Youngest Grandson

Youngest Grandson is Square’s precocious Hexagon grandson, whose question about what 3^3 might mean in geometry anticipates the idea of a third dimension that Square initially rejects. His later response to Square’s attempt to teach him turns that early insight into a revealing contrast: mathematical imagination opens a door, but fear of authority closes the lesson before Square can guide him through it.

Identity and Early Promise

Youngest Grandson is one of Square’s two orphan grandchildren. At the turn of the millennium, Square describes him as a promising young Hexagon, unusually brilliant and perfectly angular. His education already includes practical Sight Recognition: Square and the boy’s uncles rotate on their centres at varying speeds, while the child identifies their positions. His answers have been so satisfactory that Square rewards him with elementary arithmetic applied to geometry.

The lesson proceeds through a concrete construction. Square joins nine one-inch squares into a larger square three inches on each side, showing how the area can be calculated by squaring the side length: 3^2 represents nine square inches. The boy follows the demonstration but notices that Square has also taught him to raise numbers to the third power. He asks what 3^3 means geometrically. Square answers that it means nothing in geometry, which has only two dimensions, and explains how the motion of a point produces a line and the parallel motion of a line produces a square.

The Question of 3^3

Rather than accepting Square’s boundary between meaningful geometry and nonsense, Youngest Grandson extends the lesson’s own logic. If a point moved through three inches makes a line represented by 3, and a line moved parallel to itself makes a square represented by 3^2, then, he proposes, a square moved somehow parallel to itself might make another figure represented by 3^3. He cannot say how that motion works or what the resulting figure would be, but the analogy is precise enough to expose a possibility Square’s doctrine excludes.

Square responds not with a geometric answer but with irritation, sending the boy to bed and telling him that talking less nonsense would help him remember more sense. The child leaves disgraced. Yet his question remains with Square through the end of 1999: while sitting beside his wife and reflecting on the year ahead, Square cannot shake the thought raised by the boy’s “prattle.” As he turns the half-hour glass northward for the final time in the old millennium, he repeats his dismissal that the boy is a fool. The recurrence makes the dismissal less secure than it sounds; Square is still occupied by the very suggestion he has rejected.

Square’s First Attempt to Teach Him

After Square encounters the Sphere and learns that Flatland is a plane and that a higher dimension can disclose what a two-dimensional observer cannot see, the boy’s earlier question acquires new force. The Sphere’s explanation likewise depends on the limits of a realm: a three-dimensional being can appear in Flatland only as a section, just as Square had appeared as a line to the King of Lineland. Later, from above Flatland, Square sees the interiors of houses and creatures, and even the depths of mines and caverns. These experiences provide the knowledge he wants to pass on, though not yet the means to make it persuasive.

Back in his household, Square considers whom it is safe to teach. His Pentagonal sons are respectable physicians but, in his judgment, lack the mathematical aptitude his purpose requires. He worries that their patriotism and reverence for the Circles could lead them to report him to the Prefect for advancing the seditious heresy of a third dimension. The young Hexagon seems a better pupil: mathematically inclined, docile, and, because he is a child, apparently unaware of the Council’s new proclamation against those who profess revelations from another world.

Square first reassures his wife without mentioning the three-dimensional world, then summons his grandson. He carefully secures the door and takes out their mathematical tablets, which Square calls Lines. Resuming the previous lesson, he reviews how a point moving in one dimension produces a line and a line moving in two dimensions produces a square. He then invites the boy to repeat his conjecture that a square might move upward, rather than northward, to produce a new figure in three dimensions.

The Proclamation and Failed Lesson

As Square begins, the herald’s proclamation is heard again in the street. It announces the Council’s resolution ordering the arrest, imprisonment, or execution of anyone who perverts the people’s minds with delusions or claims revelations from another world. Youngest Grandson understands the danger more quickly than Square expects. Although still a boy and raised to revere the Circles, he waits until the proclamation ends, then bursts into tears and disavows his earlier idea. He insists that he was joking, that he knew nothing of the new law, and that he never said anything about a third dimension or about moving upward rather than northward.

The boy’s retreat is both comic and consequential. He repeats that upward motion without northward motion is absurd, then laughs at the notion. Square loses his temper and tries to demonstrate with a movable square lying at hand. But he cannot show how it moves upward; his explanation trails into an admission that it must move somewhere, somehow, and he shakes the square purposelessly. Youngest Grandson laughs harder, decides that Square is joking rather than teaching, unlocks the door, and runs out. Square’s first attempt to make a disciple of the Gospel of Three Dimensions ends without converting the pupil whose earlier reasoning had pointed closest to it.

Narrative Significance

Youngest Grandson matters less as a developed independent actor than as the hinge between Square’s inherited certainty and his later revelation. His question arises from the orderly rules Square himself teaches: the progression from point to line to square invites a further step, even when Flatland’s accepted geometry forbids it. The child does not possess a demonstrated understanding of three-dimensional space, but his analogy gives Square a first, unwelcome hint that the limits of accepted knowledge may be limits of imagination rather than of reality.

The two lessons also frame a change in their circumstances. In the first, Square can dismiss the question as nonsense and send the boy away; in the second, the Council’s proclamation has made the same subject dangerous, and the grandson’s compliance with authority overwhelms his earlier curiosity. Square’s own failure is exposed in the demonstration: he can now assert “Upward, not Northward,” but cannot translate what he has seen into a convincing movement within Flatland. The unlocked door and the boy’s laughter leave the lesson unresolved, while showing how private instruction, mathematical analogy, and political fear collide.