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Line 1: {{Infobox Town DE\| In [[geometry]], a '''Coxeter-Dynkin diagram''' is a [[Graph (mathematics)\|graph]] representing a relational set of [[mirror]] (or [[reflection]]al [[hyperplane]]s) in space for a [[Kaleidoscope\|kaleidoscopic]] construction. Each node in the graph represents one mirror, and each edge represents the [[dihedral angle]] between two mirrors. name = Dillenburg\| name_local =\| image_coa = Wappen Dillenburg.png\| image_map = KarteDillenburg.png\| state = [[Hesse]]\| regbzk = [[Gießen (region)\|Gießen]]\| district = [[Lahn-Dill-Kreis]]\| population = 24,533\| population_as_of = 31/12/2004\| population_ref =\| pop_dens = 292\| area = 83.88\| elevation = 233\| lat_deg = 50\| lat_min = 44\| lat_hem = N\| lon_deg = 8\| lon_min = 17\| lon_hem = E\| postal_code = 35661–35690 (''old:'' 6340)\| area_code = 02771\| licence = LDK\| mayor = Michael Lotz ([[Christian Democratic Union (Germany)\|CDU]])\| website = [http://www.dillenburg.de Stadt Dillenburg]\| }} '''Dillenburg''' is a town in Hesse's [[Gießen (region)\|Gießen]] region in [[Germany]]. The town was formerly the seat of the old Dillkreis district, which is now part of the [[Lahn-Dill-Kreis]]. The town lies on the German-[[Netherlands\|Dutch]] holiday road called the ''Oranier-Route'', joining towns, cities and regions associated with the [[House of Orange]]. ~~As a graph itself, the diagram represents [[Coxeter group]]s.~~ == Geography == The diagram can also represent [[polytopes]] by adding [[ring]]s (circles) around nodes. The rings express information on whether a generating point is on or off the mirror. Specifically a mirror is ''active'' (creates reflections) only when points are off the mirror, so adding a ring means a point is off the mirror and creates a reflection. === Location === Dillenburg lies on the eastern edge of the [[Westerwald]] range in the narrow valley of the river [[Dill (river)\|Dill]], which flows from Hesse-Westphalia border to [[Wetzlar]], emptying into the [[Lahn]]. <gallery> Image:Dillenburg03.jpg\|Downtown Dillenburg Image:Dillenburg01.jpg\|View of the town looking north from the Wilhelmsturm Image:Dillenburg02.jpg\|View of the town looking south from the same vantage point Image:db_manderbach_25.jpg\|Dillenburg-Manderbach </gallery> === Neighbouring communities === Dillenburg borders in the north on the community of [[Eschenburg]], in the east on the community of [[Siegbach]], in the south on the town of [[Herborn]], and the community of [[Breitscheid (Hesse)\|Breitscheid]], and in the west on the town of [[Haiger]] (all in the Lahn-Dill-Kreis). === Constituent communities === Edges are labeled with an [[integer]] ''n'' (or sometimes more generally a [[rational number]] ''p/q'') representing a [[dihedral angle]] of ''180/n''. If an edge is unlabeled, it is assumed to be ''3''. If ''n'' is two the angle is 90 degrees and the mirrors have no interaction, and the edge can be left unmarked. Two parallel mirrors can be marked with an infinity (∞) symbol. Dillenburg is divided into the centres of Donsbach, Eibach, Frohnhausen, Manderbach, Nanzenbach, Niederscheld and Oberscheld. ==== Eibach ==== In general ''n'' mirrors can be represented in an n-[[simplex]] graph where all ''n(n-1)/2'' edges are drawn. In practice interesting sets of mirrors will have a number of right angles, and such edges can be left undrawn. Eibach has some 1,450 inhabitants. The village, whose livelihood was once based on [[mining]], lies among the other constituent communities of Nanzenbach, Oberscheld and Niederscheld. Its healing [[spring (hydrosphere)\|spring]], whose water is heavy with [[iron]], makes the village a favourite among locals. At [[Easter]]time, it is decorated. ~~[[Image:Example fundamental ___domain coxeter dynkin diagram.png\|320px\|thumb]]~~ ~~[[Image:Example2 fundamental ___domain coxeter dynkin diagram.png\|320px\|thumb]]~~ [[Polytopes]] and [[tessellation]]s can be generating using these mirrors and a single generator point. Mirror images create new points as reflections. [[Edge]]s can be created between points and a mirror image. [[Face (geometry)\|Faces]] can be constructed by cycles of edges created, etc. ==== Manderbach ==== ~~Examples:~~ Manderbach lies on a sunny [[plateau]] 3 km north of the main town of Dillenburg. A single node represents a single mirror. All points off the mirror are drawn with a ring. Connecting such an off mirror point to its reflection creates a [[digon]] or [[edge]] perpendicular to the mirror. * Two unattached nodes represent two [[perpendicular]] mirrors. If both nodes are ringed, a [[rectangle]] can be created, or a [[square]] if the point is equal distance from both mirrors. * Two nodes attached by an ''n'' edge can create an [[polygon\|n-gon]] if the point is on one mirror, and a ''2n-gon'' if the point is off both mirrors. * Three mirrors in a triangle form images seen in a traditional [[kaleidoscope]] and be represented by 3 nodes connected in a triangle. Repeating examples will have edges labeled as (3 3 3), (2 4 4), (2 3 6), although the last two can be drawn in a line with the ''2'' edge ignored. These will generate [[Tiling by regular polygons\|uniform tilings]]. * Three mirrors with a common point can generate [[uniform polyhedron]]s, including rational numbers is the set of [[Schwarz triangle]]s. * Three mirrors with one perpendicular to the other two can form the [[Prism (geometry)\|uniform prisms]]. <div style="height: 265px; width: 95%; overflow: auto; text-align: left; border:solid 1px;" title="braglist - zum scrollen"> In general all regular n-[[polytope]]s, represented by [[Schläfli symbol]] symbol {p,q,r,...} can have their [[fundamental ___domain]]s represented by a set of ''n'' mirrors and a related in a '''Coxeter-Dynkin diagram''' in a line of nodes and edges labeled by p,q,r... [[Image:Manderbach_Panorama-Südblick.JPG\|1300px]] </div> <center>Panoramic views over Manderbach</center> <div style="height: 265px; width: 95%; overflow: auto; text-align: left; border:solid 1px;" title="braglist - zum scrollen"> [[Image:Manderbach_1.jpg\|1300px]] </div> ==== Niederscheld ==== ~~Examples~~ Niederscheld is a village with about 3000 inhabitants, lying 2 km from the main town of Dillenburg. The name comes from a small brook called the Schelde that rises between Oberscheld and Tringenstein, and empties into the Dill at Niederscheld. The village's greatest hallmarks are the old [[blast furnace]] and the Adolfshütte [[industrial park]]. Towards the end of the [[Second World War]], the village suffered comparatively heavy damage from [[Allies of World War II\|Allied]] [[air raid]]s. * [[Image:Dynkins-100.png]] - {p,q} - a regular polyhedron or tiling Niederscheld had been appointed as a target because at the Adolfshütte were built parts for the V2-Rocket. * [[Image:Dynkins-1000.png]] - {p,q,r} - a regular [[polychoron]] or honeycomb == ~~See also~~History == Dillenburg had its first documentary mention in 1254. Dillenburg was the ancestral seat of the Orange branch of the [[House of Nassau]]. Dillenburg Castle was build on top of the peak now called the Schlossberg in the late [[13th century\|13th]] or early [[14th century]]. There are no pictures of this castle, however, as it was wooden, and was destroyed in the Dernbach Feud. * [[Coxeter group]] * [[Uniform polytope]] * [[Wythoff construction]] From his stately home in exile, [[William the Silent\|William I of Orange-Nassau]], who was born in Dillenburg, organized the Dutch resistance against [[Spain]] ([[1567]]-[[1572]]), which still occasions regular Dutch royal visits to the town to this day. The land was administered by the presidents of the House of Nassau-Dillenburg. One of the last presidents was Georg Ernst Ludwig Freiherr von Preuschen von und zu [[Liebenstein]] (born 1727 in [[Diethardt]]; died 1794 in [[Bad Ems]]). In the [[Seven Years' War]], the stately home was destroyed ([[1760]]), and Wilhelmstraße (a street) was built out of the remains. In 1875, the Wilhelmsturm (tower), views from which can be seen in this article, was completed on the Schlossberg. It is today the town's landmark. The "casemates" under the former stately home are among the biggest defensive works in [[Europe]]. They have been partly excavated and may be toured. ~~== References ==~~ * [[Harold Scott MacDonald Coxeter\|Coxeter]] ''Regular Polytopes'' (1963), Macmillian Company In the [[19th century]] came the [[Industrial Revolution]] with the building of the [[Gießen]]-[[Cologne]] [[railway]] line – called the Sieg-Dill line – and the use of [[iron ore]] found on the Lahn, Dill and Sieg. Many mines, [[Foundry\|foundries]] and metalworking operations came into being in the region. In this time, many railway branchlines were built from Dillenburg to, among other places, Gönnern and Ewersbach. These lines have all been abandoned now. The line to Gönnern was abandoned in 1987 and torn up. The railway depot, so useful in the time of steam traction, was shut down in 1983. ** ''Regular Polytopes'', Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Section 11.3 Representation by graphs) * [[Harold Scott MacDonald Coxeter\|Coxeter]] ''The Beauty of Geometry: Twelve Essays'', Dover Publications, 1999, ISBN 99-35678 (Chapter 3: Wythoff's Construction for Uniform Polytopes) In the [[Second World War]], Dillenburg became a target of Allied attacks with its now closed [[Classification yard\|marshalling yard]]. Ore mining became ever less profitable and in 1968, the last blast furnace, in Oberscheld, ceased operations. ==== Eibach ==== Eibach's history began in "Nassau times" in the [[13th century]]. In 1313, the village had its first documentary mention. In the Second World War, it was left unscathed. In 2004, the healing spring was renovated, and a brineworks was built. ==== Manderbach ==== [[Image:Wappen 180.jpeg\|thumb\|100px\|Manderbach's arms]] Manderbach had its first documentary mention in 1225, making it older than the main town of Dillenburg (1254). The two former villages – nowadays parts of Dillenburg – Frohnhausen and Manderbach, had much in common in their early history. Here the two noble families von Hunsbach and von Selbach both held sway. As in Frohnhausen, there was also a great fire in Manderbach – albeit 148 years before Frohnhausen's – which, having been started by a [[lightning strike]], burnt 38 houses down within an hour and a half on [[29 April]] [[1630]]. ==== Nanzenbach ==== The Name Nanzenbach was mentioned the first time 8.05.1325 in a document. In this document the „die Nantzenbecher“ as the "inhabitants of Nanzenbach ". This allows the village the first reference at May 8th 1325. See more on [http://www.nanzenbach.de] === Population development === <small>(in each case on [[31 December]])</small> [[1998]] - 25,053 [[1999]] - 25,124 [[2000]] - 25,092 [[2001]] - 25,017 [[2002]] - 24,923 [[2003]] - 24,681 [[2004]] - 24,533 == Coat of arms == The oldest town seals, dating from the 15th to 19th centuries, show the same composition as Dillenburg's current civic [[coat of arms]]. The arms were conferred officially in 1907 and confirmed in 1934. The lion inside the gateway is the Lion of Nassau. [http://www.ngw.nl/int/dld/d/dillenbu.htm] == Culture and sightseeing == === Museums === ''Wilhelmsturm'' (tower) with the Orange-Nassau Museum * "Villa Grün" museum of economic history * The "Casemates", old defensive structures. * ''Hessisches Landgestüt'' (≈ Hessian State Stud Farm) with coach museum in the Orangery. "Living Museum" about the [[horse]]. === Buildings === [[Image:Manderbach_Kirche.jpg\|thumb\|Manderbach Church]] * ''Wilhelmsturm'' (tower) built in 1872 - 1875 * The "Casemates", old defensive structures.from the 16th century * The Evangelische Town Church from 1491 * The ''Dillturm'' (tower) from 1597 * The old rectory from 1531 - 1533 * The ''Untertor'' (Lower Gate) from 1344 (alterations in 1594 and 1737) * Manderbach Church === Parks === In Donsbach is a wildlife park. === Hiking trails === The following trails go through or begin in Dillenburg: * The ''Rothaarsteig'' from Dillenburg to [[Brilon]] * The ''Schlösserweg'' from Dillenburg to [[Düsseldorf\|Düsseldorf -Benrath]] * The ''Dillweg'' from [[Haiger]] to [[Wetzlar]] * The ''Uplandweg'' from Dillenburg to [[Salzkotten]] === Regular events === * Jazz-Weekend, in June * ''Kirschenmarkt'' (cherry market), in June * Aquarena-Nacht, in July * Hubertus-Markt, in October * Hengstparade des Hessischen Landgestüts (stallion parade) * Maypole Festival in Eibach, at the beginning of May * ''Rocknacht'' music festival in Eibach, in summer. === Other === * Brineworks and healing spring, Eibach == Economy and infrastructure == === Transport === The bypass on Federal Highway (''Bundesstraße'') B277 opened in April 2007. It is a [[tunnel]] under the Schlossberg, bypassing the historic Old Town with its [[Timber framing\|half-timbered]] houses and it was one of Germany's biggest tunnel projects. As a result of the bankruptcy of the contracter for the works, Walter Bau, completion of the project was delayed by more than a year. === Established businesses === * [[Deutsche Post AG]] * [[E.ON]] Mitte (OT Oberscheld) * Funkenerosionstechnik Hartwig Hermann * INDEN Design * [[Isabellenhütte Heusler]] GmbH & Co. KG * Linde & Wiemann * Ströher-Keramik * [[ThyssenKrupp Nirosta]] Dillenburg Works * TSR Recycling GmbH & Co. KG Dillenburg Branch * [[Volksbank]] Dill eG * Weber Kunststofftechnik === Media === * Dill-Post * Dill-Zeitung === Public institutions === * [[Police station]] * [[Fire brigade]] * Dill-Kliniken ([[hospital]]) * [[Deaconate]] * [[German Red Cross]] Dillkreis chapter * Lahn-Dill Youth Office * Lahn-Dill ''Jugendbildungswerk'' * ''Lebenshilfe'' for the mentally handicapped ''Kreisvereinigung für den ehem. Dillkreis e.V. '' (District association for the former Dill district) * Lahn-Dill Social Office === Education === * Gewerbliche Schulen ([[vocational school]]) * Goldbachschule (Haupt- and Realschule) * Juliane-von-Stolberg-Schule ([[primary school]]) * Johann-von-Nassau-Schule ([[Hauptschule\|Haupt]]- and [[Realschule]]) * Kaufmännische Schulen (vocational school) * Kindergartens ([[Evangelical Church in Germany\|Evangelical]], [[Catholic]], municipal, ''Arbeiterwohlfahrt'' [German workers' welfare]) * Lahn-Dill-Akademie ([[Folk high school]]) * Otfried-Preußler-Schule für Praktisch Bildbare ([[special school]]) * Roteberg-Schule (primary school) * Schelderwald-Schule (primary school and Hauptschule) * Wilhelm-von-Oranien-Schule ([[Gymnasium (school)\|Gymnasium]]) == Personalities == === Famous natives of Dillenburg === [[Otfried Hans Freiherr von Meusebach]] alias ''John O. Meusebach'' (1812-1897), founder of [[Fredericksburg, Texas]] and Texas Senator [[Karl Heinz Gasser]], German politician [[Maria Kliegel]], German cellist [[Rolf Krenzer]], writer of children's books and composer [[Moritz von Nassau]], Dutch field marshal, called the ''Brasilianer'' ("Brazilian") [[Ernst Casimir van Nassau-Dietz]], ancestor of Kings of the Netherlands [[John VI of Nassau-Dillenburg]], German aristocrat [[Maurice of Nassau]] (Also known as Maurits van Oranje-Nassau/Moritz von Oranien), [[Stadtholder]] of [[Holland]], [[Zeeland]], [[Diocese of Utrecht\|Utrecht]], [[Guelders]] and [[Overijssel]], also [[Prince of Orange]] [[William the Silent\|William I of Orange-Nassau]], leader in the Dutch war of independence against Spain Prof. Dr. [[Melanie Tatur]], German political scientist and sociologist [[Wilhelm Zepper]], reformed theologian; court chaplain and professor in [[Herborn]] === People who worked in Dillenburg === [[Georg Ludwig Hartig]] worked from [[1797]] to [[1806]] as state forest superintendent in Dillenburg * [[Maximilian Mörlin]] Evangelical theologian and reformer == External links == * [http://www.dillenburg.de Dillenburg] * [http://mathworld.wolfram.com/Coxeter-DynkinDiagram.html MathWorld] * [http://www.bahnhof-dillenburg.de Website about Dillenburg railway station and railway guide line 445] * [http://www.feuerwehr-dillenburg.de Dillenburg fire station] *{{dmoz\|World/Deutsch/Regional/Europa/Deutschland/Hessen/Landkreise/Lahn-Dill-Kreis/Städte_und_Gemeinden/Dillenburg\|Dillenburg}} == Reference == {{de}} {{Towns and municipalities in Lahn-Dill-Kreis}} [[Category:Towns in Hesse]] ~~{{geometry-stub}}~~ [[~~Category~~de:~~polytopes~~Dillenburg]] [[eo:Dillenburg]] [[fr:Dillenburg]] [[nl:Dillenburg (stad)]] [[pl:Dillenburg]] [[ru:Дилленбург]] [[sv:Dillenburg]] [[vo:Dillenburg]]

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