Geometry: Shapes & Type in Mathematics - 98thPercentile

The term "geometry" is derived from the ancient Greek word "geometria," meaning measurement (-metria) of earth or land (geo). It is a branch of mathematics that explains the relationship between shape, size, and numbers. 

 

Ancient civilizations like the Indus Valley civilization and Babylonia, in roughly 3000 BC, are credited for the foundation of rules and formulas of geometry, suitable for planning, constructing, astronomy, and solving mathematical problems, using the principles of length, area, angle, and volume.

 

what is geometry

 

In the middle ages, mathematicians and philosophers from different cultures continued to use geometry to create a model of the universe. Studies by the French mathematician and philosopher René Descartes (1596-1650) of coordinate systems to define the positions of the points in 2D and 3D space led to the birth of the field of analytical geometry.

 

The conceptual idea of the surface of a sphere, where the axioms of Euclidean geometry are inapplicable, was known. Still, the discovery of non-Euclidean geometry clarified broader fundamental principles that combined numbers and geometry. 

 

In 1899, notable German mathematician David Hilbert (1862-1943) developed advanced-level axioms, establishing a new era around the 20th and 21st centuries, where axioms were applied to various mathematical cases.

 

Timeline of Geometry

 

  • 3000 BC – Practical geometry of the ancient world, like Pyramids

 

  • 300 BC – Spherical geometry

The concept was compiled in a book named "spherics" by Greek astronomer Theodosius of Bithynia (169-100 BC) that amalgamates the earlier work by Euclid (325-265 BC) and Autolycus of Pitane (360-290 BC) on spherical astronomy.

 

For astronomical mapping, it calculates areas and angles on spherical surfaces, such as a star or planetary positions. 

 

  • 500 BC: Pythagoras of Samos named the "Pythagoras theorem," which calculates the hypotenuse (the longest side) of a right-angled triangle from the lengths of the other edges, which would sum to 180 degrees or two right angles.

 

  • 4th century BC:Geometric tools to measure, sketch, and build geometric forms and constructions, can be traced back to the ancient Egyptians and Greeks.

The Greek philosopher Plato (428-347 BC) stated that the tools of geometry should be limited to a straightedge and a compass. Some examples are building a line that is twice as long as another line or a line that divides an angle into two halves.

 

  • 360 BC: Platonic solids

Introduced by Plato, known as the five regular convex polyhedra (polygonal bodies), where five shapes can be formed by joining similar faces along the edges, creating a tetrahedron (four faces), a cube(six), the octahedron(eight), the dodecahedron(twelve), icosahedron (twenty faces).

 

  • 240 BC – Archimedean solids

Developed by Greek mathematician Pappus of Alexandria (320 AD), Archimedean solids are described as 13 convex polyhedrons, which are uniform polygons with congruent edges and corners. 

 

  • 1619 – Kepler's polyhedron

German mathematician Johannes Kepler (1571-1630) discovered a new class of polyhedra, known as the star-polyhedron ("Kepler-Poinsot polyhedral"), stated the small stellated dodecahedron, the great stellated dodecahedron, the great dodecahedron, and the great icosahedron.

 

  • 1637 – Analytical geometry

"La Geométrie," a book of coordinate systems, gave birth to the study of geometric forms and their attributes in the domain of analytical geometry or coordinate geometry, or Cartesian geometry. René Descartes is the "father" of analytical geometry because of his contribution to this field.

 

  • 1858 – Topology

Topology studies the features of geometrical objects and spaces that remain unchanged while continuously deformed (Möbius strip). Leonard Euler, a Swiss mathematician, is the father of modern topology. 

1882 – The discovery of the Klein bottle by German scientist Felix Klein (1849-1925), which has a one-sided surface without any surface borders, thus proving geometry beyond three dimensions. The Klein bottle is a mathematical persona of a character with no standard boundaries, as a loop that is twisted and linked to itself. It can't be immersed in three-dimensional space without intercrossing itself. 

 

  • 20th century – Fractal geometry

Computers have led to the discovery of fractals, equations of detailed models that are repetitive at different scales and produce shapes like the Mandelbrot set displayed in a graphical form.

Benoit Mandelbrot (1970s), a mathematician born in Poland, is widely recognized as the pioneer of fractal geometry, which is commonly applied in many modern disciplines, such as physics, biology, and development of computer graphics as well as the research of chaotic systems.

 

Computational geometry allowed us to solve problems such as the four-color theorem, highlighting various regions within a complex mapping by four colors. Francis Guthrie (1852) introduced the four-color theorem, and Kenneth Appel and Wolfgang Haken(1976) verified it by computer. They used them in the study of cartography mapping and other spatial systems.

 

Types of Geometry

types of geometry

 

  • Euclidean geometry 

It analyzes shapes in two and three dimensions according to the rules established by Euclid.

 

  • Non-Euclidean geometry

Use of various axioms: Notable examples are hyperbolic geometry (use of axioms for parallel lines) and elliptic geometry (use of axioms for the sum of the angles in a triangle).

 

  • Projective geometry

It deals with geometric shapes, and its alternation of length-to-width ratios (the ratios of distances) remains the same. They are used to analyze perspective drawings and graphics forms (art, architecture, and photography).

 

  • Topological geometry

It relates to the unchangeable property of geometric objects when subjected to continuous deformations like stretching or bending—often used to investigate sub-atomic structures and cosmological characteristics.

 

  • Differential geometry

It is a calculus-based research field in geometry, including curves and 3D surfaces. They are applied in investigating spatial phenomena and studying the properties of dynamic physical systems.

It has numerous applications in engineering, surface qualities, designing, and analyzing complex systems like airplanes and cars.

 

  • Algebraic geometry

Applications of algebraic geometry include Algebraic curves, Algebraic surfaces, and Algebraic varieties of high-dimensional spaces expressed in algebraic equations such as parabolas, planes, ellipsoids, lines, circles, or the curvature of space-time.

 

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