55, 89, …
  • Answer: 144, 233
  • Pattern: Fibonacci sequence, each term is the sum of the two preceding terms.
• J, F, M, A, M, …
  • Answer: J, J
  • Pattern: Abbreviation for months of the year, repeating.
• -14, -10, -6, -2, 2, …
  • Answer: 6, 10
  • Pattern: Arithmetic sequence, adding 4 to the previous term.
• 15, 10, 14, 10, 13, …
  • Answer: 9, 12
  • Pattern: Alternating pattern. The first, third, and fifth terms decrease by 1 each time. The second and fourth terms are constant.

Mathematical Language

• Mathematics uses precise symbols and rules to express ideas and relationships clearly.
• Mathematical language includes counterparts of nouns, pronouns, and verbs.
• Nouns: Numbers represent quantities or values.
  • Example: 5 represents a specific quantity.
• Pronouns: Variables represent unknown or changing quantities.
  • Example: 'x' can represent any number.
• Verbs: Relation symbols show relationships between mathematical objects.
  • Equality: = (e.g., 5 = 5)
  • Inequality: <, >, ≤, ≥ (e.g., 3 < 7)
  • Membership: ∈ (e.g., 2 ∈ {1, 2, 3})

Translating Statements into Mathematical Language

• The sum of x and y
  • Mathematical Expression: x + y
  • Explanation: Represents the operation of adding two variables.
• Density D is equal to the ratio of the mass m per volume v
  • Mathematical Sentence: D = m/v
  • Explanation: Establishes a relationship between density, mass, and volume.
• The speed of car x is 10 kilometers per hour less than the speed of car y
  • Mathematical Sentence: x = y - 10
  • Explanation: Expresses a relationship between the speeds of two cars.
• The product of length l, width w, and height h
  • Mathematical Expression: l * w * h
  • Explanation: Represents the multiplication of three variables.
• The square of the hypotenuse c is equal to the sum of the square of the legs a and b
  • Mathematical Sentence: c² = a² + b²
  • Explanation: Represents the Pythagorean theorem.

Sets (Roster Form)

• A = {x such that x is an integer, where x is greater than or equal to 4 but less than 17}
  • A = {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}
• B = {x such that x is a primary or secondary color}
  • B = {red, blue, yellow, green, orange, purple}
• C = {x such that x is an abbreviation of a day in a week}
  • C = {M, T, W, Th, F, Sa, Su}
• D = {x such that x is a prime number less than 20}
  • D = {2, 3, 5, 7, 11, 13, 17, 19}
• E = {x such that x is a squared integer, where x is greater than or equal to 1 but less than or equal to 10}
  • E = {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}

Sets (Builder Form)

• A = {2, 4, 6, 8, 10}
  • A = {x | x is an even integer, 2 ≤ x ≤ 10}
• B = {January, February, March, April, May}
  • B = {x | x is a month of the year, x ∈ {January, February, March, April, May, June, July, August, September, October, November, December}}
• C = {1, 1, 2, 3, 5}
  • C = {F(n) | n ∈ N, n ≤ 5} (where F(n) is the nth Fibonacci number)
• D = {1, 8, 27, 64, 125}
  • D = {x³ | x is an integer, 1 ≤ x ≤ 5}
• E = {9, 12, 15, 18, 21}
  • E = {3n | n is a positive integer, 3 ≤ 3n ≤ 21}

Symbols

• • : Plus, Addition
• − : Minus, Subtraction
• × or · : Times / Dot, Multiplication
• ÷ or / : Division, Division
• = : Equals, Equality
• ≠ : Not equal to, Two values are not equal
• <: Less than, A is smaller than B
• > : Greater than, A is larger than B
• ≤ : Less than or equal to, A is less than or equal to B
• ≥ : Greater than or equal to, A is greater than or equal to B

Algebra Symbols

• x, y, z : Variables, Unknown values
• a² : Exponent, a squared = a × a
• √a : Square root, The number that when squared gives a
• ∛a : Cube root, The number that when cubed gives a
• ∑ : Sigma, Summation
• ∏ : Pi (Product), Multiplication over a set

Set Theory Symbols

• ∈ : Element of, x ∈ A means x is in set A
• ∉ : Not element of, x ∉ A means x is not in set A
• ⊂ : Subset, A ⊂ B means all elements of A are in B
• ⊆ : Subset or equal, A ⊆ B includes the possibility A = B
• ∪ : Union, A ∪ B is all elements in A or B
• ∩ : Intersection, A ∩ B is elements common to both
• ∅ : Empty set, A set with no elements

Geometry Symbols

• ∠ : Angle, Used to denote an angle (e.g. ∠ABC)
• ° : Degree, Angle measure
• ∥ : Parallel, Lines are parallel
• ⊥ : Perpendicular, Lines are at 90°
• π : Pi, 3.14159..., ratio of circumference to diameter

Statistics & Probability Symbols

• P(A) : Probability, Probability of event A
• μ : Mu, Mean (average) of a population
• σ : Sigma, Standard deviation of a population
• x̄ : x-bar, Mean of a sample
• n : Sample size, Number of elements in a sample

Logic & Misc Symbols

• ∴ : Therefore, Used to show conclusion (∴ A = B)
• ∵ : Because, Shows reasoning (∵ A = B)
• ⇒ : Implies, If A, then B
• ⇔ : If and only if, A is true exactly when B is true
• ∃ : There exists, There is at least one
• ∀ : For all, Applies to all members of a set

Fibonacci Sequence

• The Fibonacci sequence is where each number is the sum of the two preceding numbers.
• Introduced by Leonardo Pisano Bigollo.
• The sequence is: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,...
• The Fibonacci sequence can be used to estimate the Golden Ratio (phi or Φ).

Golden Ratio

• The Golden Ratio is a mathematical ratio commonly found in nature
• Aesthetically pleasing to the eye when used in design.

Finding the 20th term in the Fibonacci Sequence Example

• Let n = the term you are looking for
• X20= (Φ)²⁰ − (1−Φ)²⁰ / √5
• X20 = (1.618034)²⁰ − (1−1.618034)²⁰ / √5
• X20 = 15,127 / √5
• X20 ≈ 6,755 (Rounded off)

Predicting Phenomena with Mathematics

• Mathematics can help predict the behavior of nature and phenomena in the world, including calamities (volcanic eruptions, hurricanes, earthquakes, floods, wildfires, landslides etc.)
• Mathematical tools create models that correspond to measurable and observable real-world phenomena.
• Weather forecasting applies science and technology to predict atmospheric conditions.

Weather Forecasting Tools

• Thermometer: Measures current ambient air temperature.
• Barometer: Measures air pressure, indicating clearing conditions (rising) or potential storms (falling).
• Anemometer: Measures wind speed and direction, typically mounted on a rooftop.
• Computer Models: Sophisticated programs that extrapolate changes in pressure zones using data on temperature and pressure.
• Weather Satellites: Provide pictures of weather systems from space, with geostationary satellites offering continuous coverage.

Natural Phenomena

• Fire Rainbows: Optical phenomenon formed by ice crystals in high-altitude cirrus clouds.
• Rogue Waves: Large, dangerous surface waves that occur far out at sea
• Algal Blooms: Rapid multiplication of algae in waterways with high nutrient levels.
• Ice Circles: Thin, circular slabs of ice that rotate slowly in the water, believed to form in eddy currents.
• Penitentes: Tall, thin blades of hardened snow or ice found at high altitudes.
• Aurora Borealis: Natural light display in the sky caused by the collision of energetic charged particles with atoms in the thermosphere.
• Halos: Rings of light caused by ice crystals bending light from the sun or moon.
• Fire Whirls (Fire Tornadoes): Spinning columns of flames.
• Blood Falls: A flow of iron-rich saltwater in Antarctica, colored red by iron oxide.
• Desert Roses: Special form of gypsum that develops in dry, sandy places that occasionally flood.
• Giant Permafrost Explosions: Occur when frozen methane trapped in permafrost turns into gas, building up pressure and causing the ground to explode.
• Eye of the Sahara (Richtat Structure): Eroded remains of a giant dome of rock in Mauritania.
• Waterspouts: Swirling towers of wind that climb from the water to the sky, similar to tornadoes.
• Spotted Lake: A lake in Canada with polka dot-like spots formed by mineral deposits left behind as the water evaporates in the summer.
• Nacreous Clouds: High-altitude clouds that form near the poles and reflect sunlight, appearing colorful.

Female Mathematicians and Their Contributions

• Hypatia of Alexandria (c. 360–415 AD)
  • Philosopher, mathematician, and astronomer in ancient Alexandria.
  • Expanded on the works of Diophantus and taught geometry and astronomy.
• Ada Lovelace (1815–1852)
  • Considered the world’s first computer programmer.
  • Created detailed notes on Babbage’s Analytical Engine, including the first algorithm.
• Sofia Kovalevskaya (1850–1891)
  • Russian mathematician and the first woman to receive a doctorate in mathematics.
  • Developed Kovalevskaya’s Theorem and published papers on Saturn’s rings.
• Emmy Noether (1882–1935)
  • Important mathematician of the 20th century.
  • Established Noether’s Theorem and contributed to abstract algebra.
• Mary Cartwright (1900–1998)
  • British mathematician who contributed to non-linear differential equations and chaos theory.
  • Developed the Cartwright-Littlewood Theorem and worked on radio waves.
• Grace Hopper (1906–1992)
  • American mathematician and computer scientist.
  • Developed the first compiler and played a key role in the development of COBOL.
• Maryam Mirzakhani (1977-2017)
  • Iranian mathematician and the first woman to win the Fields Medal.
  • Contributed to hyperbolic geometry and ergodic theory.