Geometry: Is there a 3D shape in which all points are equidistant from one another, apart from a tetrahedron?
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Geometry: Is there a 3D shape in which all points are equidistant from one another, apart from a tetrahedron?
In three dimensions, the only regular polyhedron where all points (or vertices) are equidistant from each other is the tetrahedron, which is a type of pyramid with four triangular faces. However, when considering shapes beyond the tetrahedron, we can also think about higher-dimensional analogs.
In 3D, if we are looking for shapes where all vertices are equidistant, we encounter the case of a regular tetrahedron, where each vertex is equidistant from the others. Beyond triangles and tetrahedrons, there isn’t a three-dimensional shape that has this property for all its vertices. As soon as you move to other polyhedra, such as cubes or octahedrons, the distances between vertices are no longer the same.
In summary, in three dimensions, apart from the tetrahedron, there isn’t a shape where all points are equidistant from each other in the same way. If we consider higher dimensions, such as a 4D shape like a 5-cell (or tetrahedral), all vertices are equidistant, but in three dimensions, the tetrahedron is the primary example fulfilling this condition.