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Polygon

In elementary geometry, a polygon is a plane figure, described by a finite number of straight line segments connected to form a closed polygonal chain or polygonal circuit. The solid plane region, the bounding circuit, or the two together, may be called a polygon; the segments of a polygonal circuit are called its edges or sides, the points where two edges meet are the polygon's vertices or corners. The interior of a solid polygon is sometimes called its body. An n-gon is a polygon with n sides. A simple polygon is one. Mathematicians are concerned only with the bounding polygonal chains of simple polygons and they define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons defined for different purposes; the word polygon derives from the Greek adjective πολύς "much", "many" and γωνία "corner" or "angle".

It has been suggested. Polygons are classified by the number of sides. See the table below. Polygons may be characterized by their convexity or type of non-convexity: Convex: any line drawn through the polygon meets its boundary twice; as a consequence, all its interior angles are less than 180°. Equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex: a line may be found which meets its boundary more than twice. Equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple: the boundary of the polygon does not cross itself. All convex polygons are simple. Concave: Non-convex and simple. There is at least one interior angle greater than 180°. Star-shaped: the whole interior is visible from at least one point, without crossing any edge; the polygon must be simple, may be convex or concave. All convex polygons are star-shaped. Self-intersecting: the boundary of the polygon crosses itself.

The term complex is sometimes used in contrast to simple, but this usage risks confusion with the idea of a complex polygon as one which exists in the complex Hilbert plane consisting of two complex dimensions. Star polygon: a polygon which self-intersects in a regular way. A polygon can not be both star-shaped. Equiangular: all corner angles are equal. Cyclic: all corners lie on a single circle, called the circumcircle. Isogonal or vertex-transitive: all corners lie within the same symmetry orbit; the polygon is cyclic and equiangular. Equilateral: all edges are of the same length; the polygon need not be convex. Tangential: all sides are tangent to an inscribed circle. Isotoxal or edge-transitive: all sides lie within the same symmetry orbit; the polygon is equilateral and tangential. Regular: the polygon is both isogonal and isotoxal. Equivalently, it is both equilateral, or both equilateral and equiangular. A non-convex regular polygon is called a regular star polygon. Rectilinear: the polygon's sides meet at right angles, i.e. all its interior angles are 90 or 270 degrees.

Monotone with respect to a given line L: every line orthogonal to L intersects the polygon not more than twice. Euclidean geometry is assumed throughout. Any polygon has as many corners; each corner has several angles. The two most important ones are: Interior angle – The sum of the interior angles of a simple n-gon is π radians or × 180 degrees; this is because any simple n-gon can be considered to be made up of triangles, each of which has an angle sum of π radians or 180 degrees. The measure of any interior angle of a convex regular n-gon is 180 − 360 n degrees; the interior angles of regular star polygons were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra: for a regular p q -gon, each interior angle is π p radians or 180 p degrees. Exterior angle – The exterior angle is the supplementary angle to the interior angle. Tracing around a convex n-gon, the angle "turned" at a corner is external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360°.

This argument can be generalized to concave simple polygons, if external angles that turn in the opposite direction are subtracted from the total turned. Tracing around an n-gon in general, the sum of the exterior angles can be any integer multiple d of 360°, e.g. 720° for a pentagram and 0° for an angular "eight" or antiparallelogram, where d is the density or starriness of the polygon. See orbit. In this section, the vertices of the polygon under consideration are taken to be, ( x 1

Hubert Bonnet

Hubert Bonnet is a Switzerland-based Belgian businessman and art collector. Hubert Bonnet, son of Pierre Bonnet and Berthe Germeau, was born to an influential family in the Belgian steel industry. At the time, the Clabecq Forge, located twenty kilometres from Brussels, specialised in the production of steel, more railway lines. Hubert's mother, the sole heir of Eugène Germeau, the businessman at the helm of Clabecq before the World War II, served as vice-president to the board of directors for many years. With her husband, Pierre Bonnet, they played a significant part in all the major transformations, including those during times of crisis. In 1974, while the Forges were in the hands of the Dessy family, the Germeau-Bonnet sold their stake to Cobepa alse, the holding company of the French group Paribas, in which they became one of the Belgian shareholding families until the takeover bid by BNP PARIBAS in 2000. Hubert Bonnet grew up between Knokke. After spending his first years at the Cardinal Mercier College in Braine-L'Alleud, he began his postgraduate course at the European Business School in Brussels.

He has always been involved in the business world, choosing the career path of BBA – Bachelor in Business Administration. He moved on to the University of Montreux in Switzerland before completing his training with an MBA at the University of Dallas in Texas. Passionate about travel and discovery since childhood, Hubert Bonnet benefitted from his studies abroad through enriching encounters and by developing new areas of interest, it was during this era that his passion for art was born, influenced by the audacity of creations by American artists. Qualifications in hand, the 27-year-old Hubert Bonnet moved to New York City and began his professional life as a trader for Cadogan Management. After the death of his mother in 1996, he returned to Belgium and became a director of Cobepa until the takeover bid by BNP Paribas in 2000. While continuing his activities in the financial sector, he ventured into real estate and created his first company. With H Group, he created prestigious projects, acquiring unique properties and improving them with the savoir-faire of renowned architects.

His clients were private investors, concerned about the security of their investments. Today, after twenty years of operation, H Group has added its signature to a number of residences in Belgium, notably in the Prince d'Orange district of Uccle, Châtelain in Ixelles, in Zoute, the Dominican Republic, Paris and Saint-Tropez. In Switzerland, his country of residence, he is involved in the development of real estate projects in the Valais canton and in Geneva. Since 2015, Hubert Bonnet has provided a new activity to complete his real estate offer. With Bibi Home, he offers the rental of prestigious locations and a complete service to his customers; the concept includes an artistic element, since outside high season he offers these prestigious residences to artists for workshops to research and create. In addition to real estate, Hubert Bonnet formed the company Meaunet Financière H2O, both operating in the financial sector. In 2005, he became shareholder and director of Groupe Josi. Having been passionate about art for many years, Hubert Bonnet is a collector who travels the world seeking out new talent, visiting art galleries, private viewings and international fairs, this aesthete is appreciative of minimal and conceptual art.

In 2012, while looking for a place to store his works, Hubert Bonnet discovered an 800m2 former coal warehouse, just a few steps from the lakes of Ixelles. This location became the CAB, an independent art centre dedicated to the promotion of contemporary Belgian and international art; each year it offers two thematic exhibitions – one from September to December and one from April to June – whereby it offers artists visibility in a place, neither an institution nor a gallery. As director of the centre, Hubert Bonnet is involved in all levels of decision making, from the theme of the exhibition to the choice of artists. To reduce the dangers facing children around the world, Hubert Bonnet set up the Bibi Foundation, based in Switzerland, which carries out development assistance independently or jointly with other similar bodies; the BIBI Foundation supports the NGO ODDY-C. Hubert Bonnet supports Womanity, an association committed to empowering women and girls in developing countries to shape their futures and accelerate progress in their communities.

Hubert Bonnet is a member of the Entrepreneurs' Organization and the Alumni of the European Business School

Orville Emil Bloch

Orville Emil Bloch was a United States Army officer and a recipient of the United States military's highest decoration—the Medal of Honor—for his actions in World War II. Bloch joined the Army from Streeter, North Dakota in February 1942, by September 22, 1944 was serving as a first lieutenant in Company E, 338th Infantry Regiment, 85th Infantry Division. On that day, near Firenzuola, Italy, he led three soldiers in an attack on enemy positions which resulted in the capture of nineteen prisoners and the silencing of five machine gun nests. For these actions, he was awarded the Medal of Honor five months on February 10, 1945. Bloch served in the Korean War, reached the rank of colonel before retiring in 1970, he died at age 68 and was buried in Evergreen-Washelli Memorial Park, Washington. Bloch's official Medal of Honor citation reads: For conspicuous gallantry and intrepidity at risk of life above and beyond the call of duty. 1st Lt. Bloch undertook the task of wiping out 5 enemy machinegun nests that had held up the advance in that particular sector for 1 day.

Gathering 3 volunteers from his platoon, the patrol snaked their way to a big rock, behind which a group of 3 buildings and 5 machinegun nests were located. Leaving the 3 men behind the rock, he attacked the first machinegun nest alone charging into furious automatic fire, kicking over the machinegun, capturing the machinegun crew of 5. Pulling the pin from a grenade, he held it ready in his hand and dashed into the face of withering automatic fire toward this second enemy machinegun nest located at the corner of an adjacent building 15 yards distant; when within 20 feet of the machinegun he hurled the grenade, wounding the machinegunner, the other 2 members of the crew fleeing into a door of the house. Calling one of his volunteer group to accompany him, they advanced to the opposite end of the house, there contacting a machinegun crew of 5 running toward this house. 1st Lt Bloch and his men opened fire on the enemy crew, forcing them to abandon this machinegun and ammunition and flee into the same house.

Without a moment's hesitation, 1st Lt. Bloch, rushed through the door into a hail of small-arms fire, firing his carbine from the hip, captured the 7 occupants, wounding 3 of them. 1st Lt. Bloch with his men proceeded to a third house where they discovered an abandoned enemy machinegun and detected another enemy machinegun nest at the next corner of the building; the crew of 6 spotted 1st Lt. Bloch the instant. Without a moment's hesitation he dashed toward them; the enemy fired pistols wildly in his direction and vanished through a door of the house, 1st Lt. Bloch following them through the door, firing his carbine from the hip, wounding 2 of the enemy and capturing 6. Altogether 1st Lt. Bloch had single-handedly captured 19 prisoners, wounding 6 of them and eliminating a total of 5 enemy machinegun nests, his gallant and heroic actions saved his company many casualties and permitted them to continue the attack with new inspiration and vigor. List of Medal of Honor recipients List of Medal of Honor recipients for World War II This article incorporates public domain material from websites or documents of the United States Army Center of Military History.

"Orville Emil Bloch". Claim to Fame: Medal of Honor recipients. Find a Grave. Retrieved 2008-02-26. "Medal of Honor recipients - World War II". Medal of Honor citations. United States Army Center of Military History. June 8, 2009. Retrieved 2008-02-26