Conditions for equidistant points in space

Question

Given a set of n distinct points in space, where n is a natural number greater than 1, what is the geometric structure that results from connecting each point to every other point with straight lines of equal length?

Answer 1

Results of prelim investigation (given the constraint that all points need to be equidistant from each other)

1. With 1 point: A point (0 dimensions)
2. With 2 points: A straight line (1 dimension)
3. With 3 points: An equilateral triangle - 3 sides with equal lengths (2 dimensions)
4. With 4 points: A regular terahedron - 4 equilateral triangles as 4 faces (3 dimensions)
5. With 5 points: I don't know what to call this geometrical object, let's name it $T_1$ , but I know it has 5 tetrahedrons as its "sides" ($5 \times 4 = 20$ equilateral triangular sides)
6. With 6 points: This will have 6 of $T_1$, which is $6 \times 5 \times 4 = 120$ equilateral triangles
   n) With $n$ points, this object will have $^nP_{n - 3}$ equilateral triangles.