Mathemaniac | Theorema Egregium: why all maps are wrong @mathemaniac | Uploaded 1 year ago | Updated 2 hours ago
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Website for files download (remember to use the password shown in the video!): mathemaniac.co.uk/download
The Mercator projection is the standard world map, but it famously makes Greenland and Africa the same size, but in reality, Greenland is so much smaller. Gall-Peters projection aims to solve exactly this area mismatch problem, but the shape resulted is horrible, and actually unsuitable for any navigation, unlike Mercator. Can we make a world map that preserves both areas (like Gall-Peters) and angles (like Mercator)? No, and the reason why is Theorema Egregium, the subject of the video.
Traditionally, Theorema Egregium was proved with a lot of tedious calculations, and somehow magically, you can compute the curvature with the "first fundamental form", whatever that means. It took until more than a century later than its original discovery that a geometric proof was found, and is presented here.
Theorema Egregium, more traditional proof, going through first and second fundamental forms: dpmms.cam.ac.uk/~gpp24/dgnotes/dg.pdf
Tristan Needham's book on visual differential geometry: amazon.co.uk/Visual-Differential-Geometry-Forms-Mathematical/dp/0691203709
Video chapters:
00:00 Introduction
02:40 Chapter 1: Curvature
10:32 Chapter 2: Spherical areas
17:34 Chapter 3.1: Gauss map preserves parallel transport
22:15 Chapter 3.2: Geodesics preserved
27:16 Chapter 3.3: Parallel transport preserved
31:46 Chapter 3.4: Area = holonomy on sphere
36:43 Chapter 4: Tying everything together
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.
Social media:
Facebook: facebook.com/mathemaniacyt
Instagram: instagram.com/_mathemaniac_
Twitter: twitter.com/mathemaniacyt
Patreon: patreon.com/mathemaniac (support if you want to and can afford to!)
Merch: mathemaniac.myspreadshop.co.uk
Ko-fi: ko-fi.com/mathemaniac [for one-time support]
For my contact email, check my About page on a PC.
See you next time!
Head to squarespace.com/mathemaniac to save 10% off your first purchase of a website or domain using code mathemaniac.
Website for files download (remember to use the password shown in the video!): mathemaniac.co.uk/download
The Mercator projection is the standard world map, but it famously makes Greenland and Africa the same size, but in reality, Greenland is so much smaller. Gall-Peters projection aims to solve exactly this area mismatch problem, but the shape resulted is horrible, and actually unsuitable for any navigation, unlike Mercator. Can we make a world map that preserves both areas (like Gall-Peters) and angles (like Mercator)? No, and the reason why is Theorema Egregium, the subject of the video.
Traditionally, Theorema Egregium was proved with a lot of tedious calculations, and somehow magically, you can compute the curvature with the "first fundamental form", whatever that means. It took until more than a century later than its original discovery that a geometric proof was found, and is presented here.
Theorema Egregium, more traditional proof, going through first and second fundamental forms: dpmms.cam.ac.uk/~gpp24/dgnotes/dg.pdf
Tristan Needham's book on visual differential geometry: amazon.co.uk/Visual-Differential-Geometry-Forms-Mathematical/dp/0691203709
Video chapters:
00:00 Introduction
02:40 Chapter 1: Curvature
10:32 Chapter 2: Spherical areas
17:34 Chapter 3.1: Gauss map preserves parallel transport
22:15 Chapter 3.2: Geodesics preserved
27:16 Chapter 3.3: Parallel transport preserved
31:46 Chapter 3.4: Area = holonomy on sphere
36:43 Chapter 4: Tying everything together
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos.
Social media:
Facebook: facebook.com/mathemaniacyt
Instagram: instagram.com/_mathemaniac_
Twitter: twitter.com/mathemaniacyt
Patreon: patreon.com/mathemaniac (support if you want to and can afford to!)
Merch: mathemaniac.myspreadshop.co.uk
Ko-fi: ko-fi.com/mathemaniac [for one-time support]
For my contact email, check my About page on a PC.
See you next time!
