Spherical Geometry

In this video, we investigate some of the basic properties of Spherical Geometry. Almost all of what is taught in high schools is, specifically, Euclidean geometry. Spherical Geometry is based on a different set of axioms, so many of the ideas that are taken for granted are not true in this geometry.
(If you found this video helpful and want to say "thanks," you can buy me a coffee here: ko-fi.com/tomteachesmath)
The Geogebra applet used in this video can be found at: www.geogebra.org/m/WRvnNKHe#c...
TIMESTAMPS:
0:00 Introduction and historical background
3:50 "Lines" in Spherical Geometry
6:22 "Segments" in Spherical Geometry
6:46 Other comparisons between spherical and Euclidean geometry
12:15 Application of spherical geometry
13:22 Other important takeaways and general ideas

[@hiyasingh8484]

Thank you very much !!!! This video was extremely helpful

[@LaReine30]

This was very helpful, thank you! Please add the link to the geogebra great circle. Thanks!

[@ronaldmorris8390]

Continue the series!

[@raymondallenjhangiani7670]

Beautifully explained 🙏 thank you sir ! 👍

[@TomTeachesMath]

You’re very welcome!!

[@flyorwalk1743]

Thank you for this.

[@TomTeachesMath]

My pleasure!

[@moriah5880]

I'm very curious as to why, in spherical geometry, great circles are considered lines, but small circles are not. Do you happen to know why only great circles are classified as a line?

[@TomTeachesMath]

Great question. Basically, a "straight" line would always result in a great circle, because your line would have to "turn" right or left to create a smaller circle. This link can help you visualize it: www.geogebra.org/m/WRvnNKHe#material/gC7nCaFR

[@lliaolsen728]

Is it possible to make a spherical triangle with only one point? Lets call this triangle (A) Which is a triangle that grows larger and larger, wraps around the sphere and all the points converge on each other? or is there a point where the spherical triangle inverts itself. Meaning the spherical triangle can only grow to a certain size on the sphere, before the outside of the triangle switches and becomes the inside, and the triangle slowly shrinks. This type of triangle will be called (B) Then within that realm of thought... can we mistake one triangle for another. Say if we think the spherical triangle is (A) but it's actually a (B), will that throw off our measurements or expectations, Especially on cosmic scales?

[@TomTeachesMath]

Interesting question. I believe I understand what you're asking. From everything I've read & researched, Spherical Triangles have a sum of angles that is greater than 180 _but less than 540_ . So, I guess technically once the triangle is taking up more than half the circle it isn't a triangle anymore? You can play with it here: www.geogebra.org/m/WRvnNKHe#material/CAun8d7k But you're exactly right, at some point, the triangle "flips" and the inside is considered the outside. My hunch is that technically it doesn't fit the spherical definition of a triangle at that point, so we wouldn't consider it a triangle. Great question! Sorry I don't have a better answer.

[@karatsurba4791]

Thank you Teacher. Are great circles always along the longitude ? If so, why not along the latitude as well. (Dang, i'm looking at a ball 🏀 n can visualize great circles along the latitude )

[@TomTeachesMath]

It can, but the only “latitude” line that would be a great circle is the equator.

[@karatsurba4791]

@@TomTeachesMath thank you .... hmmm ... I just learnt that great circles pass through the center of the sphere. So, that makes my question irrelevant.

[@kaizenyasou6963]

Thank you 🤍

[@TomTeachesMath]

You’re welcome!!!

[@paxer]

There are some loose ends. What is the definition of an angle exactly? This probably calls for some limits because the angle is only well defined at a point and if you follow a great circle the angle changes

[@williamcompitello2302]

Good point.

[@betaorionis2164]

Yes, the angle is only defined at one point and yes, the angle changes if you follow a great circle. Sometimes, people get confused about angles in curved surfaces, because they tend to think in terms of segments of a given length, which obviously don't fit a curved surface. Well, no, you don't need any length to define an angle, 1/1000 of a millimeter (technically in maths, an infinitessimal of length) will be enough to define it, because the angle only exists at one given point.

[@scientist169]

Please continue this series

[@TomTeachesMath]

I'm glad you enjoyed the video! I actually just made this video as a lesson for my students, so I was never intending to do a full series.

[@zenith6427]

@@TomTeachesMath That's is a pity, I was looking forward to more; I loved it

[@TomTeachesMath]

Dang. Now y’all have me considering doing some more on this! If I’m ever able to get caught up with my own grading and planning maybe I’ll make another. I’m glad you liked it!!

[@zenith6427]

@@TomTeachesMath Thank you. This actually made my day!

[@yiutungwong315]

So π = 2 in Sphere Geometry and Riemann Mathematical Paradox...

[@StringerNews1]

You're the smartest man in the whole flat earth.

[@user-nf9nl5tr5r]

Iam learning this from class 7

[@TomTeachesMath]

Cool. I don’t know what “class 7” means, but I hope you’re finding it interesting!

[@homelesshendrix]

Anyone who thinks traveling in a strait line will take them back where they came from is imprisoned by their own beliefs. Earth is not a planet.