# What is a 3-faced 3-D object?

#### EnumaElish

Homework Helper
Is there a 3-D object with 3 faces? What is it called?

#### Ki Man

My mind cant comprehend the existence of such a shape

#### dtl42

Thats like asking for a 2 sided 2D object.

A politician?

#### disregardthat

A cylinder? Or must the faces be flat..?

#### Dale

Mentor
No such object exists with flat sides.

#### ice109

three lines can only intersect in one point, coincide or be parallel so yes, no such object exists.

#### Gib Z

Homework Helper
There is an infinite number of 3 dimensional objects with 3 faces, just as there are an infinite number 2 dimensional objects with 2 sides.

#### uman

A tetrahedron with its base removed?

#### Werg22

There is an infinite number of 3 dimensional objects with 3 faces, just as there are an infinite number 2 dimensional objects with 2 sides.
Playing Captain Obvious these days? :tongue2:

#### EnumaElish

Homework Helper
A cylinder? Or must the faces be flat..?
Good answer; but I was asking about flat surfaces. I understand that no such object exists. Is there a theorem about the min. number of flat surfaces that a 3-D object must have?

#### Werg22

I think it's best we first clarify what a closed 3 dimensional object is for our purposes. I would say that a simple closed 3 dimensional object is a collection of planar surfaces with properties among which is that a surface in that collection connects to at least as many surfaces as it has vertices. The simplest planar figure is the triangle; since it has 3 sides, it is not possible to meet the said property with only 3 planar faces, hence there is no such 3 dimensional object.

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#### Ben Niehoff

Gold Member
Is there a 3-D object with 3 faces? What is it called?
A trihedron, of course! :P

#### Gib Z

Homework Helper
A tetrahedron with its base removed?
Getting the right idea =] No one had stated the shape has to be closed.

#### masnevets

Good answer; but I was asking about flat surfaces. I understand that no such object exists. Is there a theorem about the min. number of flat surfaces that a 3-D object must have?
Hello,

What you are thinking of is a 3 dimensional (convex) polytope. I assume you mean codimension 1 faces (i.e. 2 dimensional faces). Technically, edges and vertices are also called faces. In this case, the minimum number of faces is 4 (a tetrahedron). In general, an n dimensional polytope needs to have at least n+1 facets.

#### rbzima

Hello,

What you are thinking of is a 3 dimensional (convex) polytope. I assume you mean codimension 1 faces (i.e. 2 dimensional faces). Technically, edges and vertices are also called faces. In this case, the minimum number of faces is 4 (a tetrahedron). In general, an n dimensional polytope needs to have at least n+1 facets.
Darn, I was going to say that. Furthermore, when you consider the convex hull of this polytope, the convex hull can extend in two dimensions; but there needs to be at least one point above the plane of the other points in order to achieve what you desire. Otherwise it's simply a 2-dimensional face.

#### mgb_phys

Homework Helper
Wouldn't a Mobius strip have 3 faces? ie. 1 'face' and 2 edges.

#### dst

Wouldn't a Mobius strip have 3 faces? ie. 1 'face' and 2 edges.
Good thinking. Why wouldn't it? Make the edges thicker = 3 faces.

But obviously no flat surfaced object could have 3 faces. (politicians aside)

#### titaniumx3

Are there geometries where such an object could be constructed or is that a silly question to ask?

#### masnevets

Wouldn't a Mobius strip have 3 faces? ie. 1 'face' and 2 edges.
It only has 1 'face' and 1 edge though. And a Möbius strip is 2-dimensional (as a manifold).

#### mgb_phys

Homework Helper
Of course - the edges are connected - dumb of me.
So it's possible to have a 2 sided 3D object but not a 3 sided one ?

#### Huckleberry

Of course - the edges are connected - dumb of me.
So it's possible to have a 2 sided 3D object but not a 3 sided one ?
A cylinder has 3 sides if you include it's curved surface. Both 'sides' of a mobius strip are curved into each other so you only have 2 sides. With only flat surfaces I think n+1 is the minimum number.

#### mathwonk

Homework Helper
just start with a closed 3 dimensional object like a sphere. then triangulate it with three faces, using curved edges of course.

#### mathwonk

Homework Helper
actually i am not sure what you mean by three dimensional object as i have described a 2 dimensional manifold. a three dimensional object should have some 3 dimensional faces.

i guess i thought you meant a closed surface that does not embed in 2 space, which is the lay persons notion of a three dimensional object.

so to answer the question we need a definition of a three dimensional object.

i guess i could modify ,my example by taking the cone over it with vertex at the center of the sphere, i.e. make it into a ball example.

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