SUMMARY / RELATED TOPICS

Doubling the cube

Doubling the cube known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first; as with the related problems of squaring the circle and trisecting the angle, doubling the cube is now known to be impossible using only a compass and straightedge, but in ancient times solutions were known that employed other tools. The Egyptians and the Greeks were aware of the problem and made many futile attempts at solving what they saw as an obstinate but soluble problem. However, the nonexistence of a solution was proven by Pierre Wantzel in 1837. In algebraic terms, doubling a unit cube requires the construction of a line segment of length x, where x3 = 2; this is because a cube of side length 1 has a volume of 13 = 1, a cube of twice that volume has a side length of the cube root of 2. The impossibility of doubling the cube is therefore equivalent to the statement that 3√2 is not a constructible number.

This is a consequence of the fact that the coordinates of a new point constructed by a compass and straightedge are roots of polynomials over the field generated by the coordinates of previous points, of no greater degree than a quadratic. This implies that the degree of the field extension generated by a constructible point must be a power of 2; the field extension generated by 3√2, however, is of degree 3. We begin with the unit line segment defined in the plane. We are required to construct a line segment defined by two points separated by a distance of 3√2, it is shown that compass and straightedge constructions would allow such a line segment to be moved to touch the origin, parallel with the unit line segment - so equivalently we may consider the task of constructing a line segment from to, which entails constructing the point. The tools of a compass and straightedge allow us to create circles centred on one defined point and passing through another, to create lines passing through two defined points.

Any newly defined point either arises as the result of the intersection of two such circles, as the intersection of a circle and a line, or as the intersection of two lines. An exercise of elementary analytic geometry shows that in all three cases, both the x- and y-coordinates of the newly defined point satisfy a polynomial of degree no higher than a quadratic, with coefficients that are additions, subtractions and divisions involving the coordinates of the defined points. Restated in more abstract terminology, the new x- and y-coordinates have minimal polynomials of degree at most 2 over the subfield of ℝ generated by the previous coordinates. Therefore, the degree of the field extension corresponding to each new coordinate is 2 or 1. So, given a coordinate of any constructed point, we may proceed inductively backwards through the x- and y-coordinates of the points in the order that they were defined until we reach the original pair of points and; as every field extension has degree 2 or 1, as the field extension over ℚ of the coordinates of the original pair of points is of degree 1, it follows from the tower rule that the degree of the field extension over ℚ of any coordinate of a constructed point is a power of 2.

Now, p = x3 − 2 = 0 is seen to be irreducible over ℤ – any factorisation would involve a linear factor for some k ∈ ℤ, so k must be a root of p. By Gauss's Lemma, p is irreducible over ℚ, is thus a minimal polynomial over ℚ for 3√2; the field extension ℚ:ℚ is therefore of degree 3. But this is not a power of 2, so by the above, 3√2 is not the coordinate of a constructible point, thus a line segment of 3√2 cannot be constructed, the cube cannot be doubled; the problem owes its name to a story concerning the citizens of Delos, who consulted the oracle at Delphi in order to learn how to defeat a plague sent by Apollo. According to Plutarch it was the citizens of Delos who consulted the oracle at Delphi, seeking a solution for their internal political problems at the time, which had intensified relationships among the citizens; the oracle responded that they must double the size of the altar to Apollo, a regular cube. The answer seemed strange to the Delians and they consulted Plato, able to interpret the oracle as the mathematical problem of doubling the volume of a given cube, thus explaining the oracle as the advice of Apollo for the citizens of Delos to occupy themselves with the study of geometry and mathematics in order to calm down their passions.

According to Plutarch, Plato gave the problem to Eudoxus and Archytas and Menaechmus, who solved the problem using mechanical means, earning a rebuke from Plato for not solving the problem using pure geometry. This may be why the problem is referred to in the 350s BC by the author of the pseudo-Platonic Sisyphus as still unsolved; however another version of the story says that all three found solutions but they were too abstract to be of practical value. A significant development in finding a solution to the problem was the discovery by Hippocrates of Chios that it is equivalent to finding two mean proportionals between a line segment and another with twice the length. In modern notation, this means that given segments of lengths a and 2a, the duplication of the cube is equivalent to finding segments of lengths r and s so that a r

Gawthorpe Hall

Gawthorpe Hall is an Elizabethan country house on the banks of the River Calder, in the civil parish of Ightenhill in the Borough of Burnley, England. Its estate extends with the Stockbridge Drive entrance situated there. Since 1953 it has been designated a grade; the hall is run by the National Trust in partnership with Lancashire County Council. In 2015 the Hall was given £500,000 funding from Lancashire County Council for vital restoration work needed on the south and west sides of the house. Gawthorpe Hall's origins are in a pele tower, a strong fortification built by the Shuttleworths in the 14th century as a defence against invading Scots; the Shuttleworths occupied Shuttleworth Hall near Hapton from the 12th century. The Elizabethan house was dovetailed around the pele tower from plans drawn up by Richard Shuttleworth but carried out after his death by his brother the Reverend Lawrence Shuttleworth; the foundation stone was laid on 26 August 1600. The architect is not recorded, but the house is attributed to Robert Smythson.

In 1604 Richard Stone, from Carr House in Bretherton, imported Irish panel boards and timber and stored 1,000 pieces in the tithe barn at Hoole until they were needed. The mottoes of the Kay-Shuttleworths are Kynd Kynn Knawne Kepe. Mottoes are found around the top of the tower; the initials KS, Kay-Shuttleworth occur in decoration throughout the house, on the front door and plaster roundels on the ceiling in the main dining room. An early occupant was Colonel Richard Shuttleworth. Colonel Shuttleworth was High Sheriff of Lancashire for 1637, Member of Parliament for Preston and commander of the Parliamentarian Army of the Blackburn Hundred during the Civil War. After his death Gawthorpe was leased to tenants, the Shuttleworths preferring to live at Forcett Hall near Richmond. After Forcett was sold the Shuttleworths returned to Gawthorpe. In 1818 barrister, Robert Shuttleworth died and his daughter Janet inherited the estate at an early age, her mother remained at Gawthorpe to protect her inheritance.

In 1842 Janet married Sir James Kay of Rochdale, who adopted the surname Kay-Shuttleworth and commissioned Sir Charles Barry to carry out restoration and improvements to the house in the 1850s. Sir James was made a baronet in 1849 and served as High Sheriff of Lancashire for 1864. Charlotte Brontë, a family friend, visited the house. In 1953 Charles Kay-Shuttleworth, 4th Baron Shuttleworth, left Gawthorpe to live at Leck Hall near Kirkby Lonsdale and in 1970, after the death of Rachel Kay-Shuttleworth, Gawthorpe was gifted to the National Trust; the National Trust described the hall as "an Elizabethan gem in the heart of industrial Lancashire". Nicholas Cooper described the hall's plan as an early example in which the main stair is accessible from the main entrance, a feature that became standard; the hall has a collection of 17th and 18th century portraits on permanent loan from the National Portrait Gallery and is notable for its textiles, collected by the last resident family member Rachel Kay-Shuttleworth, about a fifth of which are on display.

The porch was rebuilt by Sir Charles Barry in 1851 who replaced the round-headed archway over the door with a four-centred arch on columns set on raised plinths and installed a three-light mullioned window above it to create a tile-floored vestibule. A stone plaque displaying the Shuttleworth and Kay-Shuttleworths arms carved by Thomas Hurdeys in 1605 was retained; the Kay motto was inscribed on the Shuttleworth's on the inside. The door's decorative ironwork was designed by Pugin and made by Hardman's of Birmingham in 1851 at a cost of £17 1s 6d; the interior is decorated with a carved stone panel bearing Sir James Kay-Shuttleworth's arms and two ceremonial sheriff's javelins and a black oak sword-chest dated to about 1500. The entrance hall was extended at its east end and reordered when the 17th-century mezzanine bedroom, a low-ceilinged pantry and the buttery were removed in the 1850s; the fireplace's stone over-mantel was used in the vestibule. The fireplace was given a marble surround, incorporating family initials in 1856 and an iron grate with lions-head dampers was supplied in 1852.

A Renaissance-style panelled and arcaded openwork wooden screen was constructed in 1851 by William Horne. Oak panelling was installed framing two internal windows between, a Jacobean panel and above it was a gallery for family portraits. An Edwardian photograph shows the hall with a billiard table, upholstered bobbin-turned chairs, two wicker chairs and a Glastonbury armchair; the entrance hall was converted into a kitchen in 1945. The archway blocked, the screen dismantled, panelling removed and an internal window made into a serving hatch. Only the fireplace and geometrical ceiling were left intact; the room was made into a study. In 1986 the screen was reconstructed, surviving woodwork re-installed and missing pieces re-carved and some stonework was repaired. Portraits from the mid 17th century, include four on loan from the National Portrait Gallery, commemorating Roundheads imprisoned in Windsor Castle. There are portraits of Lord and Lady Derby, of their contemporaries. Furniture includes a hutch cupboard inlaid with holly and bog oak from 1630 on a late 17th-century cupboard, two panel-back carved armchairs and a blanket chest.

An ornate eight-day bracket clock from about 1725 is signed by Louis Mynuel. The 17th-century Great Hall was used for formal dinners, performing plays and dancing and from 1816 became the family dining room, it was refurnish

Flow Country

The Flow Country is a large, rolling expanse of peatland and wetland area of Caithness and Sutherland in the North of Scotland. It is the largest expanse of blanket bog in Europe, covers about 4,000 km2, it is an area of deep peat, dotted with bog pools and a important habitat for wildlife, as well as climate change mitigation. As peat is made up of the remains of plants, which are themselves made up of carbon, it locks up large stores of carbon for thousands of years; this carbon would otherwise contribute to global warming. The Flow Country is being considered as a potential World Heritage Site on account of its unparalleled blanket bog habitat, it could be part of the Global Peatlands Initiative. Named after the Old Norse word'floi' meaning'wet' or'marshy', the Flow Country is home to a rich variety of wildlife, is used as a breeding ground for many different species of birds, including greenshank, dunlin and golden plover. Birds of prey found in the Flow Country include the hen harrier. One of the most prevalent plant species of the Flow Country is sphagnum moss, which can store large amounts of water, form peat - the building block of a blanket bog.

Carnivorous plants such as roundleaved sundew, greater sundew, butterwort feed on the multitude of insects that inhabit the Flow Country. Large mammals such as red deer, the less common, roe deer roam the Flow Country all year round and can be heard roaring during the Autumn rutting season; the Flow Country was badly damaged between 1979 and 1987 through the planting of non-native conifer forests and the cutting of thousands of miles of drains. The trees dried out the peat, changing the habitat and destroying its value for birds and other wildlife; the trees were planted on land bought by Fountain Forestry who recognised that this would be attractive to wealthy investors who could claim planting grants and tax relief against all their other income, at a time of high personal taxation. The ploughing of the bogs and tree planting helped to reduce local unemployment, among the highest in the United Kingdom: however in 1987 the Nature Conservancy Council launched a report in London, critical of the foresters.

The Conservative Party, in government at the time decided to disband the NCC and create a separate Scottish agency now called Scottish Natural Heritage. However, in 1988 Nigel Lawson, The Chancellor of the Exchequer, recognised that a tax break was doing enormous harm to the last real wilderness in the United Kingdom and scrapped the forestry tax reliefs; this halted further planting and encouraged the Forestry Commission to adopt a much broader approach that respects existing landscapes. In an effort to restore the damage, the Royal Society for the Protection of Birds have bought a large area in the centre of the Flow Country and have created the Forsinard Flows national nature reserve. More than 20 km2 has been bought back from Fountain Forestry and the young trees felled and allowed to rot in the plough furrow in the hope and expectation that, in 30 to 100 years, the land will revert to peat bog; the RSPB is a leading partner in the Flows to the Future Project, an ambitious, far-reaching project aiming to restore vast areas of the Flow Country and increase public and visitor awareness of the importance of the Flow Country.

The project funded the iconic. Around 1500 km2 of the Flow Country is protected as both a Special Protection Area and Special Area of Conservation under the name Caithness and Sutherland Peatlands; the Flow Country is on the 2006 UK "tentative list" as a possible UNESCO World Heritage Site. The application was re-listed in 2010 and Stuart Housden, the director of RSPB Scotland and a member of the Rural Development Council, said, "This unspoilt landscape has thousands of hectares of ancient blanket peat land, making this the most important area of its type in the Northern Hemisphere. A global badge would help us to get this special place in better heart; this area was sorely threatened in the 1980s and early 1990s by inappropriate afforestation, but just in time a conservation campaign mounted by the RSPB stopped the worst excesses. Action is now underway to restore the area." The Far North Line connects into Forsinard station serving the area. The Flow Country website RSPB Forsinard Flows Reserve RSPB Flow Country Appeal Peat bogs on SNH