T-Sphere
This activity belongs to the GeoGebra book GeoGebra Principia.
A T-sphere is formed by the points in space that are T-equidistant from its center. It takes on the shape of a regular E-octahedron (which is not T-regular, as T-regular triangles do not exist) with its diagonals parallel to the axes. It can also be generated by T-rotating a T-circle around the horizontal or vertical diameter.
The loci that appear in taxicab geometry invite us to ask: what does "something" need to have in order to be that "something"? What characterizes an object? For instance, is the E-sphere round because its points are equidistant from its center, or is it round due to the uniformity in the direction that Euclidean distance possesses?
Author of the construction of GeoGebra: Rafael Losada.