This is such an underrated channel! I would have used homogenous coordinates for the hexagonal tiling, to match with the matrix representations of the spherical and hyperbolic tilings.
@@Number_Cruncher Of course! By embedding the tiling in 3D, specifically the plane z=1, translations in addition to rotations/reflections can be described by 3x3 matrix multiplication, where the third column is dedicated to the translation. Since it's hard to type this stuff in a youtube comment, I threw together some LaTeX to demonstrate: quicklatex.com/cache3/36/ql_c4d3ae28b27f18bfed835349c1d93e36_l3.png Equations (1)-(3) show how to rewrite the formula at 2:59 in homogeneous coordinates, and (4)-(6) give the homogeneous versions of M_a, M_b, and M_c. You can verify that the properties at 3:19 hold, so these matrices form a Coxeter group like the 2x2 ones.
Thank you very much. This is nice and easy. I was afraid of some projective geometry, when I heard homogeneous coordinates. I'll add your improvements to the video description. I wish, I had known this before:-)
This is such an underrated channel! I would have used homogenous coordinates for the hexagonal tiling, to match with the matrix representations of the spherical and hyperbolic tilings.
May I ask you to write a few more details on your idea of homogeneous coordinates?
@@Number_Cruncher Of course! By embedding the tiling in 3D, specifically the plane z=1, translations in addition to rotations/reflections can be described by 3x3 matrix multiplication, where the third column is dedicated to the translation. Since it's hard to type this stuff in a youtube comment, I threw together some LaTeX to demonstrate:
quicklatex.com/cache3/36/ql_c4d3ae28b27f18bfed835349c1d93e36_l3.png
Equations (1)-(3) show how to rewrite the formula at 2:59 in homogeneous coordinates, and (4)-(6) give the homogeneous versions of M_a, M_b, and M_c. You can verify that the properties at 3:19 hold, so these matrices form a Coxeter group like the 2x2 ones.
Thank you very much. This is nice and easy. I was afraid of some projective geometry, when I heard homogeneous coordinates. I'll add your improvements to the video description. I wish, I had known this before:-)
@@Number_Cruncher I found your video through the Polytope Discord, a community of (mostly) math people. Would you consider joining?
discord.gg/zMRu7T4
Biggest understatement: “That’s All for Today” love this Dude
So beautiful.. It's ART!
Beautiful, concise, illuminating! Thank you very much!
Thanks. I've wanted to understand this better for a while now.
Reminds me of flame fractals. :)
Great work!
wonderfuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuul < amazing< thank you
Exquisite
Thankyou.
It's nice to see that you seem to enjoy these videos.
@@Number_Cruncher They are wonderful for those best educated visually. Although, also beneficial to others. Thankyou again.
nice video!
this si brilliant. I have a question. How do You animate The tesallation spherical i'm manim ?
The three dimensional animations weren't made in manim. At the time the video was made I didn't understand the 3d possibilities in manim well enough.
Oh, this time I almost understood! 😲
If you don't mind, leave a comment about what's left unclear.