In geometry, a regular icosahedron is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, the one with the most sides, it has five equilateral triangular faces meeting at each vertex. It is represented by its Schläfli symbol, or sometimes by its vertex figure as 3.3.3.3.3 or 35. It is the dual of the dodecahedron, represented by, having three pentagonal faces around each vertex. A regular icosahedron is a convex deltahedron and a gyroelongated pentagonal bipyramid and a biaugmented pentagonal antiprism in any of six orientations; the name comes from Greek εἴκοσι, meaning'twenty', ἕδρα, meaning'seat'. The plural can be either "icosahedrons" or "icosahedra". If the edge length of a regular icosahedron is a, the radius of a circumscribed sphere is r u = a 2 ϕ 5 = a 4 10 + 2 5 = a sin ⁡ 2 π 5 ≈ 0.951 056 5163 ⋅ a OEIS: A019881and the radius of an inscribed sphere is r i = ϕ 2 a 2 3 = 3 12 a ≈ 0.755 761 3141 ⋅ a OEIS: A179294while the midradius, which touches the middle of each edge, is r m = a ϕ 2 = 1 4 a = a cos ⁡ π 5 ≈ 0.809 016 99 ⋅ a OEIS: A019863where ϕ is the golden ratio.

The surface area A and the volume V of a regular icosahedron of edge length a are: A = 5 3 a 2 ≈ 8.660 254 04 a 2 OEIS: A010527 V = 5 12 a 3 ≈ 2.181 694 99 a 3 OEIS: A102208The latter is F = 20 times the volume of a general tetrahedron with apex at the center of the inscribed sphere, where the volume of the tetrahedron is one third times the base area √3a2/4 times its height ri. The volume filling factor of the circumscribed sphere is: f = V 4 3 π r u 3 = 20 3 2 π ≈ 0.605 461 3829, compared to 66.49% for a dodecahedron. A sphere inscribed in an icosahedron will enclose 89.635% of its volume, compared to only 75.47% for a dodecahedron. The midsphere of an icosahedron will have a volume 1.01664 times the volume of the icosahedron, by far the closest similarity in volume of any platonic solid with its midsphere. This arguably makes the icosahedron the "roundest" of the platonic solids; the vertices of an icosahedron centered at the origin with an edge-length of 2 and a circumradius of ϕ + 2 ≈ 1.9 are described by circular permutations of: where ϕ = 1 + √5/2 is the golden ratio.

Taking all permutations results in the Compound of two icosahedra. Note that these vertices form five sets of three concentric, mutually orthogonal golden rectangles, whose edges form Borromean rings. If the original icosahedron has edge length 1, its dual dodecahedron has edge length √5 − 1/2 = 1/ϕ = ϕ − 1; the 12 edges of a regular octahedron can be subdivided in the golden ratio so that the resulting vertices define a regular icosahedron. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle similarly subdividing each edge into the golden mean along the direction of its vector; the five octahedra defining any given icosahedron form a regular polyhedral compound, while the two icosahedra that can be defined in this way from any given octahedron form a uniform polyhedron compound. The locations of the vertices of a regular icosahedron can be described using spherical coordinates, for instance as latitude and longitude. If two vertices are taken to be at the north and south poles the other ten vertices are at latitude ±arctan ≈ ±26.57°

The 97th Military Police Battalion is a Military Police Battalion of the United States Army based at Fort Riley, Kansas. Activated in Europe during World War II, the unit provided military police support and during the Korean War, the 97th Military Police Battalion conducted internment operations throughout the duration of the conflict. Since the Battalion has honorably served the country in several conflicts to include Vietnam, Operation Iraqi Freedom, most returning in July 2010, Operation Enduring Freedom. On order, the 97th Military Police Battalion deploys worldwide to conduct Military Police operations in support of Forces Command; the battalion conducts continuous Law and Order operations in support of the Fort Riley military community. The battalion is subordinate to 89th Military Police Brigade, it is headquartered at Kansas. Over 800 Soldiers assigned to the 97th MP BN are stationed there; the Battalion consists of four Military Police companies, two Military Police Detachments and one Headquarters and Headquarters Detachment: Headquarters and Headquarters Detachment 116th Military Police Company 287th Military Police Company 300th Military Police Company 977th Military Police Company 73rd Military Police Detachment 523rd Military Police Detachment Activated in Europe on 13 June 1945 in the towards the end of World War II, the unit served in Western Europe during the end of the war before being deactivated on 12 November 1945 in France.

On 28 October 1951, the battalion was reactivated to serve in the Korean War where it operated the Prisoner of War Enclosure Number Nine in support of the UN led Prisoner of War internment operations. The facility secured 21,932 prisoners; the 97th MP BN served in the Korean War until it was deactivated on 20 March 1953. The unit was awarded the Republic of Korea Presidential Unit Citation two times for its contributions; the 97th MP BN was reactivated at Fort Lewis, Washington on 1 June 1966 and deployed to South East Asia. While based at Cam Ranh Bay, their mission was to provide Law and Order as well as Battlefield Circulation Control, it accomplished this with the 630th MP Company, 218th MP Company, 981st MP Company, the 178th MP Detachment. The battalion is credited with completing a 450 mile long convoy of engineers and supplies that started in southern Vietnam and ended in Cambodia; this was the longest convoy attempted during the conflict. The battalion's assets included V-100 Commando Armored Security Vehicles and Military Police gun jeeps mounted with M60 machine guns.

After six years of service in Vietnam, the battalion redeployed to the United States and inactivated at Oakland Army Base, California on 30 April 1972. The battalion was awarded the Meritorious Unit Commendation and the Republic of Vietnam Cross of Gallantry with Palm for its distinctive service in South East Asia. Towards the end of the Cold War, the battalion found itself activated in Mannheim, Germany on 16 September 1989, where its primary missions were military customs and control of the U. S. Confinement Facility in Mannheim, it was subordinate to the 42nd MP Group which reflagged as the 14th MP Brigade. Its customs units were the 193rd MP Company, 256th MP Company, 285th MP Company, 294th MP Company and 560th MP Company; the battalion was inactivated in September 1994. The battalion was activated at Fort Riley, Kansas on 16 October 2005; the battalion commands and controls the 116th MP Company, the 287th MP Company, the 300th MP Company, the 977th MP Company. The BN HQ deployed in September 2006 in support of Operation Iraqi Freedom where it provided command and control to 11 Military Police Companies conducting Police Transition Team missions.

The battalion redeployed in December 2007 after 15 months to reconstitute and prepare for future requirements in support of the Global War on Terrorism. The BN headquarters deployed in July 2009 to Afghanistan in support of the International Security Assistance Force and Operation Enduring Freedom IX-XI, mentoring and partnering with the Afghan National Police in Kandahar City; the battalion was ADCON to 4th BCT, 82nd Airborne Division and NATO, TACON to Task Force Kandahar, commanded by a Canadian General Officer. The battalion's mentorship mission evolved to include responsibilities as the battle space owners for Kandahar City, ISAF's strategic center of gravity, focusing on security and governance; the battalion redeployed in July 2010 to continue its support to the 1st Infantry Division and the Fort Riley community, standing ready for when the nation calls again for the Guardian Battalion to Assist and Defend our country's interests. 97th Military Police Battalion website

William Hodgson Ellis was a Professor of Applied Chemistry from 1878, Dean of the Faculty of Applied Science and Engineering in the University of Toronto from 1914, until his retirement in 1919. William Hodgson Ellis was born on November 23, 1845 in Holme Hall, Derbyshire, England, he was the son of the only daughter of Mr. Joseph Hodgson of Holme Hall and her husband, the resident physician, John Eimeo Ellis. Ellis emigrated with his family to the State of Illinois in 1857 and to Toronto in 1859. Ellis's paternal grandfather was the English missionary and author. Ellis entered University College at the University of Toronto in 1863, receiving his B. A. in 1867 with the Gold Medal in Natural Science. He obtained his M. A. in 1868 and his M. B. in 1870. Following his education in Toronto, he went to London and obtained a position on the house staff of St. Thomas' Hospital, securing his L. R. C. P. in the autumn of 1871, when he returned to Canada. He settled down to practice his profession, but was offered the position of lecturer in chemistry in Trinity College, shortly afterwards he undertook similar duties in the newly-founded School of Technology.

This latter institution became in 1877 The Ontario School of Practical Science in which he was Assistant Professor of Chemistry. In 1887 he became Professor of a position which he held till his retirement. In 1907, when the School of Practical Science became the Faculty of Applied Science and Engineering of the University of Toronto, he was made head of instruction in chemistry for the whole University. After the death of Dean John Galbraith in 1914, he was made Dean of the Faculty of Applied Science, from which position he retired in 1919. In the Medical Faculty, he held the position of Professor of Toxicology and Medical Jurisprudence from 1897 to 1913, when he was obliged to resign owing to the pressure of other work. For more than 30 years he was retained by the Attorney-General of Ontario as analyst and expert toxicologist in connection with criminal cases. During the same period he was Public Analyst for the Inland Revenue Department; the variety and pressing nature of his daily work prevented him from writing any extensive scientific treatises.

In 1915 he was honoured by the University of Toronto with the degree LL. D. and in 1917 a similar honour was conferred upon him by McGill University. In 1917 he was made Honorary Member of the Engineering Institute of Canada, he was a fellow of the Institute of Chemistry, a Fellow of the Royal Society of Canada. He was an ex-president of the Royal Canadian Institute, ex-chairman of the Canadian Section of the Society of Chemical Industry. Ellis died on 23 August 1920 while enjoying a holiday with his friend Dr. Rudolf on Lake Joseph in the Muskoka District. Ellis married Ellen Maude Mickle in Wellington, Canada, on 4 August 1875, their son, the prominent British Canadian physician and Regius Professor of Medicine, Arthur William Mickle Ellis, was born in Toronto, Canada on 4 May 1883. Their daughter, Ethel May Crooks, was born sometime in 1876. Content in this article was copied from William Hodgson Ellis at Skulepedia, licensed under the Creative Commons Attribution-Share Alike 3.0 license. "William Hodgson Ellis".

Proceedings and Transactions of the Royal Society of Canada. 15: V–VII. 1921. Works by William Hodgson Ellis at Project Gutenberg Works by William Hodgson Ellis at Faded Page