October 17, 2011

Carl Sagan on the 3D to 4D shift

as posted in video description..

In physics and mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate (the polar coordinate angle), so the circle is 1-dimensional even though it exists in the 2-dimensional plane. This intrinsic notion of dimension is one of the chief ways in which the mathematical notion of dimension differs from its common usages.

There is also an inductive description of dimension: consider a discrete set of points (such as a finite collection of points) to be 0-dimensional. By dragging a 0-dimensional object in some direction, one obtains a 1-dimensional object. By dragging a 1-dimensional object in a new direction, one obtains a 2-dimensional object. In general one obtains an n+1-dimensional object by dragging an n dimensional object in a new direction. Returning to the circle example: a circle can be thought of as being drawn as the end-point on the minute hand of a clock, thus it is 1-dimensional. To construct the plane one needs two steps: drag a point to construct the real numbers, then drag the real numbers to produce the plane.

Consider the above inductive construction from a practical point of view -- ie: with concrete objects that one can play with in one's hands. Start with a point, drag it to get a line. Drag a line to get a square. Drag a square to get a cube. Any small translation of a cube has non-trivial overlap with the cube before translation, thus the process stops. This is why space is said to be 3-dimensional.

High-dimensional spaces occur in mathematics and the sciences for many reasons, frequently as configuration spaces such as in Lagrangian or Hamiltonian mechanics. Ie: these are abstract spaces, independent of the actual space we live in. The state-space of quantum mechanics is an infinite-dimensional function space. Some physical theories are also by nature high-dimensional, such as the 4-dimensional general relativity and higher-dimensional string theories.

In mathematics, the dimension of Euclidean n-space E n is n. When trying to generalize to other types of spaces, one is faced with the question what makes E n n-dimensional?" One answer is that in order to cover a fixed ball in E n by small balls of radius ε, one needs on the order of ε−n such small balls. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension. But there are also other answers to that question. For example, one may observe that the boundary of a ball in E n looks localy like E n − 1 and this leads to the notion of the inductive dimension. While these notions agree on E n, they turn out to be different when one looks at more general spaces.

A tesseract is an example of a four-dimensional object. Whereas outside of mathematics the use of the term "dimension" is as in: "A tesseract has four dimensions," mathematicians usually express this as: "The tesseract has dimension 4," or: "The dimension of the tesseract is 4."

Although the notion of higher dimensions goes back to René Descartes, substantial development of a higher-dimensional geometry only began in the 19th century, via the work of Arthur Cayley, William Rowan Hamilton, Ludwig Schläfli and Bernhard Riemann. Riemann's 1854 Habilitationsschrift, Schlafi's 1852 Theorie der vielfachen Kontinuität, Hamilton's 1843 discovery of the quaternions and the construction of the Cayley Algebra marked the beginning of higher-dimensional geometry.

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