Can higher dimensional spheres be regularly partitioned/discretized?
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A circle can be partitioned into $ninmathbb{N}$ congruent 1-spherical line segments similar to the regular polygons.
A sphere can be partitioned into $nin{4,6,20}$ congruent 2-spherical equilateral triangles similar to the tetra-, octa-, and icosahedron.
Is that the end of the story, or is it possible to partition a glome into congruent 3-spherical tetrahedrons for some $n$s?
geometry spheres platonic-solids
asked Nov 21 at 9:27
Oppenede
356111
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A circle can be partitioned into $ninmathbb{N}$ congruent 1-spherical line segments similar to the regular polygons.
A sphere can be partitioned into $nin{4,6,20}$ congruent 2-spherical equilateral triangles similar to the tetra-, octa-, and icosahedron.
Is that the end of the story, or is it possible to partition a glome into congruent 3-spherical tetrahedrons for some $n$s?
geometry spheres platonic-solids
asked Nov 21 at 9:27
Oppenede
356111
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
A circle can be partitioned into $ninmathbb{N}$ congruent 1-spherical line segments similar to the regular polygons.
A sphere can be partitioned into $nin{4,6,20}$ congruent 2-spherical equilateral triangles similar to the tetra-, octa-, and icosahedron.
Is that the end of the story, or is it possible to partition a glome into congruent 3-spherical tetrahedrons for some $n$s?
geometry spheres platonic-solids
asked Nov 21 at 9:27
Oppenede
356111
A circle can be partitioned into $ninmathbb{N}$ congruent 1-spherical line segments similar to the regular polygons.
A sphere can be partitioned into $nin{4,6,20}$ congruent 2-spherical equilateral triangles similar to the tetra-, octa-, and icosahedron.
Is that the end of the story, or is it possible to partition a glome into congruent 3-spherical tetrahedrons for some $n$s?
geometry spheres platonic-solids
geometry spheres platonic-solids
asked Nov 21 at 9:27
Oppenede
356111
asked Nov 21 at 9:27
Oppenede
356111
asked Nov 21 at 9:27
Oppenede
356111
asked Nov 21 at 9:27
Oppenede
356111
asked Nov 21 at 9:27
Oppenede
356111
356111
add a comment |
add a comment |
1 Answer
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The question is equivalent to asking for regular polytopes whose facets are simplices. There are two infinite families: the simplices themselves (which are generalizations of tetrahedra), and the cross-polytopes (which are generalizations of octahedra). In four dimensions, there is also the 600-cell.
answered Nov 21 at 11:19
Rahul
32.9k467165
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
The question is equivalent to asking for regular polytopes whose facets are simplices. There are two infinite families: the simplices themselves (which are generalizations of tetrahedra), and the cross-polytopes (which are generalizations of octahedra). In four dimensions, there is also the 600-cell.
answered Nov 21 at 11:19
Rahul
32.9k467165
add a comment |
up vote
1
down vote
The question is equivalent to asking for regular polytopes whose facets are simplices. There are two infinite families: the simplices themselves (which are generalizations of tetrahedra), and the cross-polytopes (which are generalizations of octahedra). In four dimensions, there is also the 600-cell.
answered Nov 21 at 11:19
Rahul
32.9k467165
add a comment |
up vote
1
down vote
up vote
1
down vote
The question is equivalent to asking for regular polytopes whose facets are simplices. There are two infinite families: the simplices themselves (which are generalizations of tetrahedra), and the cross-polytopes (which are generalizations of octahedra). In four dimensions, there is also the 600-cell.
answered Nov 21 at 11:19
Rahul
32.9k467165
The question is equivalent to asking for regular polytopes whose facets are simplices. There are two infinite families: the simplices themselves (which are generalizations of tetrahedra), and the cross-polytopes (which are generalizations of octahedra). In four dimensions, there is also the 600-cell.
answered Nov 21 at 11:19
Rahul
32.9k467165
answered Nov 21 at 11:19
Rahul
32.9k467165
answered Nov 21 at 11:19
Rahul
32.9k467165
answered Nov 21 at 11:19
Rahul
32.9k467165
32.9k467165
add a comment |
add a comment |
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