Movement and section

Use in the book

The Square's dream visit to Lineland offers the first experiment. The Line can display only the portion of his body lying within it, so the intersecting portion is a Line rather than a Square, though the King’s eye sees only its end as a Point. As he moves laterally out of the realm, the King does not perceive the unfamiliar direction. He perceives continued presence followed by disappearance. The higher figure cannot bring its entire form into a space that has too few dimensions to contain it.

The Sphere explains his own appearance by referring to that encounter. His circular manifestation is a section where he intersects Flatland. As he rises, this circular section shrinks towards a Point and disappears; on returning it grows again. The Square sees the section edgewise as a changing line, not as the full circular outline available in the reader’s diagram. The Square's report that the visitor made himself smaller preserves his initial interpretation, not the actual mechanics. The Sphere does not physically contract and expand. Different sections of the same body enter the plane.

Movement supplies the generative lesson as well. A Point moving leaves a Line, a Line moving in a new independent direction makes a Square, and a Square moving out of its plane makes a Cube. The crucial condition is that each point trace a path not already traversed by another point of the figure. Simply moving a Square northward within Flatland cannot do the required work, however great the distance.

These two uses of motion complement one another without being identical. A section explains the partial manifestation of a body that already exists; generation explains how the extent of a higher-dimensional figure may be conceived from lower-dimensional elements. The Sphere moves his own body through Flatland in the first case and describes or demonstrates a construction in the second. Confusing the two makes the visit resemble shape-changing magic and obscures the mathematical lesson.

Finally, movement is not distributed equally between teacher and pupil. Once lifted out of Flatland, the Square cannot move independently. He can see his distressed Wife and imprisoned brother but cannot reach them. Dimensional access enlarges perception while creating dependence upon the guide's volition. Abbott thus makes movement a problem of agency as well as a means of geometrical explanation.

Places to read

1. §14234“I see you still; you are not moving” — An unfamiliar motion remains perceptually unavailable
2. §162345“only a slice or section of you” — The Sphere explains partial manifestation
3. §162345“as I rise in Space, so my section becomes smaller” — Sectional change is not bodily shrinkage
4. §162345“each Point shall describe a straight Line of its own” — Generation requires independent paths
5. §192345“I absolutely depended on the volition of my Guide” — New sight does not bring autonomous movement

In this edition

The diagrams and notes distinguish changing sections from changing bodily size and explain construction by parallel movement. They also retain the narrator's limited descriptions, allowing readers to see the difference between his first explanation and the geometrical account he later learns.

Line of Lines

The Square's description of his own construction from a moving Line. He turns a valid generative relation into a claim that he completes and infinitely surpasses the King. Printed form: Line of Lines (§ 14234). See Movement and section.

Luminous wake

A shining trace left by movement. The Sphere asks the Square to imagine such a trace so that familiar Lines and Squares can lead towards a generated Cube. Printed form: luminous wake (§ 162345). See Movement and section.

Section

The intersection of a higher-dimensional body with a lower-dimensional region. The Sphere appears in Flatland through a circular section, which the Square sees edgewise as a line. Printed form: section (§ 162345). See Movement and section.

Volition

The act or power of willing. The Sphere initially says that sufficient volition could let the Square leave his plane, although the later journey shows the pupil dependent on the guide's will. Printed form: volition (§ 172). See Movement and section.

The completion of your incomplete self (contextual phrase)

§14234. “the completion of your incomplete self”

The Square makes dimensional extension sound like the fulfilment of another being's deficient nature. His next explanation, that he is a Line of Lines, has a geometrical basis: a Line moving sideways can generate a Square. The moral and social judgment does not follow with equal necessity. To possess one more dimension is to have powers the King lacks; it does not establish that the King exists merely as an unfinished version of the visitor. The grand claim appears in an angry, unsuccessful attempt at instruction. It anticipates the narrator's later readiness to worship the Sphere and the Sphere's explicit correction that greater visual access does not make a person more divine.

See also. Movement and section · Omnividence and the limits of godlike sight