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pyramid,in geometry, solid figure bounded by a polygonpolygon,
closed plane figure bounded by straight line segments as sides. A polygon is convex if any two points inside the polygon can be connected by a line segment that does not intersect any side. If a side is intersected, the polygon is called concave.
..... Click the link for more information. (the base, or directrix) and the surface generated by a moving line (the generator) passing through a fixed point (vertex) and continually intersecting the perimeter of the polygon. The surface, or lateral faces, of the pyramid are triangles having as a common vertex the vertex of the pyramid; in a regular pyramid the base is a regular polygon and the lateral faces are congruent triangles. The altitude of a pyramid is the perpendicular distance from the vertex to the base. The volume of a pyramid is equal to one third the product of the altitude and the area of the base. The frustum is the portion of a pyramid between the base and a plane parallel to the base cutting the pyramid into two parts.
pyramid.The true pyramid exists only in Egypt, though the term has also been applied to similar structures in other countries. Egyptian pyramids are square in plan and their triangular sides, which directly face the points of the compass, slope upwards at approximately a 50° angle from the ground and meet at an apex. The prototype for the pyramid are the mastabasmastaba
, in Egyptian architecture, a sepulchral structure built aboveground. The mastabas of the early dynastic period (3200–2680 B.C.), such as those of the I dynasty at Sakkara, were elaborate, having many storage or offering compartments, and were quite evidently close
..... Click the link for more information. of the Old Kingdom (2680–2565 B.C.), which are rectangular in plan and have only two sloping sides. After these came the step-pyramid at SakkaraSakkara
, necropolis (burial place) of ancient Memphis, Egypt, 3 mi (5 km) from the Nile and on the border of the Libyan desert. Djoser (Zoser) had his famous step-pyramid, a precursor of the familiar straight-sided pyramid, built there in the III dynasty,
..... Click the link for more information. , built c.2620 B.C., which soon evolved into the straight-sided true pyramid. This monumental structure was developed around the IV dynasty and continued to be the favored form for royal burial through the VI dynasty.
Each monarch built his own pyramid in which his mummified body might be preserved for eternity away from human view and sacrilege. As a result of the lack of sophisticated machinery, the construction of each pyramid took many years and required measureless amounts of building materials and labor. Entrance into a pyramid is through an opening in the northern wall. A small passage, traversing lesser chambers, leads to the sepulchral room deep beneath the surface. Stone blocks forming a gable divert the weight of the great masonry masses over these chambers. Though the pyramids were usually built of rough stone blocks laid up in horizontal courses, many were constructed of mud bricks with a stone casing.
The three pyramids of Giza near Cairo, all of the IV dynasty, are the largest and finest of their kind. The Great Pyramid of KhufuKhufu
, fl. c.2680 B.C., king of ancient Egypt, founder of the IV dynasty. He was king for 23 years and was famous as the builder of the greatest pyramid at Giza.
..... Click the link for more information. or Cheops (begun c.2680 B.C.) was designated one of the Seven Wonders of the WorldSeven Wonders of the World,
in ancient classifications, were the Great Pyramid of Khufu (see pyramid) or all the pyramids with or without the sphinx; the Hanging Gardens of Babylon, with or without the walls; the mausoleum at Halicarnassus; the Artemision at Ephesus; the
..... Click the link for more information. and is the largest pyramid ever built. A solid mass of limestone blocks covering 13 acres (5.3 hectares), it was originally 756 ft (230 m) along each side of its base and 482 ft (147 m) high. It has several passages, two large chambers in addition to one beneath the ground level, and two small air chambers for ventilation.
Although not true pyramids, pyramidical structures were also built by the Mesopotamians and by the Maya of Mexico and Central America. Mesopotamian zigguratziggurat
, form of temple common to the Sumerians, Babylonians and Assyrians. The earliest examples date from the end of the 3d millenium B.C., the latest from the 6th cent. B.C.
..... Click the link for more information. was square in plan and built up in receding terraces. Mayan pyramids, built in steep, receding blocks, also were topped by ritual chambers, and in some cases, possessed an interior crypt. Stepped funeral pyramids dating from the 4th cent. B.C. were discovered in the 1990s in the Altai region of Siberia. The Romans built small pyramidical tombs of which the most famous was the Pyramid of Cestius (62 B.C.–12 B.C.) in Rome. Built of concrete faced with marble, it has an interior tomb vault and is 116 ft (35 m) high. Many modern architects have admired pyramids for their pure geometry. In the reconstruction of the LouvreLouvre
, foremost French museum of art, located in Paris. The building was a royal fortress and palace built by Philip II in the late 12th cent. In 1546 Pierre Lescot was commissioned by Francis I to erect a new building on the site of the Louvre.
..... Click the link for more information. in Paris, architect I. M. Pei added a pyramidal entrance pavilion (1987–89).
See I. E. S. Edwards, The Pyramids of Egypt (rev. ed. 1961); P. Tompkins, Secrets of the Great Pyramid (1971); K. Mendelssohn, The Riddle of the Pyramids (1974).
A Christmas pyramid is a triangular or pyramidal structure made up of shelves of unequal lengths joined along their outside edges by supporting posts or poles. Christmas decorations are displayed on each shelf, with the lowest and longest shelf often reserved for a Nativity scene. Family and friends may arrange apples, cookies, nuts, small gifts, evergreen branches, Christmas cards, stars, figurines, candles, flags, and other embellishments across the other shelves according to their taste. A star or pinecone often adorns the apex of the pyramid. In one variation of the pyramid popular in central Europe several centuries ago, a propeller sits atop a pyramid shaped like a tall, round, layer cake. A central axis pole supporting the propeller runs through each of three circular shelves. Rising heat currents from the candles on the shelves below cause the propeller to spin, which in turn causes the axis to spin and the layers of the pyramid to rotate.
Several authors view the candles as the most important ornaments on the pyramid and suggest that the decorated pyramid serves as an elaborate candlestick. Indeed, one German name for this structure, Lichtstock, means "light stick." Some authorities maintain, however, that the Lichtstock was a simple pole covered with evergreens bearing a single candle. They offer Weihnachtspyramide as the German term for the Christmas pyramid. The Italians call the pyramid a ceppo, which means "log." Some explain this odd name by noting that the ceppo, with its glowing candles, replaced the burning of the Yule log in Italy.
The Christmas pyramid originated in Germany and became a popular Christmas tradition by the seventeenth century. In early times, the pyramid was hung from the ceiling. Families garnished their pyramids with candles and figurines, for example, of soldiers and angels. Along with the paradise tree, the pyramid stands as a possible ancestor to the modern Christmas tree.
From Germany the use of pyramids spread to central Europe, Italy, and England. German settlers brought the custom to America. As early as 1747 Moravian communities in Pennsylvania were celebrating Christmas with decorated pyramids. By contrast, the first American Christmas tree dates only as far back as the early 1800s (see also America, Christmas in Nineteenth-Century; Bethlehem, Pennsyvania, Christmas in).
In Germany the Christmas tree began to replace the pyramid in the seventeenth and eighteenth centuries. The exploding popularity of the Christmas tree in the nineteenth century contributed to the declining use of the Christmas pyramid in many countries. The Italians maintained the tradition of the Christmas ceppo, perhaps because they never adopted the Christmas tree.
In Erzgebirge, a region of Germany famous for its mining industry, miners began carving fancy wooden pyramids in the nineteenth century. The miners had already developed a tradition of carving wooden candlesticks in the shape of miners and angels. The miners represented the men of the region, while the angels represented the women. Families placed groupings of these candlesticks in their windows at Christmas time, displaying one miner for every boy child in the family and one angel for every girl child. Similar wooden figurines eventually began to populate the shelves of their Christmas pyramids. Miners, Christmas trees, and scenes from the Nativity story, whirled round and round on the propeller-topped shelves. Today villages in Erzgebirge build large, motorized community pyramids, vying with one another to see which locale produces the most impressive display.
Del Re, Gerard, and Patricia Del Re. The Christmas Almanack. Garden City, N.Y.: Doubleday, 1979. Foley, Daniel J. The Christmas Tree. Philadelphia, Pa.: Chilton Company, 1960. Russ, Jennifer M. German Festivals and Customs. London, England: Oswald Wolff, 1982. Sterbenz, Carol Endler, and Nancy Johnson. The Decorated Tree. New York: Harry N. Abrams, 1982.
The German Embassy in Washington, D.C., offers a page describing the Christmas pyramid on its web site at: ch_15.html
a polyhedron with one face a polygon and the other faces triangles with a common vertex. The polygon, which may also be a triangle, is called the base, the remaining faces are called lateral faces, and the common vertex is called the vertex of the pyramid. Examples of pyramids are illustrated in Figure 1.
Pyramids are classified as triangular, quadrangular, and so on according to the number of their lateral faces. The line segment drawn perpendicularly from the vertex of the pyramid to the plane of the base is called the altitude of the pyramid, as is the length of the segment. The volume of a pyramid is given by the formula
where B is the area of the base and h is the altitude. A pyramid is said to be regular (see Figure l,b) if its base is a regular polygon and its altitude passes through the center of the base. The lateral faces of a regular pyramid are congruent isosceles triangles. The altitude of each of these triangles is called a regular pyramid’s slant height, or apothem (the projection of the slant height on the plane of the base is the apothem of the base). If a pyramid is cut into two parts by a plane parallel to its base, there result a pyramid similar to the original pyramid and a truncated pyramid.
a massive architectural structure with a square base and sloping sides meeting at an apex. Pyramids, which may be stepped or regular, are characteristic of ancient times. Pyramids were the tombs of the Egyptian pharaohs of the Old and Middle Kingdoms (from roughly 2800 to 1700 B.C.). The structures symbolized the superhuman greatness of the ruler. The largest Egyptian pyramid, that of Cheops at Gizeh, is 146.6 m high and was constructed in the 28th century B.C. In Central and South America, pyramidal structures, which often served as foundations for temples and were associated with cosmologic cults, were erected in the first millennium B.C. In ancient Roman and later in European art, the pyramidal motif was often used in memorial structures.
REFERENCESLauer, J. P. Zagadki egipetskikh piramid. Moscow, 1966. (Translated from French.)
Kink, Kh. A. Kak stroilis’ egipetskie piramidy. Moscow, 1967.
Mikhailovskii, K. Piramidy i maslaby. Warsaw, 1973.