Euler's triangulation of a polygon | Famous Math Problems 8 | NJ Wildberger

Описание: In 1751 Leonhard Euler, the greatest mathematician of modern times, posed the problem of counting the number of ways of triangulating an n-gon. When n=3, this is 1, when n=4, this is 2, when n=5 this is 5, and when n=6 this is 6. The sequence C_n, defined as the number of ways of triangulating an (n+2)-gon, is called the Catalan numbers, and appears in many different guises in combinatorics. In this lecture we derive Euler's formula for C_n, and describe just a few ways in which this fascinating sequence appears, as well as connecting it to Newton's binomial expansion for sqrt(1-4x).

This lecture has quite a few smaller problems inside that you might like to think about.

*******
Research Gate page: https://www.researchgate.net/profile/...

Blog: http://njwildberger.com/

Online courses at openlearning.com (currently Algebraic Calculus One): https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects!

Patreon:   / njwildberger   Your support would be much appreciated.

Wild Egg Maths YT channel: https://www.youtube.com/channel/UCriF...

Insights into Mathematics Playlists:

   • The Algebra of Boole, Logic and Circuit An...   (31 videos)
   • Box Arithmetic: a new framework for Mathem...   (18 videos)
   • Hypergroups and Diffusion Symmetry: an int...   (6 videos)
   • Rational Trigonometry for maths, physics a...   (4 videos)
   • Sociology and Pure Maths   (44 videos)
   • Old Babylonian mathematics and Plimpton 322   (8 videos)
   • Math Foundations   (226 videos)
   • Math Seminars N J Wildberger   (26 videos)
   • Math History (ancient to modern)   (45 videos)
   • Geometric Linear Algebra   (43 videos)
   • Algebraic Topology   (40 videos)
   • Universal Hyperbolic Geometry   (55 videos)
   • Differential Geometry   (34 videos)
   • Elementary Probability and Statistics   (8 videos)
   • Math Terminology for Incoming Uni Students   (9 videos)
   • Famous Math Problems   ( 46 videos)
   • Elementary Mathematics Explained (K-6)   (40 videos)
   • Ancient Mathematics and insights of Howard...   (7 videos)
   • Wild West Banking: A mathematician goes We...   (7 videos)
   • Playing Go: the ancient oriental board game   (19 videos)
   • Maths and Music   (21 videos)
   • Year 9 Mathematics (review fractions, deci...   (10 videos)
   • Wild Trig: An introduction to Rational Tri...   (94 videos)

Wild Egg Maths Playlists:

   • Intro to Algebraic Calculus with Box Arith...   (4 videos)
   • Classical to Quantum (for Members)   (64 videos)
   • Solving Polynomial Equations and the Geode...   (45 videos)
   • De Casteljau Bezier curves and associated ...   (20 videos)
   • Exploring q-series (for Members)   (8 videos)
   • Six: A mathematical exploration   (9 videos)
   • Algebraic Calculus One: a new foundation f...   (52 videos)
   • Advice to prospective research mathematici...   (9 videos)
   • The Hexagrammum Mysticum: a Geometric Gem ...   (14 videos)
   • Algebraic Calculus Two   (8 videos)
   • Special (Polynomial) Functions and Maxel N...   (25 videos)
   • Dynamics on Graphs: The ADE Phenomenon & B...   (30 videos)