Hexagrammum Mysticum 2 | Analytic versus synthetic approaches to geometry | Wild Egg Maths

We are aiming to understand the many remarkable features of the Hexagrammum Mysticum including 19th century work up to recent insights by John Conway and others, and we hope to go even further in certain directions. But because geometry has fallen somewhat off the mathematics education landscape in recent decades, it is worthwhile having an informal discussion about various approaches to this beautiful subject. In this talk, I give 10 reasons why you might like to think of the analytic approach to geometry above the perhaps more historical synthetic one. However that is not to say that the synthetic view is not also very valuable! This discussion is a personal one, and I am only expressing my opinions here. The Playlist "The Hexagrammum Mysticum" will be available to Members of this channel as part of the Explore Research Level Mathematics series of videos, meant for amateur mathematicians wanting to delve into real maths research. Here is the Playlist:    • Playlist   Video Contents (thanks to phi Architect) 0:00 Intro - Arguments for Analytic Approach to Geometric Investigations 0:48 Synthetic Approach 1:18 Analytic Approach 2:35 1 - Analytic approach can be modeled on a computer 3:00 2 - Implicit hidden assumptions are worrying 3:51 3 - Purely algebraic arguments tend to be more generalizable 6:05 4 - Number theoretical considerations are more clearly defined 7:38 5 - Symmetry vs anti-symmetry 9:22 6 - Less brilliance required 10:38 7 - Flexibility of the field (coordinate system) 11:26 8 - Rational Trigonometry 11:59 9 - Formulas are independent, reusable 12:34 10 - Closer to Linear Algebra ************************ Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/... My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at   / njwildberger   Your support would be much appreciated. Here are the Wild Egg Maths Playlists (some available only to Members!)    • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist   ************************************************ Here are the Insights into Mathematics Playlists:    • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist   ****************