Sunday, October 8, 2017


Four-dimensional space

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Animation of a transforming tesseract or 4-cube
This animation hints at the complexity that emerges when points in 4D are linked together to create basic four-dimensional objects. The object shown here is the 4D equivalent of a cube, known as a tesseract. To create this animation, the tesseract was rotated in four dimensions, then projected into three dimensions, and finally projected onto a two dimensional image.
A four-dimensional space or 4D space is the simplest possible generalization of the observation that one only needs three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world. For example, the volume of a rectangular box is found by measuring its length (often labeled x), width (y), and depth (z).
More than two millennia ago Greek philosophers explored in detail the many implications of this uniformity, culminating in Euclid's Elements. However, it was not until recent times that some mathematicians generalized the concept of dimensions to include more than three. The idea of adding a fourth dimension began with Joseph-Louis Lagrange in the mid 1700s and culminated in a precise formalization of the concept in 1854 by Bernhard Riemann. In 1880 Charles Howard Hinton popularized these insights in an essay titled What is the Fourth Dimension?, which was notable for explaining the concept of a four-dimensional cube by going through a step-by-step generalization of the properties of lines, squares, and cubes. The simplest form of Hinton's method is to draw two ordinary cubes separated by an "unseen" distance, and then draw lines between their equivalent vertices. This form can be seen in the accompanying animation whenever it shows a smaller inner cube inside a larger outer cube. The eight lines connecting the vertices of the two cubes in that case represent a single direction in the "unseen" fourth dimension.
Higher dimensional spaces have since become one of the foundations for formally expressing modern mathematics and physics. Large parts of these topics could not exist in their current forms without the use of such spaces.
For anyone first learning about 4D and higher spaces, it is helpful to keep in mind that a four-dimensional space just adds one number to the three we already know, and that this number can represent many different things. Calendar entries for example are usually 4D locations, such as a meeting at time t at the intersection of two streets (x and y) on some building floor (z). In list form such a meeting takes place at the 4D location (t,x,y,z). Einstein's concept of spacetime uses such a 4D space, though it has a Minkowski structure that is a bit more complicated than Euclidean 4D space.
When dimensional locations are given as ordered lists of numbers such as (t,x,y,z) they are called vectors or n-tuples. It is only when such locations are linked together into more complicated shapes that the full richness and geometric complexity of 4D and higher spaces emerges. A hint of that complexity can be seen in the accompanying animation of one of simplest possible 4D objects, the 4D cube or tesseract.

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