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Is there a simpler single polygon toroid?
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In B.M. Stewart's book Adventures Among the Toroids, toroidal shapes of many sorts are made. One of them is the ring of 8 octahedra, with 48 faces. The toroid is made with a single polygon -- the equilateral triangle.
Are there single non-regular polygons that can make a toroidal shape with fewer than 48 faces? One restriction -- all neighboring polygons must be in different planes, to prevent things like the ring of 8 cubes.
The faces should be non-intersecting. The underlying graph of edges might be one of these, maybe.
recreational-mathematics polygons polyhedra solid-geometry
asked Jul 28 at 2:40
Ed Pegg
9,13932486
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up vote
4
down vote
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In B.M. Stewart's book Adventures Among the Toroids, toroidal shapes of many sorts are made. One of them is the ring of 8 octahedra, with 48 faces. The toroid is made with a single polygon -- the equilateral triangle.
Are there single non-regular polygons that can make a toroidal shape with fewer than 48 faces? One restriction -- all neighboring polygons must be in different planes, to prevent things like the ring of 8 cubes.
The faces should be non-intersecting. The underlying graph of edges might be one of these, maybe.
recreational-mathematics polygons polyhedra solid-geometry
asked Jul 28 at 2:40
Ed Pegg
9,13932486
Am I looking at it wrongly, or does the second example graph have some faces which are triangles and some which are quadrilaterals?
– Peter Taylor
Jul 31 at 10:53
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
In B.M. Stewart's book Adventures Among the Toroids, toroidal shapes of many sorts are made. One of them is the ring of 8 octahedra, with 48 faces. The toroid is made with a single polygon -- the equilateral triangle.
Are there single non-regular polygons that can make a toroidal shape with fewer than 48 faces? One restriction -- all neighboring polygons must be in different planes, to prevent things like the ring of 8 cubes.
The faces should be non-intersecting. The underlying graph of edges might be one of these, maybe.
recreational-mathematics polygons polyhedra solid-geometry
asked Jul 28 at 2:40
Ed Pegg
9,13932486
In B.M. Stewart's book Adventures Among the Toroids, toroidal shapes of many sorts are made. One of them is the ring of 8 octahedra, with 48 faces. The toroid is made with a single polygon -- the equilateral triangle.
Are there single non-regular polygons that can make a toroidal shape with fewer than 48 faces? One restriction -- all neighboring polygons must be in different planes, to prevent things like the ring of 8 cubes.
The faces should be non-intersecting. The underlying graph of edges might be one of these, maybe.
recreational-mathematics polygons polyhedra solid-geometry
asked Jul 28 at 2:40
Ed Pegg
9,13932486
edited Jul 28 at 2:47
asked Jul 28 at 2:40
Ed Pegg
9,13932486
asked Jul 28 at 2:40
Ed Pegg
9,13932486
asked Jul 28 at 2:40
Ed Pegg
9,13932486
9,13932486
Am I looking at it wrongly, or does the second example graph have some faces which are triangles and some which are quadrilaterals?
– Peter Taylor
Jul 31 at 10:53
add a comment |Â
Am I looking at it wrongly, or does the second example graph have some faces which are triangles and some which are quadrilaterals?
– Peter Taylor
Jul 31 at 10:53
Am I looking at it wrongly, or does the second example graph have some faces which are triangles and some which are quadrilaterals?
– Peter Taylor
Jul 31 at 10:53
Am I looking at it wrongly, or does the second example graph have some faces which are triangles and some which are quadrilaterals?
– Peter Taylor
Jul 31 at 10:53
add a comment |Â
1 Answer
1
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Let $y=2, x=sqrt5+2 sqrt2approx 2.79793 $, Then the following toroid with green $y$ and blue $x$ lengths is made from 24 identical triangles.
Took less than 12 hours for someone to build it.
The net, with green points the 6 outer vertices:
Is there anything smaller than 24 faces? Here's something larger.
Let $y=2, -127 + 124 x^2 - 26 x^4 - 4 x^6 + x^8=0, xapprox 2.31498614558$. Then the following toroid with green $y$ and blue $x$ lengths is made from 32 identical triangles.
answered Aug 2 at 23:02
Ed Pegg
9,13932486
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
Let $y=2, x=sqrt5+2 sqrt2approx 2.79793 $, Then the following toroid with green $y$ and blue $x$ lengths is made from 24 identical triangles.
Took less than 12 hours for someone to build it.
The net, with green points the 6 outer vertices:
Is there anything smaller than 24 faces? Here's something larger.
Let $y=2, -127 + 124 x^2 - 26 x^4 - 4 x^6 + x^8=0, xapprox 2.31498614558$. Then the following toroid with green $y$ and blue $x$ lengths is made from 32 identical triangles.
answered Aug 2 at 23:02
Ed Pegg
9,13932486
add a comment |Â
up vote
2
down vote
Let $y=2, x=sqrt5+2 sqrt2approx 2.79793 $, Then the following toroid with green $y$ and blue $x$ lengths is made from 24 identical triangles.
Took less than 12 hours for someone to build it.
The net, with green points the 6 outer vertices:
Is there anything smaller than 24 faces? Here's something larger.
Let $y=2, -127 + 124 x^2 - 26 x^4 - 4 x^6 + x^8=0, xapprox 2.31498614558$. Then the following toroid with green $y$ and blue $x$ lengths is made from 32 identical triangles.
answered Aug 2 at 23:02
Ed Pegg
9,13932486
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Let $y=2, x=sqrt5+2 sqrt2approx 2.79793 $, Then the following toroid with green $y$ and blue $x$ lengths is made from 24 identical triangles.
Took less than 12 hours for someone to build it.
The net, with green points the 6 outer vertices:
Is there anything smaller than 24 faces? Here's something larger.
Let $y=2, -127 + 124 x^2 - 26 x^4 - 4 x^6 + x^8=0, xapprox 2.31498614558$. Then the following toroid with green $y$ and blue $x$ lengths is made from 32 identical triangles.
answered Aug 2 at 23:02
Ed Pegg
9,13932486
Let $y=2, x=sqrt5+2 sqrt2approx 2.79793 $, Then the following toroid with green $y$ and blue $x$ lengths is made from 24 identical triangles.
Took less than 12 hours for someone to build it.
The net, with green points the 6 outer vertices:
Is there anything smaller than 24 faces? Here's something larger.
Let $y=2, -127 + 124 x^2 - 26 x^4 - 4 x^6 + x^8=0, xapprox 2.31498614558$. Then the following toroid with green $y$ and blue $x$ lengths is made from 32 identical triangles.
answered Aug 2 at 23:02
Ed Pegg
9,13932486
edited 18 hours ago
answered Aug 2 at 23:02
Ed Pegg
9,13932486
answered Aug 2 at 23:02
Ed Pegg
9,13932486
answered Aug 2 at 23:02
Ed Pegg
9,13932486
9,13932486
add a comment |Â
add a comment |Â
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Am I looking at it wrongly, or does the second example graph have some faces which are triangles and some which are quadrilaterals?
– Peter Taylor
Jul 31 at 10:53