Insights into Mathematics | Synthetic versus analytic approaches to Geometry | Hexagrammum Mysticum | Wild Egg Maths @njwildberger | Uploaded 2 years ago | Updated 21 hours ago
While ancient Greek geometry, as embodied by Euclid, was built up in a step by step synthetic fashion, with proofs based on constructions and prior results, the modern European direction has been ever more in the direction laid out by Descartes and Fermat. This is based on establishing an appropriate coordinate system, thereby reducing points to ordered pairs (or in 3D triples) of numbers, and reducing curves (or in 3D surfaces) to equations. This is generally referred to as an analytic approach.

Which approach is better? Of course there are arguments to be made both ways. But for the modern student and aspiring investigator of mathematics, often with a prior course in Linear Algebra, I will argue that the analytic approach is the most suitable -- generally speaking! In this video I will lay out 10 reasons why.

This is part of a series of lectures moving towards the Hexagrammum Mysticum, that marvel of 19th century geometry built up from Pascal's remarkable extension of Pappus' theorem to conics in the plane. To access the entire Playlist, at

youtube.com/playlist?list=PLzdiPTrEWyz5j1BJdXBw1MFst_nQAqzZ_

please join the Wild Egg Maths channel by becoming a Member.

Members are able to access also the Solving Polynomial Equations Playlist (yes we really do solve this age-old problem!), the Dynamics on Graphs Playlist, and the Advice for Prospective Research Mathematicians Playlist, where we are currently expanding Number Theory in a big way to the two-dimensional situation of maxels, with applications to many of the most famous families of special polynomials, such as Chebyshev polynomials (of the first and second kind), spread polynomials, Bernoulli polynomials, Hermite polynomials, and much more.

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Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
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And here are some of the other Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-