Annexing annuli

Of course this condition on the area of the annexed region is not in itself enough: It matters how the annexed territory is distributed around , because all of the tubes have to keep on expanding more or less exponentially, and it might not be possible to avoid pinching tubes over one part of the hemisphere even though there is plenty of extra room somewhere around on the far side of the hemisphere.

To make sure that the territory we annex is nicely distributed, the ideal thing would be to annex an annulus, since then we could just divert our tubes radially away from the center of , and all would be well. Unfortunately, if is tangent or nearly tangent to the base of another relevant hemisphere, or to or , hostilities will arise when we try to annex territory belonging to or coveted by the other hemisphere. (See Figure 12.)