ANSWERS: 3
• In geometry, the tesseract is the 4-dimensional analog of the (3-dimensional) cube, where motion along the fourth dimension is often a representation for bounded transformations of the cube through time. The tesseract is to the cube as the cube is to the square, or more formally, the tesseract can be described as a regular convex 4-polytope whose boundary consists of eight cubical cells. According to OED, the word tesseract was coined and first used in 1888 by Charles Howard Hinton in his book A New Era of Thought, from tesseres aktines = 'four rays' in Ionic Greek, referring to the four lines from each vertex to other vertices. Alternately, some people have called the same figure a 'tetracube'. Generalizations of the cube to dimensions greater than three are called hypercubes or measure polytopes. This article focuses on the 4D hypercube, the tesseract. Hypercubes in computer architecture In computer science, the term hypercube refers to a specific type of parallel computer, whose processors, or processing elements (PEs), are interconnected in the same way as the vertices of a hypercube. Thus, an n-dimensional hypercube computer has 2n PEs, each directly connected to n other PEs. Examples include the nCUBE machines used to win the first Gordon Bell Prize, the Caltech Cosmic Cube and the Connection Machine, the latter using the hypercube topology to connect groups of processors. Hypercubes in information theory: Hypercubes are the logical representation of multidimensional data warehouses compiled for complex information cross-referencing. Instead of the usual two dimensional operational databases, Hypercubes allow you to cross reference a number of factors at once, structuring information into theoretical columns, rows and layers. In the analysis of business data for example, the dimensions (or factors) for analysis might be the correlation between product, buyer segments and advertising budget, allowing us to compare multiple factors simultaneously. Hypercubes however are not limited to the number of dimensions they can use; as the analysis grows more precise, additional dimensions are added creating new layers within the Hypercubes. Although we show business data being analyzed in this example, it can be applied to many other fields of study as well. The application of comparing multiple factors simultaneously during the analytical process makes the end users’ decisions much easier and more informed than with simple two-dimensional data analysis.
• According to OED, the word tesseract was coined and first used in 1888 by Charles Howard Hinton in his book A New Era of Thought, from tesseres aktines = 'four rays' in Ionic Greek, referring to the four lines from each vertex to other vertices. Alternately, some people have called the same figure a 'tetracube'. The standard tesseract in Euclidean 4-space is given as the convex hull of the points (±1, ±1, ±1, ±1). That is, it consists of the points: A tesseract is bounded by eight hyperplanes (xi = ±1). Each pair of non-parallel hyperplanes intersects to form 24 square faces in a tesseract. Three cubes and three squares intersect at each edge. There are four cubes, six squares, and four edges meeting at every vertex. All in all, it consists of 8 cubes, 24 squares, 32 edges, and 16 vertices. Since each vertex of a tesseract is adjacent to four edges, the vertex figure of the tesseract is a regular tetrahedron. Thus the tesseract is given Schläfli symbol {4,3,3}. The dual polytope of the tesseract is called the hexadecachoron, or 16-cell, with Schläfli symbol {3,3,4 Thanks to the website Wikipedia for that information. In conclusion, yes it is a proven concept.
• there are many kinds of 4D projections into 3space... and more than a few can be built with soap films mated to the 3D shape wire-forms. For fun you might read the short story "He Built A Crooked House" from the early 1940's by R.A. Heinlien

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