🥎Non-Euclidean Geometry Unit 1 – Intro to Non-Euclidean Geometry: History

What's the deal with Non-Euclidean Geometry?

• Non-Euclidean geometry explores geometries that deviate from Euclid's parallel postulate, which states that given a line and a point not on the line, there is exactly one line through the point parallel to the given line
• Challenges the notion that Euclidean geometry is the only valid geometric system
• Emerged in the early 19th century when mathematicians began questioning the necessity of Euclid's fifth postulate
• Led to the development of hyperbolic and elliptic geometries, which have different properties than Euclidean geometry
• Opened up new possibilities in mathematics and changed our understanding of space and the universe
• Has applications in fields such as physics, cosmology, and computer graphics
• Demonstrates that there are multiple consistent geometric systems that can describe the world around us

The OG Euclidean Geometry: A quick recap

• Euclidean geometry is based on the work of the ancient Greek mathematician Euclid, who lived around 300 BCE
• Euclid's "Elements" is a comprehensive treatise on geometry that has been used as a textbook for over 2,000 years
• Euclidean geometry is built upon five postulates, which are assumed to be true without proof
  • A straight line can be drawn between any two points
  • A straight line can be extended indefinitely in both directions
  • A circle can be drawn with any center and radius
  • All right angles are equal to one another
  • The parallel postulate: Given a line and a point not on the line, there is exactly one line through the point parallel to the given line
• Euclidean geometry deals with flat, two-dimensional surfaces and three-dimensional spaces
• Concepts such as points, lines, angles, triangles, and circles are fundamental to Euclidean geometry
• Euclidean geometry has been widely used in various fields, including architecture, engineering, and navigation

When Euclid's parallel postulate became sus

• Euclid's parallel postulate, also known as the fifth postulate, seemed less intuitive and more complex than the other four postulates
• Mathematicians attempted to prove the parallel postulate using the other postulates, believing it to be a theorem rather than an axiom
• Many attempts were made to prove the parallel postulate, but all of them ultimately failed or relied on assumptions equivalent to the postulate itself
• The Italian mathematician Giovanni Saccheri (1667-1733) tried to prove the parallel postulate by contradiction, but ended up discovering the first non-Euclidean geometry, although he didn't realize it at the time
• The German mathematician Carl Friedrich Gauss (1777-1855) privately explored the consequences of denying the parallel postulate, but never published his findings
• The Russian mathematician Nikolai Lobachevsky (1792-1856) and the Hungarian mathematician János Bolyai (1802-1860) independently developed hyperbolic geometry, which denies the parallel postulate
• The German mathematician Bernhard Riemann (1826-1866) developed elliptic geometry, another non-Euclidean geometry that differs from both Euclidean and hyperbolic geometry

Mind-bending alternatives: Hyperbolic and elliptic geometry

• Hyperbolic geometry, also known as Lobachevskian geometry, is a non-Euclidean geometry that denies the parallel postulate
  • In hyperbolic geometry, given a line and a point not on the line, there are infinitely many lines through the point that do not intersect the given line
  • Triangles in hyperbolic geometry have angle sums less than 180 degrees, and the sum decreases as the area of the triangle increases
  • Hyperbolic geometry has applications in special relativity and cosmology, as it can describe the geometry of spacetime in the presence of matter and energy
• Elliptic geometry, also known as Riemannian geometry, is another non-Euclidean geometry that differs from both Euclidean and hyperbolic geometry
  • In elliptic geometry, there are no parallel lines at all; any two lines in a plane will eventually intersect
  • Triangles in elliptic geometry have angle sums greater than 180 degrees, and the sum increases as the area of the triangle increases
  • Elliptic geometry has applications in general relativity, as it can describe the geometry of spacetime in the presence of gravitational fields
• Both hyperbolic and elliptic geometries demonstrate that Euclidean geometry is not the only consistent geometric system and that different geometries can have vastly different properties

Key players in the Non-Euclidean revolution

• Carl Friedrich Gauss (1777-1855): German mathematician who privately explored non-Euclidean geometry but never published his findings
• Nikolai Lobachevsky (1792-1856): Russian mathematician who independently developed hyperbolic geometry and published his work in the 1820s and 1830s
• János Bolyai (1802-1860): Hungarian mathematician who also independently developed hyperbolic geometry and published his findings in 1832
• Bernhard Riemann (1826-1866): German mathematician who developed elliptic geometry and introduced the concept of curved spaces in his 1854 lecture "On the Hypotheses Which Lie at the Foundations of Geometry"
• Eugenio Beltrami (1835-1900): Italian mathematician who provided models for hyperbolic and elliptic geometry, showing their consistency and independence from Euclidean geometry
• Felix Klein (1849-1925): German mathematician who developed the Klein model for hyperbolic geometry and contributed to the understanding of the connections between different geometries
• Henri Poincaré (1854-1912): French mathematician who made significant contributions to the development of non-Euclidean geometry and its applications in physics and cosmology

How Non-Euclidean geometry shook up mathematics

• The discovery of non-Euclidean geometries challenged the long-held belief that Euclidean geometry was the only valid geometric system
• It demonstrated that mathematics could have multiple, equally valid foundations and that the choice of axioms could lead to different geometric systems with their own unique properties
• Non-Euclidean geometry opened up new avenues for mathematical research and led to the development of new branches of mathematics, such as topology and differential geometry
• It also had a profound impact on the philosophy of mathematics, leading to a re-evaluation of the nature of mathematical truth and the role of intuition in mathematical reasoning
• The existence of non-Euclidean geometries showed that mathematics is not just a description of the physical world, but an abstract system that can have multiple consistent interpretations
• Non-Euclidean geometry paved the way for the development of modern abstract algebra and the study of abstract mathematical structures
• It also had significant implications for physics, as the geometry of spacetime in the presence of matter and energy is non-Euclidean, as described by Einstein's theory of general relativity

Real-world applications: Where this stuff actually matters

• General relativity: Einstein's theory of general relativity uses non-Euclidean geometry to describe the curvature of spacetime in the presence of matter and energy
  • The geometry of the universe on a large scale is believed to be non-Euclidean, with the specific geometry depending on the overall density and distribution of matter and energy
• Cosmology: Non-Euclidean geometry is essential for understanding the structure and evolution of the universe
  • Different cosmological models, such as the flat, open, and closed universes, are based on different non-Euclidean geometries
• Computer graphics and virtual reality: Non-Euclidean geometry is used in the creation of virtual environments and video games
  • Hyperbolic geometry can be used to create infinite, yet bounded, spaces that can be explored in virtual reality applications
• Crystallography: The geometry of certain crystals and quasicrystals can be described using non-Euclidean geometry
  • The study of these structures has applications in materials science and nanotechnology
• Navigation and GPS: The geometry of the Earth's surface is non-Euclidean due to its curvature
  • Navigational systems and GPS technology must take this into account to provide accurate positioning and guidance
• Coding theory and cryptography: Non-Euclidean geometry has applications in the design of error-correcting codes and secure communication protocols
  • Hyperbolic geometry can be used to create efficient codes and encryption schemes

Why should I care? The impact on modern math and science

• Non-Euclidean geometry has had a profound impact on the development of modern mathematics and science
• It has led to a deeper understanding of the nature of space and the universe, and has opened up new avenues for research and discovery
• Non-Euclidean geometry has shown that mathematics is not just a tool for describing the physical world, but a rich and abstract system with its own internal logic and structure
• The study of non-Euclidean geometry has led to the development of new mathematical techniques and concepts, such as topology, differential geometry, and group theory
• Non-Euclidean geometry has also had a significant impact on physics, particularly in the development of Einstein's theory of general relativity and the study of cosmology
• The applications of non-Euclidean geometry extend beyond pure mathematics and physics, with practical uses in fields such as computer graphics, navigation, and cryptography
• Understanding non-Euclidean geometry is essential for anyone interested in pursuing advanced studies in mathematics, physics, or related fields, as it provides a foundation for many modern theories and techniques