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There's just something special about the number 3.  Kubrick certainly seemed to think so.  His films seem to all have 3 parts and are often centered around 3 characters.  He particularly seemed to like 3-way conversations that involved tension and conflict.  Here's a few examples from some of his films.

Lolita:  Love triangle of Humbert, Quilty, and Lolita

Dr. Stangelove or How I Learned to Stop Worrying and Love the Bomb:  3 simultaneous situations - the war room, the Air Force base, and the airplane.

A Clockwork Orange:  Continual tension between Alex, Georgie, and Dim

Barry Lyndon:  Family tension between Barry, Lord Bullington, and Lady Lyndon

The Shining:  Another family dynamic of father, mother, and son

Full Metal Jacket:  Tensions between Joker, Pyle, and Hartman

Eyes Wide Shut:  Another 3 member family, this time with a daughter instead of a son.

 

Now, when it comes to 2001, Kubrick really seemed to show his obsession with the number 3.  Here's the one's I've found.  I'm sure there's more.

General

3 Titles for each chapter of the film ("The Dawn of Man", "Mission to Jupiter (18 Months Later)", and "Jupiter and Beyond the Infinite")
3 states of Man (man-ape, human being, Starchild)
2+0+0+1 = 3
3 times "Also Sprach Zarathustra" is played (Beginning, Man discovers tools, Man becomes Starchild)
3 times "Atmospheres" is played (Overture, Intermission, Through the stargate)
3 times "Requiem' is played (Monolith appearances on Earth, Moon, and Jupiter space)

Opening

3-way alignment of Moon, Earth, and Sun (this alignment of 3 objects happens at key points in the film.)
3 lead in notes in the "Also Sprach Zarathustra" music
3 opening credit titles ("MGM Presents", "A Stanley Kubrick Production", "2001: A Space Odyssey")

The Dawn of Man

3 activities of the man-apes are first shown (eating, drinking, sleeping)
Each of the 3 activities are interrupted by something (leopard, other man-ape tribe, monolith)
3-way alignment of monolith, Moon, and Sun


Earth to Moon

Orion spacecraft, space station, and the Moon or the Earth (but never both) form a triangle. They are never shown in alignment (that's for another discussion)
3 rows of "voice print identification" terminals, each with 3 screens
3 adjacent windows in space station as Floyd and Miller walk
3-way conversation in space station between Floyd, Smythslov, and Elena (a common Kubrick technique for another discussion)
3 glasses on table in front of scientists 
3 chairs along the walls inside space station 
Aries Spacecraft, Sun, and Earth (not in alignment) 
3 astronauts on lunar surface as Aries lands
3 martial arts performers on video screen in Aries
3 crewmembers in cockpit of Aries spacecraft
Aries, Moon, and Earth triangle
3 people speak at the briefing (Floyd, Halvorson, and Michaels)
3-way conversation in Moon bus between Floyd, Halvorson, and Michaels.
3-way alignment of monolith, Earth, and Sun
3 vessels take Floyd to his destination (Orion, Aries, Moon shuttle)

Mission to Jupiter

3 divisions of the viewport in front of Discovery
3 pod bay doors on Discovery
3 antenna dishes on Discovery
3 sets of 2 engines on Discovery
3 pods on Discovery
3 conscious personnel on Discovery (Bowman, Poole, and HAL)
3 hibernating personnel on Discovery
Poole is shown jogging around centrifuge 3 times
3-way conversation between reporter, Bowman, and Poole
3-way conversation between HAL, Bowman, and Poole
3 spacesuits in Discovery's pod bay
Dave pushes 3 buttons to initiate explosive bolts sequence

Jupiter and Beyond the Infinite

3-way alignment of Jupiter, large moon, and monolith (yes, there are other moons but there are 3 main objects)
Bowman, like the man-apes, eats, drinks, and sleeps before encountering the monolith.
Bowman, like the man-apes, has each activity interrupted (by space suited Bowman, breaking glass, monolith)
3-way alignment of Moon, Star-Child, and Earth

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