Mathematics in a Modern World

A collection of vocabulary terms exploring the nature of mathematics, its patterns in nature, foundational concepts, and its role across various disciplines based on the lecture notes.

Mathematics

A useful way to think about nature and our world that underscores the exploration of patterns in nature and the environment.

The Origin of Mathematics

Traced to the history and significance of patterns and numbers, involving ideas translated to objects and concepts created by humans to link the meaning of patterns.

Four Basic Concepts of Math Language

The mathematical language gives emphasis to four basic concepts: Sets, Functions, Relations, and Binary Operations.

Polya’s 4 Steps

A specific sequence of steps utilized in mathematics for problem solving and reasoning.

Regularity

The fact that the same thing is always happens in the same circumstances.

Sequential, Spatial, Temporal, and Linguistic

The four categories or forms that patterns can take.

Symmetry

Agreement in dimensions, due proportion and arrangement.

Spiral

A curve which emanates from a point, moving farther away as it revolves around the point.

Meander

One of a series of regular sinuous curves, bends, loops, turns or windings in the channel of a river, stream or other watercourse.

Wave

A disturbance that transfers energy through matter or space, consisting of oscillations or vibrations of a physical medium or a field.

Foam

A substance formed by trapping pockets of gas in a liquid or solid, such as a sponge or the head of a glass of beer.

Tessellation

The tiling of a plane using one or more geometric shapes (tiles) with no overlaps and no gaps.

Fracture or Cracks

The separation of an object or material into two or more pieces under the action of stress.

Normal Tensile Crack

A crack where the displacement develops perpendicular to the surface of displacement.

Shear Crack

A displacement that develops tangentially to the surface of displacement, also referred to as a slip band or dislocate.

Stripes

Patterns made by a series of bands or strips, often of the same width and color along the length.

Fractal

A never-ending, infinitely complex pattern that is self-similar across different scales and represents images of dynamic systems.

Affine Transformation

The mathematical process consisting of rotation, reflection and scaling.

Fibonacci numbers

Described as nature's numbering system.

Fibonacci sequence

An integer sequence where terms are found using the formula: $F_n = F_{n-1} + F_{n-2}$.

Pure Mathematics (Einstein)

According to Albert Einstein, this is the 'poetry of logical ideas.'

Nature of Mathematics (AAAS 1990)

Relies on both logic and creativity, pursued for both practical purposes and intrinsic interest according to the American Association for the Advancement of Science.

Logic (D’ Alembert)

Described as practical logic because in geometry, rules in reasoning are applied in the simplest and sensible manner.

Mathematics in Music (Leibnitz)

A hidden exercise in arithmetic of a mind unconscious of dealing with numbers.

Mathematics in Social Networks

Utilizes graph theory, text analysis, multidimensional scaling, and cluster analysis.

Mathematics in Political Science

Involves election results, mathematical models for conflict resolution, and game theory.

Geography

A scientific and mathematical description of our earth and its universe, including dimensions, magnitude, and ecology.