Posts tagged: hyperbolic geometry




This is what it would look like to spin around in 3 dimensional hyperbolic space tiled by regular right angled dodecahedron.

Created with Curved Spaces.




This is what it would look like to do a front flip in 3 dimensional hyperbolic space tiled by regular right angled dodecahedron.

Created with Curved Spaces.




Flying though a tiling of regular right angled dodecahedron in three dimensional hyperbolic space. There is more of an explanation here.

Created with Curved Spaces.


This is a tiling of regular right angled dodecahedron in three dimensional hyperbolic space shown through 4 iterations. 12 generators of the tiling are reflections in each of the 12 dodecahedron’s faces.

An ideal rotation around the point at infinity in the disc and upper half plane model of the hyperbolic plane







Rotation around the symmetry axis of 4th order in the disk and upper half plane models

A Moebius transformation can be used to transform the disk model to the upper half plane model of the hyperbolic plane.

Hyperbolic translation along the horizontal axis in the disc model, and along both horizontal and vertical directions in the band model

An ideal rotation around the point at infinity in the disc and band model of the hyperbolic plane







Rotation around the symmetry axis of 4th order in the disk and band models

This is a hyperbolic tiling with right angled regular pentagons shown in the disk model and the band model. The animation shows how one model is transformed to the other using the complex map The points +1 and -1 get sent to +∞ and -∞, respectively, and the points +i and -i remain fixed.



This is a hyperbolic tiling with right angled regular pentagons shown in the disk model and the band model. The animation shows how one model is transformed to the other using the complex map



The points +1 and -1 get sent to +∞ and -∞, respectively, and the points +i and -i remain fixed.


This is a hyperbolic tessellation with Schläfli symbol {4, 12} shown in the Poincaré disk model and in the band model. The animation displays a rotation of the hyperbolic plane. 

Hyperbolic ring tilings make use of conformal maps.


Hyperbolic ring tilings make use of conformal maps.

nminusone:

Homology Sphere - Poincaré Dodecahedral Space


(via                                                                smoot)


nminusone
:

Homology Sphere - Poincaré Dodecahedral Space

(via smoot)