Mathematics in Our World: Patterns, Sequences, and Global Applications

DEFINING MATHEMATICS AND ITS GLOBAL ROLE

Mathematics is the study of pattern and structure. It functions as a fundamental pillar for the physical and biological sciences, engineering, information technology, economics, and increasingly the social sciences. As a useful way to think about nature and our world, mathematics serves as a tool to quantify, organize, and control the environment, predict phenomena, and make life easier. Patterns and occurrences exist throughout nature, life, and the universe, and mathematics helps make sense of these regularities.

Mathematics organizes patterns and regularities, predicts the behavior of nature and world phenomena, and controls occurrences for human ends. Its numerous applications across multiple disciplines make it an indispensable endeavor in the human experience.

PATTERNS AND REGULARITIES IN THE NATURAL WORLD

Patterns in nature are visible regularities of form found in the natural world and the universe. These are not merely for admiration; they are vital clues to the mathematical rules that govern natural processes. Patterns are observable in stars moving in circles across the sky each day, the yearly cycle of weather seasons, and snowflakes, which exhibit sixfold symmetry where no two flakes are exactly the same.

Biological growth and form attest to mathematical regularities. This is seen in fish patterns such as those found on the spotted trunkfish, spotted puffer, blue spotted stingray, spotted moral eel, coral grouper, redlion fish, yellow boxfish, and angel fish. Animals also exhibit regular patterns: zebras, tigers, cats, and snakes are covered in stripes; leopards and hyenas exhibit spots; and giraffes are covered in blotches.

Natural forces also create patterns that serve as clues to the rules governing the flow of water, sand, and air. These include waves across oceans, sand dunes in deserts, the formation of typhoons, and water drop ripples. Group behaviors in the animal kingdom, such as the ball of mackerel, the v-formation of geese in the sky, and the tornado formation of starlings, further illustrate these regularities. Nothing in nature happens without a reason; all patterns exist to act efficiently and possess an inherent beauty.

FUNDAMENTAL TYPES OF PATTERNS: SYMMETRY, FRACTALS, AND SPIRALS

Symmetry is a sense of harmonious and beautiful proportion or balance where an object remains invariant to various transformations, including reflection, rotation, or scaling. Bilateral Symmetry occurs when the left and right sides of an organism can be divided into approximate mirror images along a midline. This exists in insects, animals, plants, and flowers like orchids. Most animals and plant leaves exhibit bilateral or vertical symmetry.

Radial Symmetry, or rotational symmetry, occurs around a fixed center point and can be classified as cyclic or dihedral. Plants and some animal groups often display this. Five-fold symmetry is found in echinoderms like starfish ($D_5$ dihedral symmetry), sea urchins, and sea lilies. Radial symmetry suits non-moving organisms like sea anemones and jellyfish ($D_4$ dihedral symmetry). Specific botanical examples include the Bee orchid (1 reflectional and rotational), Yellow flag (3 reflectional), Cuckoo flower (3 rotational), Greater stitchwort (2 reflectional and 4 rotational), and Wild garlic (3 reflectional and 6 rotational).

Fractals are curves or geometric figures where each part has the same statistical character as the whole, characterized by self-similarity. This never-ending pattern repeats over and over at different scales. Examples include tree growth via repetitive branching, lightning bolts, veins in the human body, ferns, and aerial views of river systems.

Spirals, or logarithmic/growth spirals, are self-similar curves first described by Rene Descartes and later investigated by Jacob Bernoulli. They focus on a center point with circular shapes revolving around it. Examples include pine cones, pineapples, and hurricanes. Plants utilize spiral forms to maintain security while constantly growing.

THE FIBONACCI SEQUENCE AND RABBIT POPULATION GROWTH

The Fibonacci sequence is a series of numbers where each number is found by adding the two numbers before it. Named after Leonardo of Pisa (also known as Leonardo Pisano or Fibonacci) and introduced in his 1202 book Liber Abbaci (Book of Calculation), the sequence starts with 0 and 1, continuing as 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The mathematical rule is expressed as:

$X_n = X_{n-1} + X_{n-2}$

Fibonacci numbers were popularized through an exercise regarding the growth of a rabbit colony in a walled area. The production assumes each pair produces a new pair every month, and from the second month onwards, the new pair becomes productive. The growth of the colony over 12 months is documented as follows:

Month 1: 1 Adult Pair, 1 Young Pair, Total 2 Month 2: 2 Adult Pairs, 1 Young Pair, Total 3 Month 3: 3 Adult Pairs, 2 Young Pairs, Total 5 Month 4: 5 Adult Pairs, 3 Young Pairs, Total 8 Month 5: 8 Adult Pairs, 5 Young Pairs, Total 13 Month 6: 13 Adult Pairs, 8 Young Pairs, Total 21 Month 7: 21 Adult Pairs, 13 Young Pairs, Total 34 Month 8: 34 Adult Pairs, 21 Young Pairs, Total 55 Month 9: 55 Adult Pairs, 34 Young Pairs, Total 89 Month 10: 89 Adult Pairs, 55 Young Pairs, Total 144 Month 11: 144 Adult Pairs, 89 Young Pairs, Total 233 Month 12: 233 Adult Pairs, 144 Young Pairs, Total 377

THE GOLDEN RATIO, RECTANGLES, AND TRIANGLES

The Fibonacci sequence approaches the Golden Ratio asymptotically. The Golden Ratio relates two quantities where their ratio is the same as the ratio of their sum to the larger of the two quantities. It was called the "Divine Proportion" in the early 1500s in the work of Leonardo da Vinci and explored by Luca Pacioli in the 1509 book De Divina Proportione. Da Vinci likely first called it "section aurea" (Latin for Golden Section). It is represented by the symbol $\phi$ (phi) and is approximately equal to $1.618033987...$. The equation is:

$\frac{a+b}{a} = \frac{a}{b} = 1.618033987...$

A Golden Rectangle is a perfect rectangle formed using Fibonacci numbers. It can be broken down into squares the size of the next smaller Fibonacci number. Dividing these squares with arcs creates a Fibonacci spiral. Similarly, a Golden Triangle is an isosceles triangle with base angles of $72^{\circ}$. Bisecting one base angle creates another similar golden triangle. Repeating this creates whirling triangles that converge at an intersection where a logarithmic spiral can be drawn.

The progression of ratios between consecutive Fibonacci numbers illustrates the movement toward the Golden Ratio:

$2/1 = 2$$3/2 = 1.5$$5/3 = 1.666666666...$$8/5 = 1.6$$13/8 = 1.625$$21/13 = 1.615384615...$$34/21 = 1.61905$$55/34 = 1.61765$$233/144 = 1.618055556...$$377/233 = 1.618025751...$

SCIENTIFIC AND ARTISTIC APPLICATIONS OF THE GOLDEN RATIO

The Golden Ratio manifests across the universe. Flower petals often follow Fibonacci numbers: lilies have 3; buttercups have 5; chicory has 21; and daisies often have 34 or 55. Seed heads, such as those in sunflowers, produce spirals from the center outward, usually containing 34 and 55 seeds in opposing clockwise and counter-clockwise directions. Fibonacci numbers are found in tree branching and leaf counts on stems (like the sneezewort), but numbers like 4 are rarely seen.

In the human body, proportions relate to $\phi$ and the number 5, including 5 appendages to the torso, 5 fingers and toes on each, and 5 openings on the face. Human and non-human faces exhibit the Golden Ratio in the positions of the mouth and nose relative to the eyes and chin. Shells, such as those from snails and the nautilus, follow the logarithmic spiral, as does the human inner ear cochlea, goat horns, and spider webs. Spiral galaxies like the Milky Way have arms that are logarithmic spirals of approximately $12^{\circ}$. Hurricanes also follow this sequence.

Artistic applications include:

1. The Pathernon in Athens, Greece, whose exterior dimensions embody the ratio.
2. Plato's description of the five regular Platonic Solids in Timaeus: Tetrahedron, Hexahedron, Octahedron, Dodecahedron, and Icosahedron.
3. Euclid's definition of the ratio in Elements, linking it to the pentagram where each intersection and segment ratio is golden.
4. Leonardo da Vinci's use of the Golden Ratio in the Vitruvian Man, The Last Supper, Monalisa, and St. Jerome in the Wilderness.
5. Michelangelo di Lodovico Simon's use of the ratio in The Creation of Adam on the Sistine Chapel ceiling.
6. Raffaello Sanzio da Urbino (Raphael) distributing proportions in The School of Athens and using the golden triangle and pentagram in Crucifixion.
7. Works by Sandro Botticelli (Birth of Venus), George-Pierre Surat (Bathers at Assinieres, Bridge of Courbevoie, and A Sunday on La Grande Jette), and Salvador Dali (The Sacrament of the Last Supper).

ARCHITECTURAL MANIFESTATIONS OF MATHEMATICAL RATIOS

Architecture frequently utilizes these ratios for aesthetic and structural reasons:

1. The Great Pyramid of Giza (built approx. 4700 BC) has base sides of 756 feet and a height of 481 feet. The base-to-height ratio is about 1.5717, close to the Golden Ratio.
2. Notre Dame, the Gothic Cathedral in Paris (built 1163–1250), uses the ratio in key design proportions.
3. The Taj Mahal (completed 1648) utilizes proportionately reducing arches following the ratio.
4. The Cathedral of Our Lady of Chartres in France displays golden proportions.
5. The United Nation Building uses the ratio in its window configurations.
6. The Eiffel Tower (1889) follows the ratio as it narrows from the base toward the top.
7. The CN Tower in Toronto exhibits the ratio in its design; the ratio of the observation deck height (342 meters) to the total height (553.33 meters) is approximately 0.618.

NATURAL REGULARITITES AND BIOLOGICAL STRUCTURES

Natural regularities exist in multiple forms:

1. Honeycombs: Bees use regular repeating hexagons that use the least amount of wax to store honey, creating a strong structure without gaps.
2. Zebra Stripes: Alternating black and white patterns are governed by mathematical rules related to skin pigmentation chemicals.
3. Spider Webs: Intricate structures created through innate steps.
4. Nautilus Shells: These contain logarithmic spirals.
5. Tree Rings (Dendrochronology): A scientific method of dating trees based on the number of rings in the core.
6. Turtles: Growth rings called scutes are hexagonal. The smallest, oldest scute is in the center, while larger, newer ones are on the outside.
7. Lightning and Foam: Lightning creates fractals during storms. Foam bubbles trap pockets of gas in liquid or solid to form distinctive patterns.
8. Cracked Bark and Meanders: Cracks in tree bark show weaknesses, while meanders are regular sinuous curves in bodies of water.

ADVANCED APPLICATIONS IN VARIOUS PROFESSIONAL FIELDS

Mathematics is applied across numerous specialized sectors:

1. Forensic Science: Differential and integral calculus are used to clarify blurred images. Optimization techniques maximize or minimize surface areas, volume, profit, and cost, while projectile motion is analyzed during crime scene reconstruction.
2. Medicine: Drugs are designed by modeling the geometry and motion of proteins. Math also aids in developing cures for diseases.
3. Fluid Dynamics: Engineers employ numerical analysis for phenomena involving heat, electricity, magnetism, and quantum mechanics.
4. Information Technology: Computers rely on logic, set theory, graph theory, discrete probability, recursion, and number theory. Operational Research and numerical analysis are vital to computer science development.
5. Cryptography: Combining math and computer science, it secures ATM cards, computer passwords, and electronic commerce via information theory and security engineering.
6. Archaeology: Statistical techniques manage survey data, seek human behavior patterns, and assist in carbon dating artifacts.
7. Social Sciences: Economics, sociology, psychology, and linguistics use calculus, probability, game theory, and network theory.
8. Economics: Utilizes matrices, probability, and statistics. Models may be stochastic, deterministic, linear, or non-linear, and involve differential or integral equations.
9. Political Science: Analysts use mathematical models and game theory for conflict resolution and to study voting patterns and party switching.
10. Music and Arts: Musical rhythm involves permutations and combinations. Music theorists apply set theory, abstract algebra, and number theory to understand musical structure and hearing.