Desmos 4d sphere full#
It is only when such locations are linked together into more complicated shapes that the full richness and geometric complexity of higher-dimensional spaces emerge. Single locations in Euclidean 4D space can be given as vectors or n-tuples, i.e., as ordered lists of numbers such as ( x, y, z, w). Einstein's concept of spacetime has a Minkowski structure based on a non-Euclidean geometry with three spatial dimensions and one temporal dimension, rather than the four symmetric spatial dimensions of Schläfli's Euclidean 4D space. Einstein's theory of relativity is formulated in 4D space, although not in a Euclidean 4D space. Large parts of these topics could not exist in their current forms without using such spaces.
Higher-dimensional spaces (greater than three) have since become one of the foundations for formally expressing modern mathematics and physics. The eight lines connecting the vertices of the two cubes in this case represent a single direction in the "unseen" fourth dimension. This can be seen in the accompanying animation whenever it shows a smaller inner cube inside a larger outer cube. The simplest form of Hinton's method is to draw two ordinary 3D cubes in 2D space, one encompassing the other, separated by an "unseen" distance, and then draw lines between their equivalent vertices. In 1880 Charles Howard Hinton popularized it in an essay, " What is the Fourth Dimension?", in which he explained the concept of a " four-dimensional cube" with a step-by-step generalization of the properties of lines, squares, and cubes. Schläfli's work received little attention during his lifetime and was published only posthumously, in 1901, but meanwhile the fourth Euclidean dimension was rediscovered by others. The general concept of Euclidean space with any number of dimensions was fully developed by the Swiss mathematician Ludwig Schläfli before 1853. published in 1754, but the mathematics of more than three dimensions only emerged in the 19th century. The idea of adding a fourth dimension appears in Jean le Rond d'Alembert's "Dimensions". This concept of ordinary space is called Euclidean space because it corresponds to Euclid's geometry, which was originally abstracted from the spatial experiences of everyday life. For example, the volume of a rectangular box is found by measuring and multiplying its length, width, and height (often labeled x, y, and z). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world. NLERP is equivalent (in this case) and may be faster, but it would have taken a few more equations to implement in Desmos.Four-dimensional space ( 4D) is the mathematical extension of the concept of three-dimensional space (3D). The arcs between the points are generated using SLERP (, derivation at ). The only points displayed are those on the "camera side" of the sphere, which is done using dot product Mathematically, this is done using geometric algebra in a manner isomorphic to using quaternions. These are randomly generated by choosing an axis (in the same we we chose a random point) and rotating about it with an angle equal to T (from 0 to τ). The points move around in circles on the sphere.
This (x,y,z) distribution is spherically symmetric, so we can divide by its distance from the origin to normalize it to lie on the unit sphere To randomly generate points on the sphere, we first take (x,y,z) sampled from three independent normal distributions. To see the underlying sphere and great circles, unhide lines 76 and 77 respectively.
Essentially, this is just a bunch of points moving in circles (not necessarily great circles) around a unit sphere, and nearby points are connected using arcs. Inspired by made by fadaaszhi, which was in turn inspired by a random GIF in the Discord.
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