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33-555: 1909-1944 Major-General Lim Bo Seng was born on 27 April 1909 in Nan-an, Fukien, China. He came to Singapore at the age of 16. After studying at Raffles Institution and Hongkong University, he inherited his father's business in Malaya. Since 1937 he became prominent in anti-Japanese activities. When Singapore fell in 1942, he went to Chungking and on instructions from the Chinese government, he joined

66-613: A number of octagonal churches in Norway . The central space in the Aachen Cathedral , the Carolingian Palatine Chapel , has a regular octagonal floorplan. Uses of octagons in churches also include lesser design elements, such as the octagonal apse of Nidaros Cathedral . Architects such as John Andrews have used octagonal floor layouts in buildings for functionally separating office areas from building services, such as in

99-415: A regular octagon is 135 ° ( 3 π 4 {\displaystyle \scriptstyle {\frac {3\pi }{4}}} radians ). The central angle is 45° ( π 4 {\displaystyle \scriptstyle {\frac {\pi }{4}}} radians). The area of a regular octagon of side length a is given by In terms of the circumradius R , the area is In terms of

132-480: A right angle at the centre of the circle which connects its vertices. Its area can thus be computed as the sum of eight isosceles triangles, leading to the result: for an octagon of side a . The coordinates for the vertices of a regular octagon centered at the origin and with side length 2 are: Coxeter states that every zonogon (a 2 m -gon whose opposite sides are parallel and of equal length) can be dissected into m ( m -1)/2 parallelograms. In particular this

165-491: Is r16 and no symmetry is labeled a1 . The most common high symmetry octagons are p8 , an isogonal octagon constructed by four mirrors can alternate long and short edges, and d8 , an isotoxal octagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular octagon. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only

198-408: Is a hexadecagon , {16}. A 3D analog of the octagon can be the rhombicuboctahedron with the triangular faces on it like the replaced edges, if one considers the octagon to be a truncated square. The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°. If squares are constructed all internally or all externally on the sides of an octagon, then

231-596: Is a zig-zag skew octagon and can be seen in the vertices and side edges of a square antiprism with the same D 4d , [2 ,8] symmetry, order 16. The regular skew octagon is the Petrie polygon for these higher-dimensional regular and uniform polytopes , shown in these skew orthogonal projections of in A 7 , B 4 , and D 5 Coxeter planes . The regular octagon has Dih 8 symmetry, order 16. There are three dihedral subgroups: Dih 4 , Dih 2 , and Dih 1 , and four cyclic subgroups : Z 8 , Z 4 , Z 2 , and Z 1 ,

264-446: Is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular octagon , m =4, and it can be divided into 6 rhombs, with one example shown below. This decomposition can be seen as 6 of 24 faces in a Petrie polygon projection plane of the tesseract . The list (sequence A006245 in the OEIS ) defines the number of solutions as eight, by

297-751: The Intelsat Headquarters of Washington or Callam Offices in Canberra. The octagon , as a truncated square , is first in a sequence of truncated hypercubes : As an expanded square, it is also first in a sequence of expanded hypercubes: The Cenotaph, Singapore The Cenotaph is a war memorial located within the Esplanade Park at Connaught Drive , within the Central Area in Singapore 's central business district . The inscription at

330-651: The Supreme Commander of the South East Asia Command to receive the surrender of the Japanese at City Hall on 12 September 1945. On 28 December 2010, The Cenotaph was gazetted by Preservation of Monuments Board as a National Monument along with Lim Bo Seng Memorial and Tan Kim Seng Fountain at the Esplanade Park and the Singapore Conference Hall along Shenton Way . On 23 April 2013,

363-696: The World War II . The war memorial is the only structure in Singapore that commemorates an individual's efforts in World War II and was gazetted as a national monument on 28 December 2010. In 1946, The Lim Bo Seng Memorial Committee ; a group formed by various Chinese clan associations, the United Chinese Library and the Singapore Chinese Chamber of Commerce; was established to raise funds for

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396-419: The apothem r (see also inscribed figure ), the area is These last two coefficients bracket the value of pi , the area of the unit circle . The area can also be expressed as where S is the span of the octagon, or the second-shortest diagonal; and a is the length of one of the sides, or bases. This is easily proven if one takes an octagon, draws a square around the outside (making sure that four of

429-733: The g8 subgroup has no degrees of freedom but can be seen as directed edges . The octagonal shape is used as a design element in architecture. The Dome of the Rock has a characteristic octagonal plan. The Tower of the Winds in Athens is another example of an octagonal structure. The octagonal plan has also been in church architecture such as St. George's Cathedral, Addis Ababa , Basilica of San Vitale (in Ravenna, Italia), Castel del Monte (Apulia, Italia), Florence Baptistery , Zum Friedefürsten Church (Germany) and

462-578: The British Commissioner-General for Southeast Asia Malcolm MacDonald laid the foundation stone for the structure at the ceremony with Lim Bo Seng's widow Gan Choo Neo and her children in present. The memorial was designed by Ng Keng Siang , the first overseas trained Singaporean architect. Inspired by Chinese Nationalist architecture (such as the Sun Yat-sen Mausoleum ) which blends Traditional Chinese motifs with modern materials;

495-543: The Cenotaph was vandalised by someone who sprayed the word "DEMOCRACY" on the monument as well as an "X" which crossed out the text "1914 to 1918". Six days later, Mohamad Khalid Mohamad Yusop was arrested and charged with one count of vandalism under the Vandalism Act . On 26 August 2013, a district court ordered Khalid to pay S$ 208 for the cost of repairs in addition to sentencing him to three months' jail and three strokes of

528-574: The Force 136 in present at the 10th death anniversary ceremony of Major-General Lim Bo Seng. On 29 June 1959 on the 15th anniversary of the death of Major-General Lim Bo Seng, Lim's widow and her eldest son Lim Leong Geok and daughter Lim Oon Geok among others laid wreaths at the Memorial at Esplanade Park and at his memorial tomb at MacRitchie Reservoir . On 28 December 2010, the Lim Bo Seng Memorial

561-479: The Memorial is a 3.6 metres (12 ft) high octagonal pagoda made of white marble with a three-tier bronze roof on a marble and concrete pedestal . guarded by four bronze lions stand guard around the pagoda. The bronze roof and the bronze lions are imported from Hong Kong . Four bronze plaques with an account of Lim's life are installed on the pedestal. There are written in English , Chinese , Tamil and Jawi ;

594-524: The age of 35. He died in the Batu Fajah prison, a martyr to the cause of a liberated Malaya and to his loyalty to his comrades. On 13 January 1946. he was buried with full military honours in Singapore. Erected by The Lim Bo Seng Memorial Committee The Lim Bo Seng Memorial is an octagonal pagoda -like war memorial at Esplanade Park , Singapore . It was erected in 1954 in honour of the late Lim Bo Seng for his heroic acts and selfless sacrifice during

627-465: The base of the Cenotaph reads: They died that we might live . The Cenotaph was built in memory of the 124 British soldiers born or resident in Singapore who gave their lives in World War I (1914–1918), with a second dedication (but no names) added in remembrance of those who died in World War II (1939–1945). The structure was designed by Denis Santry of Swan & Maclaren . The foundation stone

660-509: The eight orientations of this one dissection. These squares and rhombs are used in the Ammann–Beenker tilings . A skew octagon is a skew polygon with eight vertices and edges but not existing on the same plane. The interior of such an octagon is not generally defined. A skew zig-zag octagon has vertices alternating between two parallel planes. A regular skew octagon is vertex-transitive with equal edge lengths. In three dimensions it

693-426: The eight sides overlap with the four sides of the square) and then takes the corner triangles (these are 45–45–90 triangles ) and places them with right angles pointed inward, forming a square. The edges of this square are each the length of the base. Given the length of a side a , the span S is The span, then, is equal to the silver ratio times the side, a. The area is then as above: Expressed in terms of

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726-404: The formulas for their length: A regular octagon at a given circumcircle may be constructed as follows: A regular octagon can be constructed using a straightedge and a compass , as 8 = 2 , a power of two : The regular octagon can be constructed with meccano bars. Twelve bars of size 4, three bars of size 5 and two bars of size 6 are required. Each side of a regular octagon subtends half

759-529: The four official languages of Singapore . The memorial occupies a site measuring 100 feet (30 m) by 80 feet (24 m) near The Padang ; which was donated by the colonial government On 29 June 1954, the completed Memorial was unveiled by Sir Charles Loewen , the Commander-in-Chief of the Far East Land Forces in front of a crowd of people with Lim's widow, her children and some members of

792-402: The last implying no symmetry. On the regular octagon, there are eleven distinct symmetries. John Conway labels full symmetry as r16 . The dihedral symmetries are divided depending on whether they pass through vertices ( d for diagonal) or edges ( p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Full symmetry of the regular form

825-468: The memorial and prepared the proposals of the plan to colonial government , one of which proposed the construction of a memorial park around the late Lim Bo Seng 's grave at MacRitchie Reservoir . The colonial government rejected five of the committee's proposals, and would later grant permission in 1953 for a memorial to be built at the Esplanade based on the sixth proposal. On 3 November 1953 at 5.30 pm,

858-415: The midpoints of the segments connecting the centers of opposite squares form a quadrilateral that is both equidiagonal and orthodiagonal (that is, whose diagonals are equal in length and at right angles to each other). The midpoint octagon of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. If squares are constructed all internally or all externally on

891-412: The regular octagon in terms of the side length a is and the inradius is (that is one-half the silver ratio times the side, a , or one-half the span, S ) The inradius can be calculated from the circumradius as The regular octagon, in terms of the side length a , has three different types of diagonals : The formula for each of them follows from the basic principles of geometry. Here are

924-501: The sides of the midpoint octagon, then the midpoints of the segments connecting the centers of opposite squares themselves form the vertices of a square. A regular octagon is a closed figure with sides of the same length and internal angles of the same size. It has eight lines of reflective symmetry and rotational symmetry of order 8. A regular octagon is represented by the Schläfli symbol {8}. The internal angle at each vertex of

957-519: The span, the area is Another simple formula for the area is More often the span S is known, and the length of the sides, a , is to be determined, as when cutting a square piece of material into a regular octagon. From the above, The two end lengths e on each side (the leg lengths of the triangles (green in the image) truncated from the square), as well as being e = a / 2 , {\displaystyle e=a/{\sqrt {2}},} may be calculated as The circumradius of

990-870: The underground resistance section of 136 Force under the Supreme Allied Command, South-East Asia, on 2 November 1943 as Commanding Officer of the Malayan Chinese section. He landed from a submarine at Bagan Bator in Perak to join the British and Chinese officers already working in Malaya, with anti-Japanese forces. Later he left the jungle to work in Ipoh, but he was discovered and arrested by the Japanese Military Police on 27 March 1944. He manfully endured repeated tortures to which he ultimately succumbed on 29 June 1944, at

1023-506: The unveiling ceremony , a chaplain blessed the Cenotaph with the words, " The stone is well laid and truly laid to the Glory of God and the memory of the illustrious dead ." Against the backdrop of the sea then fronting Queen Elizabeth Walk , Governor Guillemard awarded medals of courage to those who had served in the war. In Prince Edward's entourage was Louis Mountbatten . At the end of World War II, Mountbatten returned to Singapore as

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1056-540: Was gazetted by the Preservation of Monuments Board as a National Monument along with The Cenotaph and the Tan Kim Seng Fountain at the Esplanade Park and the Singapore Conference Hall along Shenton Way . Octagonal A regular octagon has Schläfli symbol {8} and can also be constructed as a quasiregular truncated square , t{4}, which alternates two types of edges. A truncated octagon, t{8}

1089-603: Was laid by Sir Lawrence Nunns Guillemard , the Governor of the Straits Settlements , on 15 November 1920. In attendance was the visiting French Premier , Georges Clemenceau who was the French Minister of War from 1917 to 1919. The memorial was completed in 1922, and was unveiled on 31 March that year by the young Prince Edward of Wales , later King Edward VIII then Duke of Windsor, during his Asia-Pacific tour. During

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