Differences Between 2D and 3D in Geometry
W. Randolph Franklin

Here is a sampling of how 2D geometry is essentially different from 3D.

    2D 3D
    1. There are an infinite number of regular polygons, but only a finite number of regular polyhedra.
    2. Given two equal-area polygons, one can be dissected into a finite number of pieces, then reassembled into the other. This is usually not true for pairs of equal-volume polyhedra.
    3. There exists a square that is decomposable into smaller, different, squares. There is no cube that is decomposable into smaller, different, cubes.
    4.All polygons are decomposable into triangles by adding only interior edges. Not all polyhedra are decomposable into tetrahedra by adding only interior faces.
    5. For polygons, all such decompositions have the same number of triangles. Some polyhedra can be decomposed different ways into different numbers of tetrahedra.
    6.Every polygon has every interior point visible from some vertex. Some polyhedra have interior points not visible from any vertex.
    7. A 2-D Voronoi diagram's complexity is linear. A 3-D Voronoi diagram's complexity can be quadratic.
    8.Given two convex polygons, there exists an edge of one that separates them. Given two polyhedra, it is possible that none of their faces separate them.
    9.Rotations commute. Rotations usually don't commute.
    10. For each edge of a polygon, consider the half-plane of points on the inside side of that edge. Then, the polygon's interior can be expressed as a Boolean expression in those half planes, with each half plane used only once. Not in 3D.

    Dr. W. Randolph Franklin,
    Email: wrfATecse.rpi.edu
    +1 (518) 276-6077; Fax: -4403 (new)
    ECSE Dept., 6026 JEC, Rensselaer Polytechnic Inst, Troy NY, 12180 USA
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    Copyright © 1994-2002, Wm. Randolph Franklin. You may use my material for non-profit research and education, provided that you credit me, and link back to my home page.