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Only theoretically. A "point" takes up some space, but obviously they are used on the assumption that a "point" takes up no space at all. This assumption makes it difficult for them to realistically exist.

Q: Can a point practically exist in three dimensional world?

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A cone needs a three dimensional space in which to exist but it's not a solid, it's a two dimensional surface.

Cone

A triangle is a two dimensional shape with three straight lines and three vertices. A sphere is a three dimensional object with no straight lines and no vertices and such that every point on its surface is a constant distance from a fixed point. It is not possible to have a triangle sphere.

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Yes, it does exist.

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The term "statue" is used for three-dimensional works of art. Of course, two-dimensional works of art exist, but they are not usually called "statues".

A 3D triangle

There are a number of three dimensional shapes. These include the cube, the square pyramid, the prism, as well as the rhombohedron.

vertex

A cone needs a three dimensional space in which to exist but it's not a solid, it's a two dimensional surface.

A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space.

Three-dimensional things are objects that have height, width, and depth. They exist in physical space and have the ability to be viewed from multiple angles. Examples include a cube, sphere, and pyramid.

Presumably, the "three dimensional triangular plane" is actually a two dimensional plane which is "tilted" with respect to the axes. The point of intersection is simply the coordinates of the solution to the simultaneous equations for the line and the plane.