In geometry, three-dimensional space included the three planes of space in which all matter in the physical universe existed, encompassing the X-, Y-, and Z-axes.

A hologram was a three-dimensional projection of light and energy. (VOY: "Lifesigns")

To utilize all three dimensions in navigation, a ship's heading was given in measurements called degrees on a sphere, so when degrees are given on both the horizontal and vertical plane, a ship can head in any direction in all three dimensions. (TNG: "Datalore")

During the Battle of the Mutara Nebula in 2285, Spock noted how Khan Noonien Singh patterns indicated "two-dimensional thinking" in three-dimensional space. To use this to their advantage, Admiral James T. Kirk lowered the USS Enterprise on the z-axis by 10,000 meters. (Star Trek II: The Wrath of Khan)

When the USS Enterprise-D was caught in the wake of two-dimensional lifeforms, all of their attempts to escape were based on the laws of a three-dimensional universe. During the event, Picard, was curious with how a two-dimensional entity could have access to a three-dimensional universe. (TNG: "The Loss")

When Jean-Luc Picard was reintroduced to the Borg Queen, he was confused because he thought that the ship and all the Borg on it were destroyed, including her. She belittled Picard by how small he had become for thinking in such three-dimensional terms. (Star Trek: First Contact)

During their attempt to combat the Kazon, the idea was passed around to create holographic ships in space as false targets. The idea, promoted by The Doctor, explained that "projecting the illusion of a large, three-dimensional object has been a trick of magicians for centuries." He then suggested that it could be done by "installing holo-emitters along the hull with parabolic mirrors to enlarge the images as they are reflected into space." (VOY: "Basics, Part I")

In 2377, B'Elanna Torres mocked Tom Paris created a three-dimensional movie marathon in the holodeck, noting the irony that he went "to all this trouble to program a three-dimensional environment that projects a two-dimensional image, and now you're asking me to wear these (3D glasses) to make it look three-dimensional again?" To Paris' amusement, he replied, "Great, isn't it?" (VOY: "Repression")

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