Some concave volumes are perceived as compositions of joined objects. Joined objects are optically undetermined, being compatible with three possible solutions: an empty intersection volume bounded by two indented objects; an intersection volume entirely filled-in by parts of two indented objects; a totally convex object joined to an indented object (the solution preferred by most perceivers). Knowledge of material penetrability does not predict 3D amodal completion. To identify relevant factors we selected different compositions and ran a multiple-regression analysis of completion choices. After eliminating colour, we considered six spatial factors: relative position along the vertical, relative size, orientation, proximity to the observer, minimization of completed parts, support relationship. The first two factors played a major role. The comparison of choices by upright versus inverted observers indicated that relative position along the vertical depends more on gravitational than egocentric coordinates. In conclusion, when relative spatial position and relative size cooperate, amodal completion of intersecting solids generates strong effects, which can overcome logical expectations.