In geometry, a hypercube is an n-dimensional analogue of a square and a cube. It is a closed, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in n dimensions is equal to n. An n-dimensional hypercube is more referred to as an n-cube or sometimes as an n-dimensional cube; the term measure polytope is used, notably in the work of H. S. M. Coxeter who labels the hypercubes the γn polytopes; the hypercube is the special case of a hyperrectangle. A unit hypercube is a hypercube; the hypercube whose corners are the 2n points in Rn with each coordinate equal to 0 or 1 is called the unit hypercube. A hypercube can be defined by increasing the numbers of dimensions of a shape: 0 – A point is a hypercube of dimension zero. 1 – If one moves this point one unit length, it will sweep out a line segment, a unit hypercube of dimension one. 2 – If one moves this line segment its length in a perpendicular direction from itself.
3 – If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a 3-dimensional cube. 4 – If one moves the cube one unit length into the fourth dimension, it generates a 4-dimensional unit hypercube. This can be generalized to any number of dimensions; this process of sweeping out volumes can be formalized mathematically as a Minkowski sum: the d-dimensional hypercube is the Minkowski sum of d mutually perpendicular unit-length line segments, is therefore an example of a zonotope. The 1-skeleton of a hypercube is a hypercube graph. A unit hypercube of n dimensions is the convex hull of the points given by all sign permutations of the Cartesian coordinates, it has an edge length of 1 and an n-dimensional volume of 1. An n-dimensional hypercube is often regarded as the convex hull of all sign permutations of the coordinates; this form is chosen due to ease of writing out the coordinates. Its edge length is 2, its n-dimensional volume is 2n; every n-cube of n > 0 is composed of elements, or n-cubes of a lower dimension, on the -dimensional surface on the parent hypercube.
A side is any element of -dimension of the parent hypercube. A hypercube of dimension n has 2n sides; the number of vertices of a hypercube is 2 n. The number of m-dimensional hypercubes on the boundary of an n-cube is E m, n = 2 n − m, where = n! M!! and n! Denotes the factorial of n. For example, the boundary of a 4-cube contains 8 cubes, 24 squares, 32 lines and 16 vertices; this identity can be proved by combinatorial arguments. There are ways of choosing. But, each side is counted 2 m times since it has that many vertices, we need to divide by this number; this identity can be used to generate the formula for the n-dimensional cube surface area. The surface area of a hypercube is: 2 n s n − 1; these numbers can be generated by the linear recurrence relation E m, n = 2 E m, n − 1 + E m − 1, n − 1, with E 0, 0 = 1, undefined elements = 0. For example, extending a square via its 4 vertices adds one extra line per vertex, adds t
The 2007–08 Hong Kong First Division League season was the 96th since its establishment. The first match was played on 2 September 2007 with South China lost to Kitchee 1–2. In this season, the First Division League was composed of 10 teams. Workable were promoted from the Second Division while Eastern demoted to Third Division League by rule, was invited by HKFA to play in the First Division League after securing sufficient sponsorship. Note: Here is the home stadium list of the teams: Dongguan Stadium – Lanwa Redbull Hong Kong Stadium & Mong Kok Stadium – Rest of the teams The following 10 clubs are competing in the Hong Kong First Division League during the 2007–08 season. All times are Hong Kong Time. 19 goalsDetinho 12 goalsGiovane 11 goalsMaxwell 10 goalsWang Xuanhong 8 goalsRodrigo Junior 7 goalsTomy Annan Rafeal 6 goalsLeko Paulo Chan Siu Ki Goran Stankovski Joel 5 goalsAldo Villalba Chao Pengfei Julius Akosah Roberto Fronza 4 goalsLiang Zicheng Chan Yiu Lun Fábio Godfred Anibal Pacheco Orzuza Lin Zhong Schutz Au Yeung Yiu Chung 3 goalsLam Ka Wai Marcio Ronan Batoum Roger Diego Poon Yiu Cheuk Anderson da Silva Jaimes McKee Itaparica Kwok Kin Pong Li Haiqiang Only scorers with 3 goals or above are listed here.
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Breachwood Green Mill is a Grade II listed tower mill at King's Walden, England, converted to residential accommodation. A windmill was recorded at Kings Walden in 1329. Another is mentioned in 1762; the first mention of the mill was in 1861. His son William had been born at Kings Walden in 1859. Dellow was succeeded at the mill by his son, who worked the mill until 1900; the mill had lost its sails by 1930, the cap was a bare frame by 1936. The mill was converted to residential accommodation in 1998. Recent photographs of the mill show that the brick tower has been clad in weatherboarding with the result that the mill now resembles a many-sided smock mill. Breachwood Green Mill is a five storey tower mill; the tower is 24 feet outside diameter at the base with brickwork 2 feet thick. It is 42 feet high to curb level; the dome shaped cap was winded by a fantail and there were four Patent sails. The great spur wheel was of cast iron and the mill drove two pairs of millstones. William Dellow 1859- William Dellow Jr -1900Reference for above:- Windmill World webpage on Breachwood Green Mill