Projective & Fractal Geometry: Global Cross-Cultural Perspectives
Encyclopedia/1. The Cosmos & The Natural World/1. Mathematics & Formal Systems/01. Geometry & Spatial Systems • Curated by Admin Timeline.sg
A comparative timeline of projective and fractal geometry, integrating global contributions from ancient foundations to modern breakthroughs, highlighting diverse cultural perspectives and key mathematical innovations.
Chronological Storyline (41 Milestones)
300 BCE
Euclid's Elements Published
Euclid's Elements establishes foundational geometric principles, though not projective, it underpins later developments in geometry. #geometry #history
Euclid's Elements Published By University of Pennsylvania Museum of Archaeology and Anthropology - https://openn.library.upenn.edu/Data/0016/html/e2748.html, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=169166171
320 CE
Pappus's Collection Contains Projective Theorem
Pappus of Alexandria writes his Collection, including Pappus's hexagon theorem, a key result in projective geometry. #projective #geometry
Pappus's Collection Contains Projective Theorem By Pappus Alexandrinus - https://www.christies.com/en/lot/lot-6069499, Public domain, https://commons.wikimedia.org/w/index.php?curid=132586197
600 CE
Indian Kolam Patterns Display Recursive Geometry
Kolam, ritual floor drawings in South India, exhibit recursive geometric patterns that prefigure fractal concepts. #fractal #india #art
Indian Kolam Patterns Display Recursive Geometry By SKsiddhartthan - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=67189832
700 CE
Islamic Geometric Patterns in Architecture
Islamic artisans create intricate geometric tilings in mosques and palaces, incorporating self-similarity and complex symmetries. #geometry #islamic #art
Islamic Geometric Patterns in Architecture By Ian Alexander - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=45782132
1021 CE
Ibn al-Haytham Advances Perspective Theory
Ibn al-Haytham's Book of Optics describes the camera obscura and lays groundwork for linear perspective, influencing projective geometry. #optics #projective
Ibn al-Haytham Advances Perspective Theory By Adolph Boÿ, engraved by Jeremias Falck. Used as the frontispiece to Johannes Hevelius, Selenographia, 1647 - https://www.loc.gov/resource/rbctos.2017rosen1321/?sp=9&r=0.059,0.617,0.327,0.182,0, Public domain, https://commons.wikimedia.org/w/index.php?curid=165125519
1202 CE
Fibonacci Introduces Arabic Numerals to Europe
Fibonacci's Liber Abaci introduces Hindu-Arabic numerals, enabling advanced geometric calculations later used in projective and fractal studies. #mathematics #history
Fibonacci Introduces Arabic Numerals to Europe By Unknown author, Public domain, https://commons.wikimedia.org/w/index.php?curid=720501
1425 CE
Brunelleschi Develops Linear Perspective
Filippo Brunelleschi demonstrates geometric perspective in Florence, a key precursor to projective geometry. #perspective #art #geometry
Brunelleschi Develops Linear Perspective By scan uploaded by Sailko, cropped by MenkinAlRire - Wikimedia Commons; Sailko; book: Elena Capretti, Brunelleschi, Giunti Editore, Firenze 2003, Public domain, https://commons.wikimedia.org/w/index.php?curid=171034830
1435 CE
Alberti's De Pictura Systematizes Perspective
Leon Battista Alberti writes De Pictura, codifying the mathematical rules of linear perspective used in projective geometry. #perspective #geometry
Alberti's De Pictura Systematizes Perspective By Leon Battista Alberti - https://archive.org/stream/dellapitturaedel00albe#page/n177/mode/2up, Public domain, https://commons.wikimedia.org/w/index.php?curid=42911315
1525 CE
Dürer Publishes Underweysung der Messung
Albrecht Dürer's treatise on geometry includes methods for perspective construction and geometric transformations. #geometry #renaissance
1550 CE
Japanese Sangaku Geometry Emerges
Sangaku, Japanese temple geometry problems, include projective and recursive geometric figures, often involving tangent circles. #japan #geometry
Japanese Sangaku Geometry Emerges By Momotarou2012 - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=26367286
Girard Desargues publishes Brouillon Project, foundational work on projective geometry introducing points at infinity and conics. #projective #geometry
Desargues' Brouillon Project Introduces Projective Geometry By Théobald Chartran - https://medium.com/language-lab/i-think-therefore-i-am-cartesian-3ebfa6afd44e, Public domain, https://commons.wikimedia.org/w/index.php?curid=187058981
1639 CE
Pascal's Theorem on Conics
Blaise Pascal discovers Pascal's theorem, a fundamental projective result about hexagons inscribed in conics. #projective #geometry
Pascal's Theorem on Conics By Wcherowi - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=22820278
1666 CE
Newton's Universal Law of Gravitation Uses Geometry
Isaac Newton develops calculus and geometric methods for planetary motion, influencing later projective representations. #geometry #physics
Newton's Universal Law of Gravitation Uses Geometry By Godfrey Kneller - File:Portrait of Sir Isaac Newton, 1689.jpg from https://exhibitions.lib.cam.ac.uk/linesofthought/artifacts, Public domain, https://commons.wikimedia.org/w/index.php?curid=132521185
1731 CE
Taylor's Linear Perspective Treatise
Brook Taylor publishes Linear Perspective, systematizing perspective drawing with projective geometric principles. #perspective #geometry
Taylor's Linear Perspective Treatise By Hans Hysing - Art UK, Public domain, https://commons.wikimedia.org/w/index.php?curid=164681398
1795 CE
Monge Found Descriptive Geometry
Gaspard Monge develops descriptive geometry, a projective method for representing 3D objects in 2D, essential for engineering and architecture. #projective #geometry
Monge Found Descriptive Geometry By François-Séraphin Delpech - http://www.sil.si.edu/digitalcollections/hst/scientific-identity/CF/by_name_display_results.cfm?scientist=Monge,%20Gaspard, Public domain, https://commons.wikimedia.org/w/index.php?curid=646122
1822 CE
Poncelet's Traité des Propriétés Projectives
Jean-Victor Poncelet publishes Traité des Propriétés Projectives, establishing projective geometry as an independent discipline with invariant properties. #projective #geometry
Poncelet's Traité des Propriétés Projectives By Smithsonian Libraries, https://library.si.edu/image-gallery/74037 (You must cite and link to, when possible, the SI Website as the source of the Content) see [3] - [1] and [2], Public domain, https://commons.wikimedia.org/w/index.php?curid=179179
1827 CE
Möbius' Barycentric Calculus
August Möbius introduces barycentric coordinates, a coordinate system for projective geometry using weighted points. #projective #geometry
Möbius' Barycentric Calculus By Adolf Neumann - http://www.portraitindex.de/documents/obj/33213645, Public domain, https://commons.wikimedia.org/w/index.php?curid=58320662
1832 CE
von Staudt's Geometrie der Lage
Karl von Staudt publishes Geometrie der Lage, a purely synthetic treatment of projective geometry without coordinates. #projective #geometry
von Staudt's Geometrie der Lage By Unknown author - http://www.rsg.rothenburg.de/, Public domain, https://commons.wikimedia.org/w/index.php?curid=5533638
1847 CE
Cayley's Analytical Geometry Advances Projective Theory
Arthur Cayley publishes work on analytical geometry, linking projective geometry with algebra and later group theory. #projective #mathematics
Cayley's Analytical Geometry Advances Projective Theory By Herbert Beraud (1845–1896) - http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Cayley.html, Public domain, https://commons.wikimedia.org/w/index.php?curid=904674
1854 CE
Riemann's Foundations of Geometry
Bernhard Riemann's Habilitationsschrift introduces Riemannian geometry, expanding projective ideas into broader geometric frameworks. #geometry #riemann
Riemann's Foundations of Geometry By Unknown author - http://www.sil.si.edu/digitalcollections/hst/scientific-identity/explore.htm according to the German Wikipedia., Public domain, https://commons.wikimedia.org/w/index.php?curid=27383
1872 CE
Klein's Erlangen Program Classifies Geometries
Felix Klein publishes the Erlangen Program, identifying projective geometry as the most fundamental geometry under group transformations. #projective #geometry
Klein's Erlangen Program Classifies Geometries By Unknown author, Public domain, https://commons.wikimedia.org/w/index.php?curid=38617
1874 CE
Cantor's Set Theory and Self-Similar Sets
Georg Cantor describes the Cantor set, an early example of a fractal with self-similarity and fractional dimension. #fractal #settheory
Cantor's Set Theory and Self-Similar Sets By Thefrettinghand at English Wikibooks - Transferred from en.wikibooks to Commons., CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=61785435
1890 CE
Peano's Space-Filling Curve
Giuseppe Peano constructs the first space-filling curve, a continuous mapping from a line to a square, challenging intuitive geometry. #fractal #curve
Peano's Space-Filling Curve By No machine-readable author provided. Arbol01 assumed (based on copyright claims). - No machine-readable source provided. Own work assumed (based on copyright claims)., CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=1776029
1904 CE
Koch Snowflake Introduced
Helge von Koch publishes the Koch snowflake, a continuous but nowhere differentiable curve with infinite perimeter, a classic fractal. #fractal #geometry
Koch Snowflake Introduced By Wrtlprnft - Own work, Public domain, https://commons.wikimedia.org/w/index.php?curid=1225322
1915 CE
Sierpinski Triangle and Carpet
Wacław Sierpiński describes the Sierpinski triangle and carpet, iconic fractals created by recursive removal of triangles or squares. #fractal #geometry
Sierpinski Triangle and Carpet By Unknown author, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=8862246
1918 CE
Hausdorff Dimension Defined
Felix Hausdorff introduces Hausdorff dimension, a key concept for measuring fractal sets with non-integer dimensions. #fractal #dimension
Hausdorff Dimension Defined By Original: Chas_zzz_brown,Shibboleth Vector: The original uploader was Wxs at Wikimedia Commons. - Own work based on: KochFlake.png, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=1898291
1926 CE
Lévy C Curve Discovered
Paul Lévy describes the Lévy C curve, a self-similar fractal curve that is everywhere continuous but nowhere differentiable. #fractal #curve
1938 CE
Weierstrass Function Exhibits Fractal Behavior
Karl Weierstrass's continuous nowhere differentiable function is later recognized as an early fractal, highlighting irregularity. #fractal #function
Weierstrass Function Exhibits Fractal Behavior By Eeyore22 - Own work, Public domain, https://commons.wikimedia.org/w/index.php?curid=5075959
1951 CE
Mandelbrot's Early Work on Scaling
Benoit Mandelbrot begins studying scaling laws and long-range dependence, later leading to fractal geometry. #fractal #scaling
Mandelbrot's Early Work on Scaling By Steve Jurvetson - https://www.flickr.com/photos/jurvetson/4770047266/, CC BY 2.0, https://commons.wikimedia.org/w/index.php?curid=155438846
1961 CE
Mandelbrot Studies Coastline Paradox
Mandelbrot investigates the coastline paradox, showing coastlines have fractal dimensions, linking geometry to natural phenomena. #fractal #nature
Mandelbrot Studies Coastline Paradox By Unknown author, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=642220
1975 CE
Mandelbrot Coins the Term 'Fractal'
Benoit Mandelbrot introduces the term 'fractal' in his book Les Objets Fractals, formalizing the study of irregular geometric shapes. #fractal #geometry
1977 CE
Mandelbrot's The Fractal Geometry of Nature
Mandelbrot publishes The Fractal Geometry of Nature, popularizing fractals and their applications across science and art. #fractal #book
1980 CE
Mandelbrot Set Visualized
The Mandelbrot set is generated using computer graphics, revealing infinite complexity and self-similarity, becoming an iconic fractal image. #fractal #computergraphics
Mandelbrot Set Visualized By Created by Wolfgang Beyer with the program Ultra Fractal 3. - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=321973
1985 CE
Barnsley's Iterated Function Systems
Michael Barnsley develops iterated function systems (IFS), a method to generate fractals using affine transformations, enabling fractal compression. #fractal #mathematics
Barnsley's Iterated Function Systems By António Miguel de Campos - self made using own JAVA animation, Public domain, https://commons.wikimedia.org/w/index.php?curid=2091769
1990 CE
Fractal Compression Patented
Barnsley and colleagues patent fractal image compression, using self-similarity to reduce file sizes, applied in digital imaging. #fractal #compression
Fractal Compression Patented By Original: Marekd Vector: Theki - Own work, Public domain, https://commons.wikimedia.org/w/index.php?curid=162766420
1995 CE
Projective Geometry in Computer Vision
Projective geometry becomes fundamental in computer vision for camera calibration, 3D reconstruction, and image rectification. #projection #computervision
2000 CE
Fractal Antennas for Wireless Communication
Fractal antenna designs are developed for multiband and miniaturized applications, leveraging self-similarity for improved performance. #fractal #engineering
Fractal Antennas for Wireless Communication By Nathan Cohen - http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&p=1&u=%2Fnetahtml%2FPTO%2Fsearch-bool.html&r=1&f=G&l=50&co1=AND&d=PTXT&s1=6452553.PN.&OS=PN/6452553&RS=PN/6452553, Public domain, https://commons.wikimedia.org/w/index.php?curid=41700498
2005 CE
African Fractal Architecture Recognized
Research highlights fractal patterns in traditional African architecture and design, showing indigenous knowledge of self-similar geometry. #fractal #africa #culture
2010 CE
Quasicrystals and Fractal Geometry
Discovery of natural quasicrystals with fractal-like ordering, connecting projective and fractal geometry in crystallography. #fractal #crystal
Quasicrystals and Fractal Geometry By J.W. Evans, The Ames Laboratory, US Department of Energy - http://www.mcbmm.ameslab.gov/high_TrappingNano.htm, Public domain, https://commons.wikimedia.org/w/index.php?curid=10094704
2015 CE
Projective Geometry in Robotics and SLAM
Projective geometry underpins simultaneous localization and mapping (SLAM) algorithms in robotics, enabling spatial reasoning. #projection #robotics
Projective Geometry in Robotics and SLAM By Unknown author, Public domain, https://commons.wikimedia.org/w/index.php?curid=2807954
2020 CE
Deep Learning for Fractal Generation
Neural networks are used to generate and classify fractals, blending machine learning with geometric self-similarity. #fractal #ai
Deep Learning for Fractal Generation By Zhang, Aston and Lipton, Zachary C. and Li, Mu and Smola, Alexander J. - https://github.com/d2l-ai/d2l-en, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=152265649
