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43-613: The characteristic pyramid -shaped peak actually consists of two pinnacles, the Großglockner and the Kleinglockner (3,770 m or 12,370 ft, from German: groß 'big', klein 'small'), separated by the Glocknerscharte col . The name Glocknerer is first documented in a 1561 map designed by the Viennese cartographer Wolfgang Lazius . The denotation Glogger is mentioned in

86-404: A Platonic solid and deltahedra , and it has tetrahedral symmetry . A pyramid with the base as circle is known as cone . Pyramids have the property of self-dual , meaning their duals are the same as vertices corresponding to the edges and vice versa. Their skeleton may be represented as the wheel graph , that is they can be depicted as a polygon in which its vertices connect a vertex in

129-533: A pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex . Each base edge and apex form a triangle , called a lateral face. A pyramid is a conic solid with a polygonal base. Many types of pyramids can be found by determining the shape of bases, either by based on a regular polygon (regular pyramids) or by cutting off the apex (truncated pyramid). It can be generalized into higher dimensions, known as hyperpyramid . All pyramids are self-dual . The word "pyramid" derives from

172-570: A 1583 description of the Tyrolean Kals legal district, then referring to the whole ridge south of the Alpine main chain. In the 1760s, the Atlas Tyrolensis listed a Glockner Berg , the prefix Gross- ("great") is not mentioned before the first expedition in 1799. According to the scholar Belsazar Hacquet (1735–1815), Glockner is possibly derived from German: Glocke (" bell "), referring to

215-671: A branch-off to the Franz-Josefs-Höhe viewpoint. It was built across the historic Hochtor Pass of the Alpine divide between 1930 and 1935 according to plans designed by engineer Franz Wallack . The pass road, Austria's highest, reaches 2,576 m (8,451 ft), and is one of the most popular tourist attractions in the country (second after Schönbrunn Palace ) with about 270,000 vehicles and 900,000 visitors every year, about 50 million since its opening. Also famous for its historic vehicle hill climb races . Computer-generated virtual panoramas Pyramid (geometry) In geometry ,

258-505: A mountainous terrain. A couloir may be a seam, scar, or fissure, or vertical crevasse in an otherwise solid mountain mass. Though often hemmed in by sheer cliff walls, couloirs may also be less well-defined, and often simply a line of broken talus or scree ascending the mountainside and bordered by trees or other natural features. Couloirs are especially significant in winter months when they may be filled in with snow or ice, and become much more noticeable than in warmer months when most of

301-604: A popular ski mountaineering tour. The Grossglockner became Austria's highest mountain, when the South Tyrolean Ortler region had to be ceded to the Kingdom of Italy according to the 1919 Treaty of Saint-Germain , which promoted its reputation as a tourist attraction . Mass tourism was decisively promoted by the scenic High Alpine Road ( Großglockner-Hochalpenstraße ) running from Heiligenblut to Bruck in Salzburg with

344-464: A popular venue for Alpinists like Hermann and Adolf Schlagintweit , who all followed the route of the first ascent. By the mid 19th century, the developing Alpine tourism began to alter the traditional agriculture economy in the Heiligenblut area. Therefore, the people of Kals tried to lay out a straight ascent from the western side, which however was not reached until Julius von Payer explored

387-439: A pyramid as a solid figure, constructed from one plane to one point. The context of his definition was vague until Heron of Alexandria defined it as the figure by putting the point together with a polygonal base. A prismatoid is defined as a polyhedron where its vertices lie on two parallel planes, with its lateral faces as triangles, trapezoids , and parallelograms . Pyramids are classified as prismatoid. A right pyramid

430-495: A pyramid is the one-third product of the base's area and the height. The pyramid height is defined as the length of the line segment between the apex and its orthogonal projection on the base. Given that B {\displaystyle B} is the base's area and h {\displaystyle h} is the height of a pyramid, the volume of a pyramid is: V = 1 3 B h . {\displaystyle V={\frac {1}{3}}Bh.} The volume of

473-408: A pyramid was recorded back in ancient Egypt, where they calculated the volume of a square frustum , suggesting they acquainted the volume of a square pyramid. The formula of volume for a general pyramid was discovered by Indian mathematician Aryabhata , where he quoted in his Aryabhatiya that the volume of a pyramid is incorrectly the half product of area's base and the height. The hyperpyramid

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516-527: A southeasterly direction and forming the boundary between the East Tyrolean municipality of Kals am Großglockner, about 8 km (5.0 mi) in the southwest at 1,324 m (4,344 ft), and Carinthian Heiligenblut , about 12 km (7.5 mi) in the northeast at 1,291 m (4,236 ft). This boundary is also the watershed between the Kals Valley and its Teischnitz and Ködnitz side valleys on

559-634: A track in the snow, had installed fixed ropes at some steeper sections up to the end of the Glocknerleitl, and even built a second refuge, called Hohenwarte Hut. The vanguard reached the Kleinglockner peak, however, according to the expedition records by the Dellach priest Franz Joseph Horasch (Orasch), only the four guides and Mathias Hautzendorfer, the local priest of the Rangersdorf parish, were able to cross

602-453: Is C n v , a symmetry of order 2 n : the pyramids are symmetrical as they rotated around their axis of symmetry (a line passing through the apex and the base centroid), and they are mirror symmetric relative to any perpendicular plane passing through a bisector of the base. Examples are square pyramid and pentagonal pyramid , a four- and five-triangular faces pyramid with a square and pentagon base, respectively; they are classified as

645-485: Is a pyramid whose base is circumscribed about a circle and the altitude of the pyramid meets the base at the circle's center; otherwise, it is oblique. This pyramid may be classified based on the regularity of its bases. A pyramid with a regular polygon as the base is called a regular pyramid. For the pyramid with an n - sided regular base, it has n + 1 vertices, n + 1 faces, and 2 n edges. Such pyramid has isosceles triangles as its faces, with its symmetry

688-602: Is counted as an independent peak in mountaineering literature. The Obere Glocknerscharte between the two peaks, at 3,766 m (12,356 ft), is the highest col in Austria, from which a couloir up to 55° in gradient and 600 m (2,000 ft) in altitude descends down to the Glocknerkees glacier, called Pallavicinirinne after the Austrian mountaineer Alfred von Pallavicini (1848–1886). It runs northeastwards and borders on

731-422: Is the generalization of a pyramid in n - dimensional space. In the case of the pyramid, one connects all vertices of the base, a polygon in a plane, to a point outside the plane, which is the peak . The pyramid's height is the distance of the peak from the plane. This construction gets generalized to n dimensions. The base becomes a ( n − 1) - polytope in a ( n − 1) - dimensional hyperplane. A point called

774-457: Is the side of Grossglockner with the least ice (people feared glaciers in these times). These valiant men, called "Glockners" in the records, did more than they were ordered to do—and probably reached the Kleinglockner summit on 23 July 1799. One month later the bishop's expedition started: a mountain hut (the first Salm Hut) had been built and the path in the Leitertal valley was prepared so that

817-402: The n - dimensional volume of the hyperpyramid. A denotes the ( n  − 1) - dimensional volume of the base and h the height, that is the distance between the apex and the ( n  − 1) - dimensional hyperplane containing the base A . Couloir A couloir ( French: [ku.lwaʁ] , "passage" or "corridor") is a narrow gully with a steep gradient in

860-568: The Academy of Ljubljana . He travelled the Eastern Alps from 1779 to 1781 and published an itinerary in 1783, describing the Glokner mountain and stating that it had not been climbed yet. He estimated the mountain's height with converted 3,793 m (12,444 ft) and left an engraving illustrating Grossglockner and Pasterze, the first known depiction of the mountain. Inspired by Hacquet's book and

903-611: The Obere Glocknerscharte and climb the Grossglockner summit. Hautzendorfer had to be persuaded to venture the step and administered the last rites in advance. The two "Glockners" are usually identified as the brothers Joseph ( Sepp ) and Martin Klotz , however, this surname is not listed in the Heiligenblut parish register. A local peasant named Sepp Hoysen is documented as a member of the second Grossglockner expedition in 1802, and

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946-508: The ancient Greek term "πυραμίς" (pyramis), which referred to a pyramid-shaped structure and a type of wheat cake. The term is rooted in the Greek "πυρ" (pyr, 'fire') and "άμις" (amis, 'vessel'), highlighting the shape's pointed, flame-like appearance. In Byzantine Greek , the term evolved to "πυραμίδα" (pyramída), continuing to denote pyramid structures. The Greek term "πυραμίς" was borrowed into Latin as "pyramis." The term "πυραμίδα" influenced

989-520: The first ascent of the Mont Blanc in 1786, the Gurk prince-bishop Count Franz Xaver of Salm (1749–1822) together with his vicar general Sigismund Ernst Hohenwart (1745–1825) and Baron Franz Xaver von Wulfen (1728–1805) started efforts for a Grossglockner expedition. They engaged two peasants from Heiligenblut as mountain guides to do the first explorations for an ascent through the Leitertal valley, which

1032-407: The star pyramid in which its base is the regular star polygon . The truncated pyramid is a pyramid cut off by a plane; if the truncation plane is parallel to the base of a pyramid, it is called a frustum . The surface area is the total area of each polyhedra's faces. In the case of a pyramid, its surface area is the sum of the area of triangles and the area of the polygonal base. The volume of

1075-807: The 3,721 m (12,208 ft) high Glocknerwand . A prominent ridge, the Stüdlgrat , named after the Prague Alpinist Johann Stüdl (1839–1925), runs from the Grossglockner away to the southwest. Together with its extension, the Luisengrat , it separates the West Face and the Teischnitzkees glacier at its foot from the South Face and its Ködnitzkees glacier. A couloir known as the Pillwaxrinne crosses

1118-639: The Alps. Even its topographic prominence , at 2,424 m (7,953 ft), is the second highest after Mont Blanc in the entire Alps (see the list of Alpine peaks by prominence ). That makes it one of the most independent peaks in the Alps. The view from the Grossglockner summit is one of the farthest of all mountains in the Eastern Alps. It ranges out to 220 km (140 mi) or, taking account of atmospheric refraction , almost 240 km (150 mi). Its view over more than 150,000 km (58,000 sq mi) of

1161-761: The Earth's surface reaches as far as the Upper Swabian Plateau in the northwest, to Regensburg and the peaks of the Bohemian Forest in the north, to Ortler in the west, to the Padan Plain in the south, and to Triglav and the Totes Gebirge range in the east. The Grossglockner rock summit, due to its high Alpine, heavily glaciated appearance, is often compared to the mountains of the Western Alps . Together with

1204-421: The Kleinglockner to the southeast it forms a distinctive double peak . There are differing views in the literature as to whether the Kleinglockner is a subpeak or a separate main summit. Due to its low topographic prominence and isolation as well as its close links in climbing history, it is counted as part of that of the Grossglockner in historic publications; however, in view of its separate climbing routes it

1247-692: The Northeast and North Faces of the Grossglockner. The latter faces are bounded by the Northwest Ridge, part of the main Glockner crest, which runs over the Grögerschneid , 3,660 m (12,010 ft), and the ridge elevations of Glocknerhorn , 3,680 m (12,070 ft), and Teufelshorn , 3,677 m (12,064 ft), to the Untere Glocknerscharte , 3,598 m (11,804 ft), which connects to

1290-663: The South Face below the Obere Glocknerscharte; most of the South Face lies east of this gully below the Kleinglockner. The east side of the Kleinglockner, the Glocknerleitl , is glaciated to just below the summit and is continued by the Kleinglocknerkees and Hofmannskees before reaching the Pasterze. The history of the climbs started with French-born natural scientist Belsazar Hacquet , from 1773 professor of anatomy at

1333-790: The Tyrolean side and the Möll Valley with the Pasterze glacier on the Carinthian side. The region around the mountain has formed part of the Grossglockner-Pasterze special protected area within the High Tauern National Park since 1986. The Glockner is the highest mountain in the Alps east of the Ortler range, about 175 km (109 mi) away, and, after Mont Blanc , has the second greatest topographic isolation of all mountains in

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1376-423: The apex is located outside the hyperplane and gets connected to all the vertices of the polytope and the distance of the apex from the hyperplane is called height. The n - dimensional volume of a n - dimensional hyperpyramid can be computed as follows: V n = A ⋅ h n . {\displaystyle V_{n}={\frac {A\cdot h}{n}}.} Here V n denotes

1419-478: The bishop could use a horse to reach it. 30 people, among them Salm, Hohenwart and Wulfen, were part of the expedition. They suffered with bad weather and a first effort failed, but on 25 August 1799 Hohenwart and at least four other people, including the two "Glockners", reached—again—the Kleinglockner, where they installed one of the first summit crosses (one of the main goals of the church expedition). Hohenwart's reports did not tell clearly that they had not touched

1462-401: The center called the universal vertex . A right pyramid may also have a base with an irregular polygon. Examples are the pyramids with rectangle and rhombus as their bases. These two pyramids have the symmetry of C 2v of order 4. The type of pyramids can be derived in many ways. The base regularity of a pyramid's base may be classified based on the type of polygon: one example is

1505-544: The evolution of the word into "pyramid" in English and other languages. A pyramid is a polyhedron that may be formed by connecting a polygonal base and a point, called the apex . Each base edge and apex form an isosceles triangle called a lateral face. The edges connected from the polygonal base's vertices to the apex are called lateral edges. Historically, the definition of a pyramid has been described by many mathematicians in ancient times. Euclides in his Elements defined

1548-403: The first and second Johnson solid if their regular faces and edges that are equal in length, and their symmetries are C 4v of order 8 and C 5v of order 10, respectively. A tetrahedron or triangular pyramid is an example that has four equilateral triangles, with all edges equal in length, and one of them is considered as the base. Because the faces are regular, it is an example of

1591-501: The first expedition up the steep glaciated Northeast Face, chopping 2,500 steps into the Pallavicinirinne in an ice climbing master stroke not repeated for 23 years. In 1879 Count Pallavicini dedicated a new iron summit cross on the occasion of the silver wedding of Emperor Franz Joseph of Austria and Empress Elisabeth ; both had visited Heiligenblut and walked to the present-day Franz-Josefs-Höhe viewpoint in 1865. The cross

1634-492: The highest point but Bishop Salm (who had reached the Adlersruhe rock at 3,454 m (11,332 ft)) was informed. Dissatisfied, he invited another, even bigger expedition the next year. On 28 July 1800, 62 people, among them the pedagogue Franz Michael Vierthaler and the botanist David Heinrich Hoppe , started again into the Leitertal valley. Four peasants and carpenters (the "Glockners" and two others who are not known) did

1677-640: The mountain's characteristic shape. It may also be a Germanised version of the Alpine Slavic word Klek ("mountain"), as maintained in the Slovene name Veliki Klek . The Grossglockner lies on the border between the Austrian states of Carinthia and Tyrol ( East Tyrol ). The peak is part of the Glocknerkamm ridge in the Glockner Group that branches off the main chain of the Alps at Eiskögele , heading in

1720-616: The ridge between Glöcknerleitl and Ködnitzkees in 1863. Johann Stüdl had a via ferrata erected along the southwestern ridge the next year and the Stüdlhütte erected at its foot in 1868. Already in 1869, most expeditions to the summit started in Kals. The first winter ascent of the Grossglockner was made on January 2, 1875 by William Adolf Baillie Grohman , a member of the Alpine Club . In 1876 Count Pallavicini and his guide Hans Tribusser undertook

1763-471: The summit) and in 1806, however, he himself never climbed beyond the Adlersruhe rock. The climbing of the Grossglockner was also described by the botanist Josef August Schultes , who explored the massif together with Count Apponyi in 1802. No further ascents were made during the Napoleonic Wars , the huts decayed and were plundered by locals. In the following Vormärz era, however, the mountain became

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1806-556: The surveyor Ulrich Schiegg mentioned one Martin Reicher as "Glockner" guide. The peasants and several other members of the expedition (among them Schiegg and his young apprentice Valentin Stanič , who climbed Mt. Watzmann for the first time some weeks later) did the ascent again the next day and finally installed the summit cross and a barometer on the Grossglockner summit. Bishop Salm undertook two more ascents in 1802 (with Hohenwart reaching

1849-526: Was installed on 2 October 1880. Pallavicini also had the Archduke John Hut erected at the former Adlersruhe resting place of Bishop Salm, today the highest situated mountain hut in Austria. The Austrian Alpine Club built the new Salmhütte and the Glocknerhaus along the alpine route from Heiligenblut. A first ascent by skiing was made in 1909 and the circumnavigation of the massif soon became

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