Monday, November 08, 2004

Tesseract

 The tesseract can be unfolded into eight cubes into 3D space, just as the cube can be unfolded into six squares into 2D space. An unfolding of a polytope is called a net. There are 261 distinct nets of the tesseract.[4] The unfoldings of the tesseract can be counted by mapping the nets to paired trees (a tree together with a perfect matching in its complement)

In geometry, the tesseract, or hypercube, is a regular, convex polychoron with eight cubical cells. It can be thought of as a 4-dimensional analogue of the cube. Roughly speaking the tesseract is to the cube as the cube is to the square.

Generalizations of the cube to dimensions greater than three are called hypercubes or measure polytopes. This article focuses on the 4D hypercube, the tesseract.

In a square, each vertex has two perpendicular edges incident to it, while a cube has three. A tesseract has four. Canonical coordinates for the vertices of a tesseract centered at the origin are (±1, ±1, ±1, ±1), while the interior of the same consists of all points (x0, x1, x2, x3) with -1 < xi < 1. This structure is not easily imagined but it is possible to project tesseracts into three or two dimensional spaces. Furthermore, projections on the 2D-plane become more instructive by rearranging the positions of the projected vertices. In this fashion, one can obtain pictures that no longer reflect the spatial relationships within the tesseract, but which nicely illustrate the connection structure of the vertices. The following examples are provided:

I would have thought artists like Dali and like Escher tried to develope and expand perspective capabilties of mind? To incorporate as much a "higher understanding" of the solid things, as we expect to understand all things around us?:)

Plato was pointing up for a reason, yet he believed in solid geometrical forms?