Mathematics in the Modern World - Orientation and Lesson Highlights

Flashcards covering the orientation, definitions of mathematics, natural patterns, sequences (Fibonacci, Arithmetic, Geometric), historical figures, and the functionality of mathematics in society.

Vision of the University

A globally recognized university in a heritage city by $2030$.

Mission of the University

To produce globally skilled and morally upright professionals instilled with rich cultural values.

Grading System

Midterm & Final Exams = $30\%$, Written and Performance = $70\%$.

Mathematics

The science that deals with the logic of shape, quantity, and arrangement; a science of pattern and order.

Pattern

The regular or repeated way in which something happens or is done.

Symmetry

When different sides of something are alike; it creates a pattern which makes nature more beautiful and fascinating.

Fractal

A detailed pattern that looks similar at any scale, repeats itself over time, and is a rough or fragmented geometric shape that can be subdivided into parts which are reduced-size copies of the whole.

Tessellations

Patterns that are formed by repeated cubes or tiles, examples of which include pineapples, turtles, and honeycombs.

Spiral

A growth pattern found in pinecone seeds, cactus plants, tree branches, and snail shells.

Stripe

A line or band that differs in color or tone from an adjacent area.

Crack

Linear openings that form in materials to relieve stress; the patterns reveal if the material is elastic or not.

Pythagoras ($c. 570-c. 495 BC$)

Explained patterns in nature like the harmonies of music as arising from number, which he took to be the basic constituent of existence.

Alan Turing

A British mathematician who predicted mechanisms of morphogenesis which give rise to patterns of spots and stripes.

Benoit Mandelbrot

A mathematician who showed how the mathematics of fractals could create plant growth patterns.

Leonardo Fibonacci

Introduced the Fibonacci sequence to the western world with his book Liber Abaci in $1202$, featuring a thought experiment on rabbit population growth.

Fibonacci Sequence

The series of numbers $0, 1, 1, 2, 3, 5, 8, 13, 21, 34, \dots$ where the next number is found by adding up the two numbers before it.

Recursive Definition of Fibonacci Sequence

The formula where $F_n = F_{n-1} + F_{n-2}$.

Arithmetic Sequence

A sequence of numbers where the difference between consecutive terms is always the same ($a_n = a_1 + (n - 1)d$).

Geometric Sequence

A list of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$).

Triangular Numbers

Numbers that count objects arranged in an equilateral triangle; the $n$-th triangular number is the sum of the $n$ natural numbers from $1$ to $n$.

Square Numbers

An integer that is the square of an integer (the product of an integer with itself, such as $9 = 3 \times 3$).

Cube Numbers

A number multiplied by itself $3$ times, denoted as a number raised to the power of $3$.

Golden Ratio ($\phi$)

An irrational number $\frac{1 + \sqrt{5}}{2}$, approximately equal to $1.61803$, also known as the divine proportion.

Relationship of Golden Ratio and Fibonacci

The ratio of two consecutive numbers of the Fibonacci series ($\frac{F_n}{F_{n-1}}$) approaches the Golden Ratio as the numbers increase.

Phidias

The Greek sculptor for whom the numeric value of the Golden Ratio, "phi", is named.

Mathematics for Organization

Using mathematical tools to make sense of available information, analyze data, and organize natural resources to develop society.

Mathematics for Prediction

Using mathematical models and frameworks to encode observations and forecast phenomena like weather, hurricanes, and volcanic eruptions.

Mathematics for Control

Using mathematical modeling to understand inputs and likely outcomes of events to prepare for or stop untoward consequences.

Global Circulation Models (GCMs)

Models that describe the interactions between oceans and the atmosphere to predict average weather conditions in the decades to come.