Uploaded May 2018 | Updated September 2026, 2 weeks ago
"Viewing a circular staircase from the top (or conversely, viewing it from the bottom) creates the optical illusion of a mathematical logarithmic spiral. This is the same type of spiral that nature prefers, as it is seen in the chambered nautilus, the horns of a ram, the winds in a hurricane, the water in a whirlpool, and the stars in a spiral galaxy." [1]
An easy computation shows that the first sentence is not mathematically true (it is a hyperbolic spiral, not a logarithmic one). However, it is (asymptotically) true in hyperbolic geometry!
In this video, the curvature of the space changes with time. We start with 0 (Euclidean space), but for just a very short moment. Then we go to -0.5 (hyperbolic space), than back to 0 and up to 0.2 (elliptic space). The inner radius of the staircase is 1, and the outer radius is 2 (even though the stairs appear very narrow in the hyperbolic geometry). The distance between two subsequent "levels" is 2.4. The moment when the curvature becomes positive is easy to recognize -- a "hole" appears in the middle, because light rays do not hit our staircase at all!
[1] Bülent Atalay, trekearth.com/gallery/Europe/Italy/Lazio/Vatican_City/Vatican/photo831156.htm
Made with the HyperRogue engine [ roguetemple.com/z/hyper ]
Some static images: twitter.com/ZenoRogue/status/996046721804374016
This is a VR video. It can be rotated, or viewed using VR equipment, including cheap options such as mobile-based VR sets or anaglyph glasses.
"Viewing a circular staircase from the top (or conversely, viewing it from the bottom) creates the optical illusion of a mathematical logarithmic spiral. This is the same type of spiral that nature prefers, as it is seen in the chambered nautilus, the horns of a ram, the winds in a hurricane, the water in a whirlpool, and the stars in a spiral galaxy." [1]
An easy computation shows that the first sentence is not mathematically true (it is a hyperbolic spiral, not a logarithmic one). However, it is (asymptotically) true in hyperbolic geometry!
In this video, the curvature of the space changes with time. We start with 0 (Euclidean space), but for just a very short moment. Then we go to -0.5 (hyperbolic space), than back to 0 and up to 0.2 (elliptic space). The inner radius of the staircase is 1, and the outer radius is 2 (even though the stairs appear very narrow in the hyperbolic geometry). The distance between two subsequent "levels" is 2.4. The moment when the curvature becomes positive is easy to recognize -- a "hole" appears in the middle, because light rays do not hit our staircase at all!
[1] Bülent Atalay, trekearth.com/gallery/Europe/Italy/Lazio/Vatican_City/Vatican/photo831156.htm
Made with the HyperRogue engine [ roguetemple.com/z/hyper ]
Some static images: twitter.com/ZenoRogue/status/996046721804374016
This is a VR video. It can be rotated, or viewed using VR equipment, including cheap options such as mobile-based VR sets or anaglyph glasses.
