It is not possible to make a sphere using only hexagons because of the geometric constraints involved in creating a curved surface without distortion. In order to cover a curved surface like a sphere, you need to use polygons that can conform to the curvature without leaving gaps or overlaps.
A hexagon has six sides, and when you try to arrange hexagons to cover a sphere, they cannot fit together perfectly without distortion. If you start with a single hexagon and try to add more hexagons around it, you will notice that there will always be gaps between the hexagons. These gaps cannot be filled with more hexagons without creating even more gaps. Alternatively, if you try to force the hexagons to fit together by bending or stretching them, you will end up distorting their shape.
This geometric limitation is known as the "Gauss-Bonnet theorem," which states that the total curvature of a surface (like a sphere) is related to the angles of the polygons used to cover it. In the case of hexagons, their angles cannot add up to the necessary curvature of a sphere without causing gaps or overlaps.
However, it is possible to make a sphere using a combination of polygons, such as a combination of hexagons and pentagons. This is known as a "geodesic dome" or a "Buckminster Fuller dome," where the hexagons and pentagons form a tessellation that can cover the sphere without gaps or overlaps.