SUMMARY / RELATED TOPICS

Tesseract

In geometry, the tesseract is the four-dimensional analogue of the cube. 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 called an eight-cell, C8, octahedroid, cubic prism, tetracube, it is the four-dimensional hypercube, or 4-cube as a part of the dimensional family of hypercubes or measure polytopes. Coxeter labels it the γ 4 polytope. According to the Oxford English Dictionary, the word tesseract was coined and first used in 1888 by Charles Howard Hinton in his book A New Era of Thought, from the Greek τέσσερεις ἀκτίνες, referring to the four lines from each vertex to other vertices. In this publication, as well as some of Hinton's work, the word was spelled "tessaract"; the tesseract can be constructed in a number of ways. As a regular polytope with three cubes folded together around every edge, it has Schläfli symbol with hyperoctahedral symmetry of order 384.

Constructed as a 4D hyperprism made of two parallel cubes, it can be named as a composite Schläfli symbol ×, with symmetry order 96. As a 4-4 duoprism, a Cartesian product of two squares, it can be named by a composite Schläfli symbol ×, with symmetry order 64; as an orthotope it can be represented by composite Schläfli symbol × × × or 4, with symmetry order 16. 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 called the regular hexadecachoron, or 16-cell, with Schläfli symbol, with which it can be combined to form the compound of tesseract and 16-cell. The standard tesseract in Euclidean 4-space is given as the convex hull of the points; that is, it consists of the points: A tesseract is bounded by eight hyperplanes. 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, four edges meeting at every vertex.

All in all, it consists of 8 cubes, 24 squares, 32 edges, 16 vertices. The construction of hypercubes can be imagined the following way: 1-dimensional: Two points A and B can be connected to become a line, giving a new line segment AB. 2-dimensional: Two parallel line segments AB and CD can be connected to become a square, with the corners marked as ABCD. 3-dimensional: Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH. 4-dimensional: Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a tesseract, with the corners marked as ABCDEFGHIJKLMNOP. It is possible to project tesseracts into three- and two-dimensional spaces to projecting a cube into two-dimensional space. Projections on the 2D-plane become more instructive by rearranging the positions of the projected vertices. In this fashion, one can obtain pictures that no longer reflect the spatial relationships within the tesseract, but which illustrate the connection structure of the vertices, such as in the following examples: A tesseract is in principle obtained by combining two cubes.

The scheme is similar to the construction of a cube from two squares: juxtapose two copies of the lower-dimensional cube and connect the corresponding vertices. Each edge of a tesseract is of the same length; this view 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. This configuration matrix represents the tesseract; the rows and columns correspond to vertices, edges and cells. The diagonal numbers say; the nondiagonal numbers say how many of the column's element occur at the row's element. The long radius of the tesseract is equal to its edge length. Only a few polytopes have this property, including the four-dimensional tesseract and 24-cell, the three-dimensional cuboctahedron, the two-dimensional hexagon. In particular, the tesseract is the only hypercube with this property; the longest vertex-to-vertex diameter of an n-dimensional hypercube of unit edge length is √n, so for the square it is √2, for the cube it is √3, only for the tesseract it is √4 2 edge lengths.

The tesseract, like all hypercubes, tessellates Euclidean space. The self-dual tesseractic honeycomb consisting of 4 tesseracts around each face ha

Perak Tengah District

The Perak Tengah District is a district in Perak, Malaysia. It is administered by Perak Tengah District Council, based at the town of Seri Iskandar. Perak Tengah District is divided into 11 mukims, which are: Bandar Blanja Bota Jaya Baru Kampung Gajah Kota Setia Lambor Layang-Layang Pasir Panjang Hulu Pasir Salak Pulau Tiga The following is based on Department of Statistics Malaysia 2010 census. List of Perak Tengah district representatives in the Federal Parliament List of Perak Tengah district representatives in the State Legislative Assembly of Perak Highways 109 and 5 are the main roads in the district, as well as 72 which goes to Parit and 73 to Batu Gajah. Rail services are not available in the district. Districts of Malaysia Rancangan Tempatan Daerah Perak Tengah 2030

Mazurkas, Op. 17 (Chopin)

Mazurkas, Op. 17 is a set of four mazurkas for piano by Frédéric Chopin and published between 1832 and 1833. A typical performance of the set lasts about fourteen minutes. Frédéric Chopin composed his Op. 17 in 1832-33 and they were published in Leipzig in 1834. During the time that Chopin was composing the mazurkas, he had settled in France, as he had become a refugee from Poland, however, he hoped he could move back to Poland when the political system had changed; this hope was never realised. Though Chopin had moved away from his homeland, he never forgot his Polish roots in his Mazurkas; the first Mazurka in the set is in B-flat major and has a time signature of 3/4. It has the tempo marking: Vivo e risoluto; the piece is soon followed by a section in question and answer. This phrase is completed with a descending sequence; this main theme repeats in two different new keys, one after the other. The Mazurka finishes with a slower section and the main theme repeated once more. There is a D. C. al fine and the whole piece starts again and finishes at the start of the final, slower section with a B-flat chord in octaves.

The second mazurka is in E minor, is in 3/4, has a tempo marking of Lento, ma non troppo. A typical performance of the E minor Eleventh lasts about two-and-a-half minutes; the Mazurka features an waltz style to it. The piece is in a homophonic texture with a single tune accompanied by chords; the piece ends with intricate arpeggio patterns and a quiet ending, contrasting with the forte beginning. The third mazurka, in A-flat major, is marked Legato assai; this piece is one of the longest mazurkas Chopin wrote, lasting about six minutes in a typical performance. The third Mazurka of this set is comparable the previous in its texture: homophonic; the piece unfolds with varied dynamics and half-way through, it changes to the key of E major but changes back to the original key for the last few bars and the coda. It doesn’t follow traditional harmonic progressions. There are no submediant chords in the entire movement. Most of the movement is composed of dominants, iii or III, vii° chords; this particular movement is in compound ternary form.

As many composers did in the romantic period, Chopin contrasts tonic by moving to the flat submediant The last mazurka in the set is in 3/4 and is marked Lento, ma non troppo. This piece lasts about five minutes in a typical performance, it is one of the more popular mazurkas of all Chopins' mazurkas. The final Mazurka of the set is more free than the others. Although it remains in the homophonic texture, the dynamic variation is much greater; the piece ends with the same four measures as it began, with no pedal, the chords played by the left hand portamento, the tone and time fading away in a perdendosi. These four measures would be sampled by Henryk Górecki in the opening of the third movement of his third symphony. Mazurka Op. 17 No.1, No.2, No.3 and No.4 played by Arthur Rubinstein Mazurka Op. 17 No.4 played by Vladimir Horowitz Mazurkas, Op.17: Scores at the International Music Score Library Project