Is there a 3-D object with 3 faces? What is it called?
My mind cant comprehend the existence of such a shape
Thats like asking for a 2 sided 2D object.
A cylinder? Or must the faces be flat..?
No such object exists with flat sides.
three lines can only intersect in one point, coincide or be parallel so yes, no such object exists.
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.
A tetrahedron with its base removed?
Playing Captain Obvious these days? :tongue2:
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?
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.
A trihedron, of course! :P
Getting the right idea =] No one had stated the shape has to be closed.
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.
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)
Are there geometries where such an object could be constructed or is that a silly question to ask?
It only has 1 'face' and 1 edge though. And a Möbius strip is 2-dimensional (as a manifold).
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.
just start with a closed 3 dimensional object like a sphere. then triangulate it with three faces, using curved edges of course.
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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