The Story of Flatland: an Adventure in Many Dimensions

The Story of Flatland: An Adventure in Many Dimensions

Adapted from Edwin A. Abbott’s original by Suzanne Fox Buchele

Mathematical Content and Learning Guide

Two-Dimensional Geometry:

̶ Closed figure

Encloses interior space

̶ Non-closed figures

̶ Triangle, Square, Pentagon, Hexagon, Septagon, Octagon, Decagon

̶ N-gon for N ≥ 3

N-sided polygon

̶ Isosceles triangle

Two equal sides

̶ Equilateral triangle

All sides (and angles) equal

̶ Angles measured in degrees and minutes

̶ Sharp (acute) angles

Less than 90○

̶ Obtuse angles

More than 90○

̶ Concave (reflex) angles

More than 180○

̶ Convexity

For any two interior points in a closed figure, if the line between them is completely contained inside the figure, then the closed figure is convex

̶ Interior angles of a triangle sum to 180○

Optional class discussion: sum of interior angles of any N-gon = 180*(N-3) = 180N-360

̶ Regular N-gon

Angles all equal, edges all equal

Optional class discussion: each interior angle of a regular N-gon = 180 – 360/N

A circle as the limit of a regular N-gon, as N goes to infinity

̶ Projection of a N-gon

In discussion of Recognition by Sight, concept of projecting a 2-D polygon onto a 1-D line

̶ Area of a closed figure

Geometric meaning of a number n squared as interior area of a square of side length n

Ways to compute area for a square, triangle, hexagon

One-Dimensional “Lineland”:

̶ 1-D entities are line segments and points

̶ 1-D movement is along the line only

̶ Line is infinitesimally thick

̶ “once a neighbor, always a neighbor” – can’t change relative position of points or line segments in 1-D space

̶ 2-D figure intersecting 1-D line results in 1-D cross-section of the 2-D figure

Zero-Dimensional “Pointland”

̶ A single point only, of infinitesimal size

Three-Dimensional Geometry

̶ Geometric meaning of a number n raised to the 3rd power as interior area of a cube of side length n

̶ 3-D figure intersecting 2-D plane results in 2-D cross-section of the 3-D figure

̶ Solids in 3-D: Sphere, Cylinder, Cone, Pyramid, Cube, Pentahedron, Hexahedron, Dodecahedron

̶ Circular cross-section of a sphere, with radius the same as the radius of the sphere, is a great circle of the sphere

Multi-Dimensional Concepts:

̶ Geometric progression of terminal points of figures in N-dimensions

· 0-D: point has 1 terminal point; 1-D: line has 2 terminal points; 2-D: square has 4 terminal points; 3-D: cube has 8 terminal points; in N-dimensions, analogous figure would have 2N terminal points

̶ Arithmetic progression of sides of figures in N-dimensions

· A side is a component of dimension N-1 in N dimensions

· 0-D: point has 0 sides; 1-D: line has 2 terminal points which are sides of dimension 0; 2-D: square has 4 lines which are sides of dimension 1; 3-D: cube has 6 squares which are sides of dimension 2; in N-dimensions, analogous figure would have 2N sides of dimension N-1

̶ Sphere in 3-D and circle in 2-D has constant curvature, whereas curvature of an ellipse or ellipsoid changes along the surface

̶ Inability to visualize dimensions higher than 3, but the ability to reason about them

Spherical Geometry:

̶ Geometry of the 2-D surface of a sphere: Two-dimensional, yet curved

· Analogous to the surface of the earth

̶ Moving from a point along one direction will eventually return you to the same point

̶ “Universe” consists of the surface of the sphere, just as in two dimensions the “Universe” is the plane, and in one dimension the “Universe” is the line

̶ Sum of angles of a triangle in spherical geometry is greater than 180○ and less than or equal to 270○

· Larger the triangle (up to quadrant of sphere), larger the sum of angles is

Concepts of perfection and imperfection, and measurement error

̶ In discussion of Flatland inhabitants, regular and almost regular figures

̶ In discussion in Afterword, assumption that sum of angles of triangles that was more than 180○ was measurement error

In Afterword, triangle in Sphereland discusses concepts of proof by example and a logically reasoned proof, developing a theory and a testable theoretical framework including logically reasoned theorems and axioms.

In Afterword, triangle in Sphereland discusses education and reason as the opposite of fear of the unknown and making mistakes, and that small actions over time can have large effects – education as empowerment.

Download the (free!) version of The Story of Flatland: An Adventure in Many Dimensions at: http://people.southwestern.edu/~bucheles/FlatlandBucheleJuly2010.pdf