Sci-fi's rule of 3

A lot of classic sci-fi share a similar narrative strategy of creating a novel out of district vignettes that all meditate on a similar theme or project the actions of one group of people into the future to demonstrate the consequent actions of another group of people.  Books that adopt this structure are typically more interested in thematics than in character.  At times, this structure can be unsatisfying when the author leaves off one narrative arc to pick it up in the future.  This structure is most noticeable in Isaac Asimov's Foundation series, though he does employ it in other novels.  In Foundation, the narrative opens with Hari Seldon's discovery of psychohistory, but quickly moves away from Seldon to jump decades and centuries into the future.  In this case, the chronological jumps make sense to the narrative and the form actually works with the function, as Seldon's psychohistory allows him to use current data to accurately predict future events to the point that he can engineer the direction of civilization in the direction he believes that it should go.
In other cases, however, the narrative device of jumping into the future is disruptive.  For example, Walter M. Miller, Jr.'s A Canticle for Leibowitz uses chronological jumps to show the development of Leibowitz's legacy through time.  Arguably, Miller is better at characterization than Asimov and he is better able to draw his reader into the narrative.  The novel opens with an acolyte encountering a man in the wilderness while he is on a retreat of bodily mortification and reflection.  The acolyte discovers holy relics of Leibowitz (who was an engineer in the pre-apocalyptic world) and struggles between his faith and his desire to forward the sanctification of Leibowitz.  The jumps into the future in this novel are jarring because the reader becomes invested in the acolyte's progress.  The acolyte dies at the end of the section and the jump forward picks up on a different protagonist and different challenges.
James Gunn's The Joy Makers picks up on the trope and splits into three distinct narrative sections.  In this novel, the study of Hedonics (human happiness turned into a science) is introduced, implemented, and then made into law.  Each of the three sections features a new plot line, new characters, and a different treatment of Hedonics.  In this case, Gunn shows a progression from a hopeful new direction for humanity through to the inevitable conclusion that assured happiness is incompatible with freedom.  There is greater unity in this novel than in many others that adopt the same structure, but it is jarring nonetheless.
The structure of novels like these rely on a different narrative logic than most novels.  In more traditional novelistic structures, the reader can track progression and resolution in a straightforward way.  Even in postmodern novels or in those that play with time and causality, there is generally still an arc that will lead through to a sort of conclusion. This diegesis has a momentum that draws the reader along and propels meaning.  The logic of a novel like The Joy Makers is based on syndesis, or a connection of different parts.  The novel presents the different sections and asks the reader to fill in the meaning between them, to synthesize the connections and discern meaning.
Many of these novels also probably began as separate stories or were published serially, so that they were originally written to stand alone.  This was the case with Miller's book.  Whatever the origin of the novels, the work of making sense of the finished novel remains the same.  The reader must still work to re-assemble the discrete plot lines to make sense of the larger thematics at work.
The rule of three, then, refers to the tendency to split these novels into three sections.  In this way, it does mirror a story arc: the initial section sets up the conceit of the novel, in The Joy Maker, it is the idea that happiness can be mathematically calculated and, thus, ensured to everyone.  The second section builds upon the topic of the original, typically to demonstrate growth of the idea from its initial stages to broader acceptance.  In this case, Hedonics is made the law of the land and Happiness rather than the pursuit of the same is considered to be an inalienable right.  Finally, the third section makes clear why this is actually undesirable.  The story arc, re-written as disconnected stories creates a sort of ironic structure that requires extra effort on the part of the reader.  The author trusts in the reader to make connections between the sections and to grasp the direction of the whole.