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Given a n-dimension regular polytope, we can choose coordinates so that the polytope is in R" with center at the origin, and every vertex is distance 2 from the origin. By surrounding each vertex and the center of with a sphere of radius 1, we obtain a configuration with one inner sphere, and some number of outer spheres, which touch the inner sphere at a single point (the midpoint between the vertex and the center). However, depending on the polytope, these outer spheres may or may not overlap. 1. For each regular 3-dimensional polyhedron, determine which of the following scenarios holds: the interior of some pair of the outer spheres overlaps some pair of the outer spheres "kiss" but there is no overlap of their interiors the outer spheres do not touch each other 2. Also determine which of the scenarios above holds for the simplex, cross polytope and hypercube in dimension n for any n. 3. In which dimensions does the best known kissing configuration come from a regular polytope? Given a n-dimension regular polytope, we can choose coordinates so that the polytope is in R" with center at the origin, and every vertex is distance 2 from the origin. By surrounding each vertex and the center of with a sphere of radius 1, we obtain a configuration with one inner sphere, and some number of outer spheres, which touch the inner sphere at a single point (the midpoint between the vertex and the center). However, depending on the polytope, these outer spheres may or may not overlap. 1. For each regular 3-dimensional polyhedron, determine which of the following scenarios holds: the interior of some pair of the outer spheres overlaps some pair of the outer spheres "kiss" but there is no overlap of their interiors the outer spheres do not touch each other 2. Also determine which of the scenarios above holds for the simplex, cross polytope and hypercube in dimension n for any n. 3. In which dimensions does the best known kissing configuration come from a regular polytope