For a regular polyhedron whose Schläfli symbol is {m,n}, the number of polygonal faces may be found by:
The Platonic solids known to antiquity are the only integer solutions for m ≥ 3 and n ≥ 3. The restriction m ≥ 3 enforces that the polygonal faces must have at least three sides.
When considering polyhedra as a spherical tiling, this restriction may be relaxed, since digons (2-gons) can be represented as spherical lunes, having non-zero area. Allowing m = 2 admits a new infinite class of regular polyhedra, which are the hosohedra. On a spherical surface, the polyhedron {2,n} is represented as n abutting lunes, with interior angles of 2π/n. All these lunes share two common vertices.
http://demonstrations.wolfram.com/DigonTilingOfAHosohedron/
The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily.
Hosohedra are spheres tiled by digons, which are spherical wedges. At vertical resolution 3 the digons tile a bicone.
Contributed by: Michael Schreiber
published: 08 Jul 2009
Looping in a tetragonal Hosohedron
We slowly turn around a horizontal axis in the 3-sphere observing the meridians of the the equatorial 2-sphere colored to form a tetragonal hosohedron.
Music: Beginning of "Depth Charge" by Metre
licensed under a Attribution-NonCommercial-ShareAlike License. (Link: https://freemusicarchive.org/music/Metre/flow-1 )
This Video is also licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/4.0/)
published: 12 Oct 2021
Journey through a 3-Sphere containing an Equatorial Heptagonal Hosohedron
Music: Beginning of "Frog Dream(Instrumental)" by Chad Crouch
licensed under a Attribution-NonCommercial License. (Link: https://freemusicarchive.org/music/Chad_Crouch/field-report-vol-vi-bayocean-instrumental)
This Video is licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/4.0/)
published: 10 Apr 2021
Turn Around in a 3-Sphere containing a pentagonal Hosohedron
We look around in a 3-sphere containing an equatorial pentagonal hosohedron.Music: Beginning of "Sweet Spot" by Scanglobe
licensed under a Attribution-NonCommercial-ShareAlike License. (Link: https://freemusicarchive.org/music/Scanglobe/Telegraph)
This Video is also licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/4.0/)
published: 18 Feb 2020
Elevator Ride through a 3-Sphere containing a equatorial octagonal Hosohedron
Music: Beginning of "The Edge of Nowhere" by Scott Holmes
licensed under a Attribution-NonCommercial License. (Link: https://scottholmesmusic.com/)
This Video is licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/4.0/)
published: 12 Mar 2020
Train Ride through a 3-Sphere containing an equatorial octahedral Hosohedron
We travel to the right through a 3-sphere divided in two by 8 digons.
Music: Beginning of "Canada" by Pictures of the Floating World
licensed under a Attribution-NonCommercial-ShareAlike License. (Link: https://freemusicarchive.org/music/Pictures_of_the_Floating_World/Canada)
This Video is also licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/3.0/us/)
http://demonstrations.wolfram.com/DigonTilingOfAHosohedron/
The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with ne...
http://demonstrations.wolfram.com/DigonTilingOfAHosohedron/
The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily.
Hosohedra are spheres tiled by digons, which are spherical wedges. At vertical resolution 3 the digons tile a bicone.
Contributed by: Michael Schreiber
http://demonstrations.wolfram.com/DigonTilingOfAHosohedron/
The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily.
Hosohedra are spheres tiled by digons, which are spherical wedges. At vertical resolution 3 the digons tile a bicone.
Contributed by: Michael Schreiber
We slowly turn around a horizontal axis in the 3-sphere observing the meridians of the the equatorial 2-sphere colored to form a tetragonal hosohedron.
...
We slowly turn around a horizontal axis in the 3-sphere observing the meridians of the the equatorial 2-sphere colored to form a tetragonal hosohedron.
Music: Beginning of "Depth Charge" by Metre
licensed under a Attribution-NonCommercial-ShareAlike License. (Link: https://freemusicarchive.org/music/Metre/flow-1 )
This Video is also licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/4.0/)
We slowly turn around a horizontal axis in the 3-sphere observing the meridians of the the equatorial 2-sphere colored to form a tetragonal hosohedron.
Music: Beginning of "Depth Charge" by Metre
licensed under a Attribution-NonCommercial-ShareAlike License. (Link: https://freemusicarchive.org/music/Metre/flow-1 )
This Video is also licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/4.0/)
Music: Beginning of "Frog Dream(Instrumental)" by Chad Crouch
licensed under a Attribution-NonCommercial License. (Link: https://freemusicarchive.org/music/Ch...
Music: Beginning of "Frog Dream(Instrumental)" by Chad Crouch
licensed under a Attribution-NonCommercial License. (Link: https://freemusicarchive.org/music/Chad_Crouch/field-report-vol-vi-bayocean-instrumental)
This Video is licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/4.0/)
Music: Beginning of "Frog Dream(Instrumental)" by Chad Crouch
licensed under a Attribution-NonCommercial License. (Link: https://freemusicarchive.org/music/Chad_Crouch/field-report-vol-vi-bayocean-instrumental)
This Video is licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/4.0/)
We look around in a 3-sphere containing an equatorial pentagonal hosohedron.Music: Beginning of "Sweet Spot" by Scanglobe
licensed under a Attribution-NonComme...
We look around in a 3-sphere containing an equatorial pentagonal hosohedron.Music: Beginning of "Sweet Spot" by Scanglobe
licensed under a Attribution-NonCommercial-ShareAlike License. (Link: https://freemusicarchive.org/music/Scanglobe/Telegraph)
This Video is also licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/4.0/)
We look around in a 3-sphere containing an equatorial pentagonal hosohedron.Music: Beginning of "Sweet Spot" by Scanglobe
licensed under a Attribution-NonCommercial-ShareAlike License. (Link: https://freemusicarchive.org/music/Scanglobe/Telegraph)
This Video is also licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/4.0/)
Music: Beginning of "The Edge of Nowhere" by Scott Holmes
licensed under a Attribution-NonCommercial License. (Link: https://scottholmesmusic.com/)
This Vi...
Music: Beginning of "The Edge of Nowhere" by Scott Holmes
licensed under a Attribution-NonCommercial License. (Link: https://scottholmesmusic.com/)
This Video is licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/4.0/)
Music: Beginning of "The Edge of Nowhere" by Scott Holmes
licensed under a Attribution-NonCommercial License. (Link: https://scottholmesmusic.com/)
This Video is licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/4.0/)
We travel to the right through a 3-sphere divided in two by 8 digons.
Music: Beginning of "Canada" by Pictures of the Floating World
licensed under a Attrib...
We travel to the right through a 3-sphere divided in two by 8 digons.
Music: Beginning of "Canada" by Pictures of the Floating World
licensed under a Attribution-NonCommercial-ShareAlike License. (Link: https://freemusicarchive.org/music/Pictures_of_the_Floating_World/Canada)
This Video is also licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/3.0/us/)
We travel to the right through a 3-sphere divided in two by 8 digons.
Music: Beginning of "Canada" by Pictures of the Floating World
licensed under a Attribution-NonCommercial-ShareAlike License. (Link: https://freemusicarchive.org/music/Pictures_of_the_Floating_World/Canada)
This Video is also licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/3.0/us/)
http://demonstrations.wolfram.com/DigonTilingOfAHosohedron/
The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily.
Hosohedra are spheres tiled by digons, which are spherical wedges. At vertical resolution 3 the digons tile a bicone.
Contributed by: Michael Schreiber
We slowly turn around a horizontal axis in the 3-sphere observing the meridians of the the equatorial 2-sphere colored to form a tetragonal hosohedron.
Music: Beginning of "Depth Charge" by Metre
licensed under a Attribution-NonCommercial-ShareAlike License. (Link: https://freemusicarchive.org/music/Metre/flow-1 )
This Video is also licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/4.0/)
Music: Beginning of "Frog Dream(Instrumental)" by Chad Crouch
licensed under a Attribution-NonCommercial License. (Link: https://freemusicarchive.org/music/Chad_Crouch/field-report-vol-vi-bayocean-instrumental)
This Video is licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/4.0/)
We look around in a 3-sphere containing an equatorial pentagonal hosohedron.Music: Beginning of "Sweet Spot" by Scanglobe
licensed under a Attribution-NonCommercial-ShareAlike License. (Link: https://freemusicarchive.org/music/Scanglobe/Telegraph)
This Video is also licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/4.0/)
Music: Beginning of "The Edge of Nowhere" by Scott Holmes
licensed under a Attribution-NonCommercial License. (Link: https://scottholmesmusic.com/)
This Video is licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/4.0/)
We travel to the right through a 3-sphere divided in two by 8 digons.
Music: Beginning of "Canada" by Pictures of the Floating World
licensed under a Attribution-NonCommercial-ShareAlike License. (Link: https://freemusicarchive.org/music/Pictures_of_the_Floating_World/Canada)
This Video is also licensed under a Attribution-NonCommercial-ShareAlike License. (https://creativecommons.org/licenses/by-nc-sa/3.0/us/)
For a regular polyhedron whose Schläfli symbol is {m,n}, the number of polygonal faces may be found by:
The Platonic solids known to antiquity are the only integer solutions for m ≥ 3 and n ≥ 3. The restriction m ≥ 3 enforces that the polygonal faces must have at least three sides.
When considering polyhedra as a spherical tiling, this restriction may be relaxed, since digons (2-gons) can be represented as spherical lunes, having non-zero area. Allowing m = 2 admits a new infinite class of regular polyhedra, which are the hosohedra. On a spherical surface, the polyhedron {2,n} is represented as n abutting lunes, with interior angles of 2π/n. All these lunes share two common vertices.