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60-567: The Ponte San Lorenzo was one of four Roman bridges in ancient Padua crossing the Medoacus (modern Bacchiglione). Located in the Via San Francesco , the three-arched bridge is today for the most part framed by surrounding buildings, which have moved closer to the river over the centuries. Only its eastern arch spanning the restricted waterway was largely visible until the middle of the 20th century, when it too disappeared from view as

120-419: A , b , c , d , e {\displaystyle a,b,c,d,e} and diagonals d 1 , d 2 , d 3 , d 4 , d 5 {\displaystyle d_{1},d_{2},d_{3},d_{4},d_{5}} , the following inequality holds: A regular pentagon cannot appear in any tiling of regular polygons. First, to prove a pentagon cannot form

180-470: A regular tiling (one in which all faces are congruent, thus requiring that all the polygons be pentagons), observe that 360° / 108° = 3 1 ⁄ 3 (where 108° Is the interior angle), which is not a whole number; hence there exists no integer number of pentagons sharing a single vertex and leaving no gaps between them. More difficult is proving a pentagon cannot be in any edge-to-edge tiling made by regular polygons: The maximum known packing density of

240-677: A battle against the Sabines the Romans set one of their wooden bridges on fire, driving the enemy back. Other early wooden bridges used post and lintel construction. Pontoon bridges were built by laying boats from side to side across a river. During Julius Caesar 's campaign in Germany , he built bridges by driving wooden piles into the stream bed from floating platforms and fixing beams at right angles across them to create trestles. Trajan built another bridge supported by stone during

300-456: A flat downstream face, though some bridges, such as a bridge in Chester , are exceptions. Two niches carrying cornices were inserted between pilasters . They were then put above the framed starlings. Roman bridges had spandrels , between which images of dolphins were often inserted. They rarely had wide spans and thick piers with bow -shaped piers that used small openings to allow for

360-448: A foundation. At first, they used heavy timbers as deep foundations in the riverbed, but a later technique involved using watertight walls to redirect the water and then laying a stone foundation in the area. To aid in the construction of a foundation, work was exclusively done during the dry season . This ensured as many piers as possible were accessible. There is some evidence that in order to construct bridges rivers were diverted. Such

420-420: A practice might have been performed by Trajan when constructing his Danube bridge. Roman engineers might have diverted rivers using rudimentary methods and tools. Sometimes dirt was added to the foundation. The foundation of a bridge could either be built above or below water level . Building the bridge above water level resulted in a need for a wider span. Bridge's tunnels and spandrels were designed to decrease

480-465: A range of sets of values, thus permitting it to form a family of pentagons. In contrast, the regular pentagon is unique up to similarity, because it is equilateral and it is equiangular (its five angles are equal). A cyclic pentagon is one for which a circle called the circumcircle goes through all five vertices. The regular pentagon is an example of a cyclic pentagon. The area of a cyclic pentagon, whether regular or not, can be expressed as one fourth

540-531: A reconstruction during the reign of Augustus (27 BC – 14 AD). The Pons Fabricius, built in 62 BC during the late Republic, is the oldest Roman bridge that is still intact and in use. The largest Roman bridge was Trajan's Bridge over the lower Danube , constructed by Apollodorus of Damascus , which remained for over a millennium the longest bridge to have been built both in terms of overall and span length. Roman engineers built stone arch or stone pillar bridges over all major rivers of their Imperium , save two:

600-444: A regular pentagon is ( 5 − 5 ) / 3 ≈ 0.921 {\displaystyle (5-{\sqrt {5}})/3\approx 0.921} , achieved by the double lattice packing shown. In a preprint released in 2016, Thomas Hales and Wöden Kusner announced a proof that this double lattice packing of the regular pentagon (known as the "pentagonal ice-ray" Chinese lattice design, dating from around 1900) has

660-533: A span of 1.3 metres (4 ft 3 in). Another bridge over the Bibey River in Galicia has a pier 1 metre (3 ft 3 in) wide, arches with a 4.3-metre (14 ft) span, 6-and-9-metre (20 and 30 ft) side arches, and an arch spanning 18.5 metres (61 ft). Wider spans increase the bridge's drainage, reduce water pressure on the spandrels , and reduced the bridge's weight. Trajan's Bridge over

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720-492: Is called a pentagram . A regular pentagon has Schläfli symbol {5} and interior angles of 108°. A regular pentagon has five lines of reflectional symmetry , and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Given its side length t , {\displaystyle t,} its height H {\displaystyle H} (distance from one side to

780-670: Is less than a semicircle. The Romans built both single spans and lengthy multiple-arch aqueducts , such as the Pont du Gard and Segovia Aqueduct . Their bridges often had flood openings in the piers, e.g. in the Pons Fabricius in Rome (62 BC), one of the world's oldest major bridges still standing. There were two main types of wooden bridge in Britain. Small timber bridges with girders , and large ones made of stone and wood. Throughout

840-497: Is of particular importance in the history of ancient technology for its flattened arches and slender piers. Its three arches span 12.8 m, 14.4 m and 12.5 m, with the span 3.7 times the rise, or, differently put, describing a segment of a circle of 113°. The profile of the structure thus considerably differs from the typical Roman semi-circular bridge arch with its value of 180°. The pier thickness of Roman bridges varies—as far as determined—between one half and one fifth of

900-447: Is the perimeter of the polygon, and r is the inradius (equivalently the apothem ). Substituting the regular pentagon's values for P and r gives the formula with side length t . Similar to every regular convex polygon, the regular convex pentagon has an inscribed circle . The apothem , which is the radius r of the inscribed circle, of a regular pentagon is related to the side length t by Like every regular convex polygon,

960-557: The Dacian Wars . Roman engineers gradually developed new techniques to build bridges, such as oval-shaped bases and pierced bases to facilitate the movement of water. Many bridges would have marble reliefs or carvings , but these bridges were likely used exclusively by government officials because of the difficulty and expense of carving marble artwork. There were three major types of Roman bridges. These were wooden, pontoon, and stone bridges. A list of Roman bridges compiled by

1020-477: The Danube featured open-spandrel segmental arches made of wood (standing on 40 metres (130 ft) high concrete piers). This was to be the longest arch bridge for a thousand years both in terms of overall and individual span length. The longest extant Roman bridge is the 790-metre (2,590 ft) Puente Romano at Mérida . When building bridges across moving bodies of water, Roman engineers would begin by laying

1080-739: The Euphrates , which lay at the frontier to the rival Persian empires , and the Nile , the longest river in the world, which was 'bridged' as late as 1902 by the British Old Aswan Dam . The largest rivers to be spanned by solid bridges by the Romans were the Danube and the Rhine , the two largest European rivers west of the Eurasian Steppe . The lower Danube was crossed by least two ( Trajan's Bridge , Constantine's Bridge ) and

1140-531: The Pons Fabricius , and even after the Fall of the Western Roman Empire , engineers copied their bridges. Roman bridge-building techniques persisted until the 18th century: for example, the prevalence of arches in bridges can be attributed to the Romans. Roman bridges were much larger than the bridges of other civilizations. They could be anywhere from 4.6 to 18.3 metres (15 to 60 ft) long. By

1200-558: The Via San Francesco , and the completely inaccessible Ponte Altinate in the Via Altinate . Both bridges also rest on segmented arches, as does the above-ground Ponte Molino . The fifth Roman bridge in town is the Ponte S. Matteo close to the church of the same name. The Ponte San Lorenzo is 53.30 m long and 8.35 m wide. The date of its construction is fixed by a bridge inscription to between 47 and 30 BC. The bridge

1260-426: The g5 subgroup has no degrees of freedom but can be seen as directed edges . A pentagram or pentangle is a regular star pentagon. Its Schläfli symbol is {5/2}. Its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio . An equilateral pentagon is a polygon with five sides of equal length. However, its five internal angles can take

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1320-757: The voussoir , stronger keystones , vaults , and superior arched bridges. Roman arched bridges were capable of withstanding more stress by dispersing forces across bridges. Many Roman bridges had semicircular arches, but a few were segmental , i.e. with an arc of less than 180 degrees. By the 2nd century BC, the Romans had further refined their bridge-building techniques, using stronger materials such as volcanic ash , lime and gypsum . Also, they began to use iron clamps to hold together bridges, constructing midstream arches, and pentagonal stones to allow for wider vaults . According to Canadian classicist John Peter Oleson , no known stone bridges existed in Italy before

1380-527: The 2nd century BC. This view is not supported unanimously: Spanish engineer Leonardo Fernández Troyano suggested that stone bridges have existed since Pre-Roman Italy . Between 150 and 50 BC, many stone Roman bridges were built, the Pons Aemilius being the first. Engineers began to use stone instead of wood to exemplify the Pax Romana and to construct longer-lasting bridges. These were

1440-608: The Pons Sublicius, the oldest bridge in ancient Rome, and they were probably common across northern Europe and the Tyrrhenian coast ; however, because of their lack of durability few have survived to the modern day. These bridges were supported by wooden trestles spanned by horizontal timbers and reinforced with struts , and they were possibly cantilevered . In order to simplify the process of cutting trees, multiple shorter timbers were used. Wooden poles were driven into

1500-611: The arch as their basic structure , and most used concrete , the first use of this material in bridge-building. Following the conquests of Tarquinius Priscus , Etruscan engineers migrated to Rome, bringing with them their knowledge of bridge-building techniques. The oldest bridge in ancient Rome was the Pons Sublicius . It was built in the 6th century BC by Ancus Marcius over the Tiber River . The Romans improved on Etruscan architectural techniques. They developed

1560-401: The circle at point P , and chord PD is the required side of the inscribed pentagon. To determine the length of this side, the two right triangles DCM and QCM are depicted below the circle. Using Pythagoras' theorem and two sides, the hypotenuse of the larger triangle is found as 5 / 2 {\displaystyle \scriptstyle {\sqrt {5}}/2} . Side h of

1620-461: The circumradius R {\displaystyle R} of a regular pentagon is given, its edge length t {\displaystyle t} is found by the expression and its area is since the area of the circumscribed circle is π R 2 , {\displaystyle \pi R^{2},} the regular pentagon fills approximately 0.7568 of its circumscribed circle. The area of any regular polygon is: where P

1680-499: The concrete. Travertine limestone and tuff were used to build Roman bridges, or they could be made of dry rubble or concrete. Often the building materials varied in smoothness , or rustication . Other bridges were made of bossed limestone combined with cornices, voussoirs and slabs. Sometimes bedrock , buttresses , and vaults were used to construct bridges. Bridges built in Iberia tended to have cylindrical vault geometry. In

1740-490: The construction used in Richmond's method to create the side of the inscribed pentagon. The circle defining the pentagon has unit radius. Its center is located at point C and a midpoint M is marked halfway along its radius. This point is joined to the periphery vertically above the center at point D . Angle CMD is bisected, and the bisector intersects the vertical axis at point Q . A horizontal line through Q intersects

1800-454: The cosine double angle formula . This is the cosine of 72°, which equals ( 5 − 1 ) / 4 {\displaystyle \left({\sqrt {5}}-1\right)/4} as desired. The Carlyle circle was invented as a geometric method to find the roots of a quadratic equation . This methodology leads to a procedure for constructing a regular pentagon. The steps are as follows: Steps 6–8 are equivalent to

1860-505: The distances from the vertices of a regular pentagon to any point on its circumcircle, then The regular pentagon is constructible with compass and straightedge , as 5 is a Fermat prime . A variety of methods are known for constructing a regular pentagon. Some are discussed below. One method to construct a regular pentagon in a given circle is described by Richmond and further discussed in Cromwell's Polyhedra . The top panel shows

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1920-455: The earliest surviving bridge featuring a pointed arch, though it is now submerged by the Keban Dam . Roman arches were unable to properly fit into the arch springings, forcing the base of the arches upwards. In the 2nd century, arches become thinner, and spandrels became flat and pierced with holes. They were constructed using a wooden frame to hold wedge-shaped blocks in place. Afterwards

1980-471: The empire for opus pontis . The Anglo-Saxons continued this practice with bricg-geworc , a literal translation of opus pontis . Built in 142 BC, the Pons Aemilius , later named Ponte Rotto (broken bridge), is the oldest Roman stone bridge in Rome , with only one surviving arch and pier. However, evidence suggests only the abutment is original to the 2nd century BC while the arch and pier perhaps date to

2040-469: The engineer Colin O'Connor features 330 stone bridges for traffic, 34 timber bridges and 54 aqueduct bridges , a substantial number still standing and even used to carry vehicles. A more complete survey by the Italian scholar Vittorio Galliazzo found 931 Roman bridges, mostly of stone, in as many as 26 different countries (including former Yugoslavia ; see right table). A segmental arch is an arch that

2100-416: The first civilization to build large, permanent bridges . Early Roman bridges used techniques introduced by Etruscan immigrants , but the Romans improved those skills, developing and enhancing methods such as arches and keystones . There were three major types of Roman bridge : wooden, pontoon, and stone. Early Roman bridges were wooden, but by the 2nd century BC stone was being used. Stone bridges used

2160-486: The first half of the 2nd century BC, blocks of stone held together with iron clamps were used to aid in the construction of bridges. Although Roman bricks were used to build many bridges, they were far more commonly used to build aqueducts. Bridges built from bricks were rare as bricks often failed to survive erosion . The brick bridges that were built were generally used by the military , and they used construction techniques called opus vittatum and opus mixtum ,

2220-561: The first large-scale bridges ever constructed. Bridges were constructed by the Roman government to serve the needs of the military and the empire's administration. Sometimes roads and bridges were used for commercial purposes, but this was rare as boats better served the needs of the Roman economy . By the 2nd century Roman techniques had declined, and they had been mostly lost by the 4th century. Some Roman bridges are still used today, such as

2280-418: The flow of water. During construction, cranes were used to move materials and lift heavy objects. Some bridges had aprons . They were used to surround piers. Usually, the aprons covered the area of the stream bed near the bridge. Agrippa used ashlar and bricks to cover the outside of bridges and concrete for footings and water channels. Ashlar was used because large amounts of wood was needed to cast

2340-535: The following version, shown in the animation: A regular pentagon is constructible using a compass and straightedge , either by inscribing one in a given circle or constructing one on a given edge. This process was described by Euclid in his Elements circa 300 BC. The regular pentagon has Dih 5 symmetry , order 10. Since 5 is a prime number there is one subgroup with dihedral symmetry: Dih 1 , and 2 cyclic group symmetries: Z 5 , and Z 1 . These 4 symmetries can be seen in 4 distinct symmetries on

2400-528: The foundation of the bridge would be put in this area. Cofferdams were constructed of many piles held together. It is possible the piles were interconnected, likely to improve positioning, waterproofness , or both. Cofferdams would have been sealed with packed clay. The cofferdams also needed to be consistently dry. In order to achieve this, engineers would use tools such as buckets to drain the water. Wooden bridges could be burned to stop an attacker, or dismantled quickly. For example, according to Livy , during

2460-509: The ground, and flat pieces of timber laid across them to create a flat surface. Other early techniques used to build wooden bridges involved barges , sometimes they were moored side by side. Workmen would raise weights, sometimes by rope, then it would fall down onto the piles. This method of construction, called pile driving , was necessary for wooden bridges to properly function. Because this technique created cofferdams , which are enclosures build to pump water out of an area. The base for

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2520-573: The latter alternating rows of bricks in opus reticulatum . Examples are bridges in Carmona , Palomas , Extremadura , and the Ponte della Chianche in Italy. One brick bridge in Ticino , Switzerland , has stone arches and brick spandrels. Bricks were sometimes used to create parts of bridges, such as vaults , piers with welding joints , and brick and mortar rubble . Early Roman bridges were wooden, including one constructed by Apollodorus and

2580-493: The middle and lower Rhine by four different bridges (the Roman Bridge at Mainz , Caesar's Rhine bridges , the Roman Bridge at Koblenz , the Roman Bridge at Cologne ). For rivers with strong currents and to allow swift army movements, pontoon bridges were also routinely employed. Judging by the distinct lack of records of pre-modern solid bridges spanning larger rivers, the Roman feat appears to be unsurpassed anywhere in

2640-399: The opposite vertex), width W {\displaystyle W} (distance between two farthest separated points, which equals the diagonal length D {\displaystyle D} ) and circumradius R {\displaystyle R} are given by: The area of a convex regular pentagon with side length t {\displaystyle t} is given by If

2700-410: The optimal density among all packings of regular pentagons in the plane. There are no combinations of regular polygons with 4 or more meeting at a vertex that contain a pentagon. For combinations with 3, if 3 polygons meet at a vertex and one has an odd number of sides, the other 2 must be congruent. The reason for this is that the polygons that touch the edges of the pentagon must alternate around

2760-441: The pentagon, which is impossible because of the pentagon's odd number of sides. For the pentagon, this results in a polygon whose angles are all (360 − 108) / 2 = 126° . To find the number of sides this polygon has, the result is 360 / (180 − 126) = 6 2 ⁄ 3 , which is not a whole number. Therefore, a pentagon cannot appear in any tiling made by regular polygons. There are 15 classes of pentagons that can monohedrally tile

2820-534: The pentagon. John Conway labels these by a letter and group order. Full symmetry of the regular form is r10 and no symmetry is labeled a1 . The dihedral symmetries are divided depending on whether they pass through vertices ( d for diagonal) or edges ( p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only

2880-627: The regular convex pentagon has a circumscribed circle . For a regular pentagon with successive vertices A, B, C, D, E, if P is any point on the circumcircle between points B and C, then PA + PD = PB + PC + PE. For an arbitrary point in the plane of a regular pentagon with circumradius R {\displaystyle R} , whose distances to the centroid of the regular pentagon and its five vertices are L {\displaystyle L} and d i {\displaystyle d_{i}} respectively, we have If d i {\displaystyle d_{i}} are

2940-586: The remaining canal was filled up to the Riviera del Ponti Romani street. The intact arches of the bridge still exist below street level and can be visited at fixed times by the public. Earthworks in 1773 and 1938, during which parts of the bridge were temporarily excavated, were used for archaeological investigations. Two further Roman bridges in Padua are obstructed from sight, the Ponte Corbo , also located in

3000-516: The responsibility of multiple local municipalities. Their shared costs prove Roman bridges belonged to the region overall, and not to any one town (or two, if on a border). The Alcántara Bridge in Lusitania , for example, was built at the expense of 12 local municipalities, whose names were added on an inscription. Later, in the Roman Empire , the local lords of the land had to pay tithes to

3060-585: The rest of the Roman world, except for northern Europe, arched bridges made of stone were common. This was likely due to the climate and rivers of the regions. Rivers were much calmer and water levels were lower in the southern parts of the Empire. This ensured foundations were easy to construct. While in the northern parts it was much harder to lay down foundations due to the high water level, muddy water, and substantial waterflow. The costs of building and repairing bridges, known as opus pontis ("bridge work"), were

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3120-444: The smaller triangle then is found using the half-angle formula : where cosine and sine of ϕ are known from the larger triangle. The result is: If DP is truly the side of a regular pentagon, m ∠ C D P = 54 ∘ {\displaystyle m\angle \mathrm {CDP} =54^{\circ }} , so DP = 2 cos(54°), QD = DP cos(54°) = 2cos (54°), and CQ = 1 − 2cos (54°), which equals −cos(108°) by

3180-599: The span. Small piers offer less resistance to the water flow, thus reducing the risk of undermined foundations. On the other hand, all piers have to be strong enough to accommodate two arch ribs. The pier thickness of the Ponte San Lorenzo measures only 1.72 m, which corresponds to no more than one eighth of the span of the central opening, a value not to be achieved again until the High Middle Ages . Roman bridge The ancient Romans were

3240-441: The square root of one of the roots of a septic equation whose coefficients are functions of the sides of the pentagon. There exist cyclic pentagons with rational sides and rational area; these are called Robbins pentagons . It has been proven that the diagonals of a Robbins pentagon must be either all rational or all irrational, and it is conjectured that all the diagonals must be rational. For all convex pentagons with sides

3300-473: The time of Augustus around the turn of the 1st millennium the maximum span of Roman bridges increased from around 24 metres (79 ft) in 142 BC to 35 metres (115 ft). The Ponte Sant'Angelo , built during the reign of Hadrian , has five arches each with a span of 18 metres (59 ft). A bridge in Alcántara has piers 1 metre (3 ft 3 in) wide, 47 metres (154 ft) high, and arches with

3360-409: The use of pointed arches . Roman piers were thick enough to support the pressure of an arch. Stone arches allowed bridges to have much longer spans. Usually, iron clamps covered in lead were used to build piers. Because of poor performance underwater, Roman piers were often destroyed over time. Bridges that survived to the modern day were often furnished with cut waters on the upstream side and

3420-790: The weight of the bridge and function as flood arches . The Pons Aemilius probably had stone piers, with wooden roadbeds and arches. They were rebuilt in stone in 142 BC, and either extended from the abutments to the piers , or vice versa. Throughout Roman history, brick or stone arches were used to support bridges' weight. Roman engineers built bridges with one long arch instead of several smaller ones. This practice made construction easier, as they only needed to build one arch on land, instead of many in water. Roman arches were semi-circular and used voussoirs with equal dimensions and conic sections with equal circumference. Later in Roman history arches started to become semi-circular . Sometimes arches were segmented , or not semicircular. This technique

3480-486: The wooden frame was removed, but the weight of the keystone , the last block to be put in place, held it together. Bridges had abutments at each end and piers in the middle, these two design features carrying most of the bridge's weight. Abutments could be constructed in the many arches of a bridge, allowing each to be built separately. Piers were usually twenty-six feet thick and framed with starlings . The late antique Karamagara Bridge represents an early example of

3540-402: The world until into the 19th century. Pentagon In geometry , a pentagon (from Greek πέντε (pente)  'five' and γωνία (gonia)  'angle' ) is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simple or self-intersecting . A self-intersecting regular pentagon (or star pentagon )

3600-508: Was invented by the Romans. Segmented arches allowed greater amounts of flood water to pass, preventing the bridge from being swept away and allowing it to be lighter. The Limyra Bridge in southwestern Turkey has 26 segmental arches with an average span-to-rise ratio of 5.3:1, giving the bridge an unusually flat profile unsurpassed for more than a millennium. The late Roman Karamagara Bridge in Cappadocia in eastern Turkey may represent

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