The film opens in a two dimensional world called Flatland populated by living squares, triangles, lines and other polygons. It is three days until the celebration of the year 3000. A Square, attorney at law, struggles to instruct his grand daughter, A Hexagon, in the art of sight recognition. The lesson is interrupted by his brother B Squares arrival. As clerk to President Circle, B Square wants to warn his big brother to stay home that afternoon while there is a meeting at the Senate of the Great Southern Republic of Flatland.
The Senate session has been called to discuss the increasing hostilities between the Chromatist movement and the government. Senator Chromatistes is the irregular polygon who leads this rebellious movement that wants to legalize the right of Flatlanders to color their sides as they see fit. Traditionally a taboo practice, the laws were previously relaxed somewhat which emboldened the Chromatists to demand legalization against the will of the majority. The Great Southern Republic distinguishes itself from its enemy, the Northern Kingdom, by its stances on Chromatism and Irregulars along with a democratic government. Relaxing the laws has already been perceived as weakness by the Northern Kingdom who are massing on the borders.
Unable to heed his brothers advice, A Square meets with his new client, the first female charged as a Chromatist and on his way home finds himself embroiled in the melee issuing forth from the Senate. President Circles soldiers have assassinated Senator Chromatistes along with his supporters thus sparking a riot that has spilled into the streets of the city. A Square barely makes it home safely where he then barricades his family against the chaos for the night.
Once asleep, though, A Square has an unusual dream in which he visits a one dimensional world, Lineland, and attempts to convince the realm's ignorant king of a second dimension. A Square finds it essentially impossible to make the King of Lineland see outside of his eternally straight line. After waking uneasily, A Square learns that the riots which killed hundreds originated in the Senate meeting that B Square was attending. A visit to B Squares home confirms that he is missing in action and sends A Square through the city now under martial law in a fruitless attempt to find B Square. Passing through the marketplace, A Square comes upon an unusual merchant selling a fascinating item called a glow point which he purchases for A Hexagon who has been sulking since being disciplined for coloring one of his sides purple.
The family prepares for another anxious night in Flatland only to be completely terrified by the sudden appearance in their home of A Sphere, CEO of Messiah, Inc. A Sphere declares that A Square is his apostle of the Three Dimensions and privately begins to explain things him. A Square resolutely refuses to entertain the concept of three dimensions to the point of becoming violent with A Sphere. In desperation, A Sphere decides the only way to convince A Square that there are three dimensions is to show him.
Suddenly and quite painfully, A Square finds himself plucked out of his dimension and into Spaceland. After his initial shock, the gospel of 3 dimensions becomes clear to A Square along with the possibility of locating his brother from his newfound point of view. Having business himself at the Great Hall, A Sphere brings A Square to look for his brother there. On their arrival, A Sphere delivers the standard millennial speech espousing three dimensions to President Circle and the Priests who are anticipating this very event. After rejecting A Spheres message and attempting to kill him, the Flatland leaders execute all who have witnessed the event except B Square who is only imprisoned for life on pain of silence while his brother watches from high above Flatland.
Realizing that time in Spaceland is short, at least for A Square, A Sphere brings him to Messiah, Inc. to finish his education on the gospel of Three Dimensions. Enthralled by the complex world of Spaceland, A Square posits further about 4 dimensions and so forth to A Sphere who dismisses it as nonsense. Meanwhile, A Squares intrusion into Spaceland has become a national emergency which prompts the Spaceland Senate to call to him to appear for a hearing in the Senate Chambers. The Senators demand an explanation from A Sphere for this serious breach of protocol in bringing a Flatlander into their midst. They insist that it will be viewed as an act of war by their enemies, the X-Axis, who will take the opportunity to launch a full scale war.
During the discussion, A Square also learns that the X-Axis considers the Great Senate as weak because they have allowed the continued existence of his own world, Flatland, which they view as an abomination. As the debate in the Senate rages, an ailing A Square tries to explain his theory of multiple dimensions to an unsympathetic crowd. Air raid sirens wail as A Square collapses from the overwhelming effects of gravity on his two dimensional body and chaos ensues.
A Sphere manages to send his dying apostle back on his way to Flatland via a mailing tube before bombs destroy Messiah, Inc. but A Squares journey back is fraught with obstacles and he ends up plunging in freefall toward Flatland and through its surface into unknown regions below where he experiences a revelation on dimensionality and infinity.
The next thing he knows, A Square finds himself in his own bed on the eve of the year 3000 and his family inform him that the government has issued a proclamation ordering the arrest of anyone claiming to proclaim the gospel of three dimensions. Undeterred, A Square hurries to see his brother in Flatland jail to discuss their new shared knowledge of the third dimension. B Square, afraid of execution, denies the experience and in a panic assaults his brother who falls temporarily into an unconscious state where he encounters the now deceased A Sphere along with the Monarch of Pointland who curiously resembles a glow point. As the Monarch drones on in his monologue of being the all in all, the one in the one, A Sphere informs his former apostle that time is short and A Square must proclaim the gospel of the three dimensions to his fellow Flatlanders although they, like the Monarch, will probably remain trapped within their own perspectives.
Returning to himself, A Square manages to escape Flatland prison and the guards that B Square has set upon his brother. Arriving at his home ahead of his pursuers, A Square informs his wife that they are going to defect to the Northern Kingdom where he might be able to spread the gospel of three dimensions to a more open minded populace. The soldiers arrive and A Square makes his escape with the help of Frau A Squares war cry that temporarily stuns them. Before he can reach the border, A Square is cornered by the soldiers whose attempt to dismantle and segment him is thwarted by the Northern Kingdom armys attack. In the fracas, things suddenly begin to disappear as though sucked down through the fabric of Flatland until only A Square remains. He too begins to disappear until there is only his eye, then a point of light, a glowing point of light, which welcomes him into another dimension.