wdr@faron.UUCP (William D. Ricker) (04/23/85)
In article <617@ahuta.UUCP> in net.arch, rkl@ahuta.UUCP (R. Kevin Laux) answered the question of what is a hypercube, which is relevant to certain exoctic SIMD multiproccessor architectures: > The hypercube is a 4 dimensional construct, also known as a tesseract. >Unfolded into 3D space it looks like a stack of 4 cubes and has another 4 >cubes attached to the 4 sides of the second cube from the bottom (or top, >whichever). ... Well, mostly true. The tesseract is the 4-cube, or hyper-cube in 4 dimensions. There are hyper cubes for all n>0, which are constructed by translating a (n-1)-cube through the n-th dimension a unit distance, connecting each node with its image. The 2**64 processor architectures are 64-cubes, which would only /look cubish/ if you looked at them in a 64-dimensional space. A 64 cube is built by translating a 63-cubes through the 64th dimension. ... {Recurse deeply} ... The 1-cube is two 0-cubes (points) connected by a unit-line in 1-space. The more familiar family members: The 2-cube, which we call a square, is two 1-cubes connected, forming four 1-cube faces. The 3-cube is two 2-cubes connected by four 1-cubes, to form six 2-cube faces. The 4-cube, a tesseract, is formed by connecting two 3-cubes node-to- corresponding node, which then has eight 3-cube hyper-faces in the 4-space in which it is embedded. A 5-cube can be similarly constructed /et/ tedius /cetera/, /ad infinitum/. Other Topological Trivia: The 0-cube is also the 0-ball. A 1-ball is the set of points in a one-dimensional space within a given radius of a certain point: a closed line segment. The 1-cube = 1-ball. The 0-sphere is the boundary of the 1-ball, two points or 0-balls!. The 2-ball is the disk, all points in 2-space within a radius. Its boundary is the 1-sphere, a circle, which is topologically identical to two lines, 1-balls, joined at their boundaries. The 3-ball is what we call a ball or spherical solid, all 3-space within a radious. Its boundary is called the 2-sphere, which is topologically equivalent to two 2-balls joined at their 1-spheres. There is likewise in 4-space a 4-ball whose 3-sphere surface is two 3-balls joined surface to surface. {Follow-up the topology to net.math. This is last bit is a bit far afield from computer architecture. Although Weiner did set up standing waves in an anulus of turtle muscle.} -- William Ricker wdr@faron.UUCP (UUCP) decvax!genrad!linus!faron!wdr (UUCP) {allegra,ihnp4,utzoo,philabs,uw-beaver}!linus!faron!wdr (UUCP) Opinions are my own and not necessarily anyone elses. Likewise the "facts".