TesseracT

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In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square.[1] Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes.

The tesseract is also called an 8-cell, C8, (regular) octachoron, octahedroid,[2] cubic prism, and tetracube.[3] It is the four-dimensional hypercube, or 4-cube as a member of the dimensional family of hypercubes or measure polytopes.[4] Coxeter labels it the γ 4 {\displaystyle \gamma _{4}} polytope.[5] The term hypercube without a dimension reference is frequently treated as a synonym for this specific polytope.

Since each vertex of a tesseract is adjacent to four edges, the vertex figure of the tesseract is a regular tetrahedron. The dual polytope of the tesseract is the 16-cell with Schläfli symbol {3,3,4}, with which it can be combined to form the compound of tesseract and 16-cell.

Each edge of a regular tesseract is of the same length. This is of interest when using tesseracts as the basis for a network topology to link multiple processors in parallel computing: the distance between two nodes is at most 4 and there are many different paths to allow weight balancing.

In this Cartesian frame of reference, the tesseract has radius 2 and 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.

An unfolding of a polytope is called a net. There are 261 distinct nets of the tesseract.[7] 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).

The tesseract can be decomposed into smaller 4-polytopes. It is the convex hull of the compound of two demitesseracts (16-cells). It can also be triangulated into 4-dimensional simplices (irregular 5-cells) that share their vertices with the tesseract. It is known that there are 92487256 such triangulations[9] and that the fewest 4-dimensional simplices in any of them is 16.[10]

The dissection of the tesseract into instances of its characteristic simplex (a particular orthoscheme with Coxeter diagram ) is the most basic direct construction of the tesseract possible. The characteristic 5-cell of the 4-cube is a fundamental region of the tesseract's defining symmetry group, the group which generates the B4 polytopes. The tesseract's characteristic simplex directly generates the tesseract through the actions of the group, by reflecting itself in its own bounding facets (its mirror walls).

This configuration matrix represents the tesseract. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole tesseract. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[11] For example, the 2 in the first column of the second row indicates that there are 2 vertices in (i.e., at the extremes of) each edge; the 4 in the second column of the first row indicates that 4 edges meet at each vertex.

The cell-first parallel projection of the tesseract into three-dimensional space has a cubical envelope. The nearest and farthest cells are projected onto the cube, and the remaining six cells are projected onto the six square faces of the cube.

The face-first parallel projection of the tesseract into three-dimensional space has a cuboidal envelope. Two pairs of cells project to the upper and lower halves of this envelope, and the four remaining cells project to the side faces.

The edge-first parallel projection of the tesseract into three-dimensional space has an envelope in the shape of a hexagonal prism. Six cells project onto rhombic prisms, which are laid out in the hexagonal prism in a way analogous to how the faces of the 3D cube project onto six rhombs in a hexagonal envelope under vertex-first projection. The two remaining cells project onto the prism bases.

The vertex-first parallel projection of the tesseract into three-dimensional space has a rhombic dodecahedral envelope. Two vertices of the tesseract are projected to the origin. There are exactly two ways of dissecting a rhombic dodecahedron into four congruent rhombohedra, giving a total of eight possible rhombohedra, each a projected cube of the tesseract. This projection is also the one with maximal volume. One set of projection vectors are u=(1,1,-1,-1), v=(-1,1,-1,1), w=(1,-1,-1,1).

The tetrahedron forms the convex hull of the tesseract's vertex-centered central projection. Four of 8 cubic cells are shown. The 16th vertex is projected to infinity and the four edges to it are not shown.

The tesseract, like all hypercubes, tessellates Euclidean space. The self-dual tesseractic honeycomb consisting of 4 tesseracts around each face has Schläfli symbol {4,3,3,4}. Hence, the tesseract has a dihedral angle of 90°.[12]

The regular tesseract, along with the 16-cell, exists in a set of 15 uniform 4-polytopes with the same symmetry. The tesseract {4,3,3} exists in a sequence of regular 4-polytopes and honeycombs, {p,3,3} with tetrahedral vertex figures, {3,3}. The tesseract is also in a sequence of regular 4-polytope and honeycombs, {4,3,p} with cubic cells.

The regular complex polytope 4{4}2, , in C 2 {\displaystyle \mathbb {C} ^{2}} has a real representation as a tesseract or 4-4 duoprism in 4-dimensional space. 4{4}2 has 16 vertices, and 8 4-edges. Its symmetry is 4[4]2, order 32. It also has a lower symmetry construction, , or 4{}×4{}, with symmetry 4[2]4, order 16. This is the symmetry if the red and blue 4-edges are considered distinct.[13]

The word tesseract was later adopted for numerous other uses in popular culture, including as a plot device in works of science fiction, often with little or no connection to the four-dimensional hypercube of this article. See Tesseract (disambiguation).

image Object or String - PIL Image/NumPy array or file path of the image to be processed by Tesseract. If you pass object instead of file path, pytesseract will implicitly convert the image to RGB mode.lang String - Tesseract language code string. Defaults to eng if not specified! Example for multiple languages: lang='eng+fra'config String - Any additional custom configuration flags that are not available via the pytesseract function. For example: config='--psm 6'nice Integer - modifies the processor priority for the Tesseract run. Not supported on Windows. Nice adjusts the niceness of unix-like processes.output_type Class attribute - specifies the type of the output, defaults to string. For the full list of all supported types, please check the definition of pytesseract.Output class.timeout Integer or Float - duration in seconds for the OCR processing, after which, pytesseract will terminate and raise RuntimeError.pandas_config Dict - only for the Output.DATAFRAME type. Dictionary with custom arguments for pandas.read_csv. Allows you to customize the output of image_to_data.CLI usage:

A series of images from Charles Howard Hinton's The Fourth Dimension (1904), a book all about the "tesseract" - a four-dimensional analog of the cube, the tesseract being to the cube as the cube is to the square. Hinton, a British mathematician and science fiction writer, actually coined the term "tesseract" which appears for the first time in his book A New Era of Thought (1888). We are not going to pretend to have given the time to his book to understand fully the concept behind these diagrams, but they are a fascinating series of images all the same (particular the coloured frontispiece featured above), and offer a glimpse into the theory of four-dimensional space which would prove so important to the development of modern physics. Although Hinton's work was an important stepping stone in understanding four-dimensional space, the real breakthrough came in a 1908 paper by Hermann Minkowski, in which four-dimensional space was thought of in non-Euclidean terms, leading to the revolutionary concept of "spacetime".

For the grand finale of this article lets drag a tesseract through a 4D space and see how it intersects with a 3D world. Just like an intersection of a 3D object with a 2D space creates a 2D object, an intersection of a 4D object with a 3D space creates a 3D object.

Tesseract uses training data to perform OCR. Most systems default to English training data. To improve OCR results for other languages you can to install the appropriate training data. On Windows and OSX you can do this in R using tesseract_download():

Alternatively you can manually download training data from github and store it in a path on disk that you pass in the datapath parameter or set a default path via the TESSDATA_PREFIX environment variable. Note that the Tesseract 4 and Tesseract 3 use different training data format. Make sure to download training data from the branch that matches your libtesseract version.

The idea of the tesseract scene alone was so daunting to the filmmakers, Nolan and his special effects team procrastinated for months before trying to tackle how it might work. After months of concepting and model building the team opted for the unusual approach of using minimal digital effects in favor of fabricating a massive set which the actors could physically manipulate. A remarkable feat considering not only the complexity of the concepts depicted, but the cost and labor of building something so large. 2b1af7f3a8