Dimension

The notion of dimension is important in mathematics because it gives a precise parameterization of the conceptual orvisual complexity of any geometric object. In fact, the concept can even be applied to abstract objects which cannot bedirectly visualized. For example, the notion of time can be considered as one-dimensional, since it can be thought ofas consisting of only ``now,'' ``before'' and ``after.'' Since ``before'' and ``after,'' regardless of how far back orhow far into the future they are, are extensions, time is like a line, a 1-dimensional object.

To see how lower and higher dimensions relate to each other, take any geometric object (like a Point, Line,Circle, Plane, etc.), and ``drag'' it in an opposing direction (drag a Point to trace out aLine, a Line to trace out a box, a Circle to trace out a Cylinder, a Disk to a solidCylinder, etc.). The result is an object which is qualitatively ``larger'' than the previous object, ``qualitative'' inthe sense that, regardless of how you drag the original object, you always trace out an object of the same ``qualitativesize.'' The Point could be made into a straight Line, a Circle, a Helix, or some other Curve, but all of these objectsare qualitatively of the same dimension. The notion of dimension was invented for the purpose of measuring this``qualitative'' topological property.

Making things a bit more formal, finite collections of objects (e.g., points in space) are considered 0-dimensional.Objects that are ``dragged'' versions of 0-dimensional objects are then called 1-dimensional. Similarly, objects which aredragged 1-dimensional objects are 2-dimensional, and so on. Dimension is formalized in mathematics as the intrinsicdimension of a Topological Space. This dimension is called the Lebesgue Covering Dimension (also knownsimply as the Topological Dimension). The archetypal example is Euclidean -space, which has topological dimension . The basic ideas leading up to this result (including the DimensionInvariance Theorem, Domain Invariance Theorem, and Lebesgue Covering Dimension) were developed byPoincaré, Brouwer, Lebesgue, Urysohn, and Menger.

There are several branchings and extensions of the notion of topological dimension. Implicit in the notion of theLebesgue Covering Dimension is that dimension, in a sense, is a measure of how an object fills space. If it takes up alot of room, it is higher dimensional, and if it takes up less room, it is lower dimensional. Hausdorff Dimension(also called Fractal Dimension) is a fine tuning of this definition that allows notions of objects with dimensionsother than Integers. Fractals are objects whose Hausdorff Dimension isdifferent from their Topological Dimension.

The concept of dimension is also used in Algebra, primarily as the dimension of a Vector Space over aField. This usage stems from the fact that Vector Spaces over the reals were the firstVector Spaces to be studied, and for them, their topological dimension can be calculated by purelyalgebraic means as the Cardinality of a maximal linearly independent subset. In particular, the dimension of aSubspace of is equal to the number of Linearly Independent Vectors needed togenerate it (i.e., the number of Vectors in its Basis). Given a transformation of ,

References

Dimensions

Abbott, E. A. Flatland: A Romance of Many Dimensions. New York: Dover, 1992.

Hinton, C. H. The Fourth Dimension. Pomeroy, WA: Health Research, 1993.

Manning, H. The Fourth Dimension Simply Explained. Magnolia, MA: Peter Smith, 1990.

Manning, H. Geometry of Four Dimensions. New York: Dover, 1956.

Neville, E. H. The Fourth Dimension. Cambridge, England: Cambridge University Press, 1921.

Rucker, R. von Bitter. The Fourth Dimension: A Guided Tour of the Higher Universes. Boston, MA: Houghton Mifflin, 1984.

Sommerville, D. M. Y. An Introduction to the Geometry of Dimensions. New York: Dover, 1958.

• McMohan's Theorem
• McNugget Number
• Mean
• Mean Cluster Count Per Site
• Mean Cluster Density
• Mean Curvature
• Mean Deviation
• Mean Distribution
• Mean Run Count Per Site
• Mean Run Density
• Mean Square Error
• Mean-Value Theorem
• Measurable Function
• Measurable Set
• Measurable Space
• Measure
• Measure Algebra
• Measure Polytope
• Measure-Preserving Transformation
• Measure Space
• Measure Theory
• Mechanical Quadrature
• Mecon
• Medial Axis
• Medial Deltoidal Hexecontahedron