Computing volume for a scutoid
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I was challenged by someone to compute the volume for a scutoid. At first I wanted to know how this shape is described mathematically, but did not find much on that so I started with what I already know:
- bottom is hexagonal
- top is pentagonal
- there is a triangle between 2 "faces"
- the 2 "faces" where the triangle cannot be in one plane
Starting from these, the first thing that I needed to do, independent of the method of computation, was to get some formula for the non-plane faces. My attempt to solve this relied on assuming that the surface satisfies Laplace equation (I thought of it as an elastic membrane that is stretched so that it touches the edges of the face I'm interested in).
Having those 2 computed, I guess one can try to integrate but I'm still traumatized by this approach, so I would make use of the method of inserting this shape into a cube like volume, generate random points and them count the ones that fall inside the shape.
Any other ideas that are not based on using a PC (solving the Laplace equations and the volume computation part)?
geometry volume
edited Aug 7 at 15:58
Ed Pegg
9,24732486
asked Aug 7 at 15:37
Victor Palea
246311
add a comment |Â
up vote
2
down vote
favorite
I was challenged by someone to compute the volume for a scutoid. At first I wanted to know how this shape is described mathematically, but did not find much on that so I started with what I already know:
- bottom is hexagonal
- top is pentagonal
- there is a triangle between 2 "faces"
- the 2 "faces" where the triangle cannot be in one plane
Starting from these, the first thing that I needed to do, independent of the method of computation, was to get some formula for the non-plane faces. My attempt to solve this relied on assuming that the surface satisfies Laplace equation (I thought of it as an elastic membrane that is stretched so that it touches the edges of the face I'm interested in).
Having those 2 computed, I guess one can try to integrate but I'm still traumatized by this approach, so I would make use of the method of inserting this shape into a cube like volume, generate random points and them count the ones that fall inside the shape.
Any other ideas that are not based on using a PC (solving the Laplace equations and the volume computation part)?
geometry volume
edited Aug 7 at 15:58
Ed Pegg
9,24732486
asked Aug 7 at 15:37
Victor Palea
246311
There are practical ways, i.e. building one and then sinking it and measuring the volume of water displaced. Can't help beyond that.
– stuart stevenson
Aug 7 at 15:43
The Wikipedia article says the name comes from scutum so I guess it's pronounced "skew-toid" not "scut-oid."
– saulspatz
Aug 7 at 15:56
You need more constraints, there are many types of scutoids, so that information alone won't help you find the volume.
– Rushabh Mehta
Aug 7 at 15:59
Ignore curved faces. Add a tetrahedron to get a hexagonal prism.
– Ed Pegg
Aug 7 at 16:00
1
@EdPegg He could use the lattice points you used in your other question, but that won't have the packing property that the scutoid has because of flat faces.
– Rushabh Mehta
Aug 7 at 16:02
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I was challenged by someone to compute the volume for a scutoid. At first I wanted to know how this shape is described mathematically, but did not find much on that so I started with what I already know:
- bottom is hexagonal
- top is pentagonal
- there is a triangle between 2 "faces"
- the 2 "faces" where the triangle cannot be in one plane
Starting from these, the first thing that I needed to do, independent of the method of computation, was to get some formula for the non-plane faces. My attempt to solve this relied on assuming that the surface satisfies Laplace equation (I thought of it as an elastic membrane that is stretched so that it touches the edges of the face I'm interested in).
Having those 2 computed, I guess one can try to integrate but I'm still traumatized by this approach, so I would make use of the method of inserting this shape into a cube like volume, generate random points and them count the ones that fall inside the shape.
Any other ideas that are not based on using a PC (solving the Laplace equations and the volume computation part)?
geometry volume
edited Aug 7 at 15:58
Ed Pegg
9,24732486
asked Aug 7 at 15:37
Victor Palea
246311
I was challenged by someone to compute the volume for a scutoid. At first I wanted to know how this shape is described mathematically, but did not find much on that so I started with what I already know:
- bottom is hexagonal
- top is pentagonal
- there is a triangle between 2 "faces"
- the 2 "faces" where the triangle cannot be in one plane
Starting from these, the first thing that I needed to do, independent of the method of computation, was to get some formula for the non-plane faces. My attempt to solve this relied on assuming that the surface satisfies Laplace equation (I thought of it as an elastic membrane that is stretched so that it touches the edges of the face I'm interested in).
Having those 2 computed, I guess one can try to integrate but I'm still traumatized by this approach, so I would make use of the method of inserting this shape into a cube like volume, generate random points and them count the ones that fall inside the shape.
Any other ideas that are not based on using a PC (solving the Laplace equations and the volume computation part)?
geometry volume
edited Aug 7 at 15:58
Ed Pegg
9,24732486
asked Aug 7 at 15:37
Victor Palea
246311
edited Aug 7 at 15:58
Ed Pegg
9,24732486
edited Aug 7 at 15:58
Ed Pegg
9,24732486
edited Aug 7 at 15:58
Ed Pegg
9,24732486
9,24732486
asked Aug 7 at 15:37
Victor Palea
246311
asked Aug 7 at 15:37
Victor Palea
246311
asked Aug 7 at 15:37
Victor Palea
246311
246311
There are practical ways, i.e. building one and then sinking it and measuring the volume of water displaced. Can't help beyond that.
– stuart stevenson
Aug 7 at 15:43
The Wikipedia article says the name comes from scutum so I guess it's pronounced "skew-toid" not "scut-oid."
– saulspatz
Aug 7 at 15:56
You need more constraints, there are many types of scutoids, so that information alone won't help you find the volume.
– Rushabh Mehta
Aug 7 at 15:59
Ignore curved faces. Add a tetrahedron to get a hexagonal prism.
– Ed Pegg
Aug 7 at 16:00
1
@EdPegg He could use the lattice points you used in your other question, but that won't have the packing property that the scutoid has because of flat faces.
– Rushabh Mehta
Aug 7 at 16:02
add a comment |Â
There are practical ways, i.e. building one and then sinking it and measuring the volume of water displaced. Can't help beyond that.
– stuart stevenson
Aug 7 at 15:43
The Wikipedia article says the name comes from scutum so I guess it's pronounced "skew-toid" not "scut-oid."
– saulspatz
Aug 7 at 15:56
You need more constraints, there are many types of scutoids, so that information alone won't help you find the volume.
– Rushabh Mehta
Aug 7 at 15:59
Ignore curved faces. Add a tetrahedron to get a hexagonal prism.
– Ed Pegg
Aug 7 at 16:00
1
@EdPegg He could use the lattice points you used in your other question, but that won't have the packing property that the scutoid has because of flat faces.
– Rushabh Mehta
Aug 7 at 16:02
There are practical ways, i.e. building one and then sinking it and measuring the volume of water displaced. Can't help beyond that.
– stuart stevenson
Aug 7 at 15:43
There are practical ways, i.e. building one and then sinking it and measuring the volume of water displaced. Can't help beyond that.
– stuart stevenson
Aug 7 at 15:43
The Wikipedia article says the name comes from scutum so I guess it's pronounced "skew-toid" not "scut-oid."
– saulspatz
Aug 7 at 15:56
The Wikipedia article says the name comes from scutum so I guess it's pronounced "skew-toid" not "scut-oid."
– saulspatz
Aug 7 at 15:56
You need more constraints, there are many types of scutoids, so that information alone won't help you find the volume.
– Rushabh Mehta
Aug 7 at 15:59
You need more constraints, there are many types of scutoids, so that information alone won't help you find the volume.
– Rushabh Mehta
Aug 7 at 15:59
Ignore curved faces. Add a tetrahedron to get a hexagonal prism.
– Ed Pegg
Aug 7 at 16:00
Ignore curved faces. Add a tetrahedron to get a hexagonal prism.
– Ed Pegg
Aug 7 at 16:00
1
1
@EdPegg He could use the lattice points you used in your other question, but that won't have the packing property that the scutoid has because of flat faces.
– Rushabh Mehta
Aug 7 at 16:02
@EdPegg He could use the lattice points you used in your other question, but that won't have the packing property that the scutoid has because of flat faces.
– Rushabh Mehta
Aug 7 at 16:02
add a comment |Â
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There are practical ways, i.e. building one and then sinking it and measuring the volume of water displaced. Can't help beyond that.
– stuart stevenson
Aug 7 at 15:43
The Wikipedia article says the name comes from scutum so I guess it's pronounced "skew-toid" not "scut-oid."
– saulspatz
Aug 7 at 15:56
You need more constraints, there are many types of scutoids, so that information alone won't help you find the volume.
– Rushabh Mehta
Aug 7 at 15:59
Ignore curved faces. Add a tetrahedron to get a hexagonal prism.
– Ed Pegg
Aug 7 at 16:00
1
@EdPegg He could use the lattice points you used in your other question, but that won't have the packing property that the scutoid has because of flat faces.
– Rushabh Mehta
Aug 7 at 16:02