Ringworld
Ringworld is a 1970 science fiction novel by Larry Niven, set in his Known Space universe and considered a classic of science fiction literature. Ringworld tells the story of Louis Wu and his companions on a mission to the Ringworld, a massive alien construct in space 186 million miles in diameter. Niven added three sequels and cowrote four prequels and a final sequel with Edward M. Lerner, the Fleet of Worlds series; the novels tie into numerous other books set in Known Space. Ringworld won the Nebula Award in 1970, as well as both the Hugo Award and Locus Award in 1971. On planet Earth in 2850 AD, Louis Gridley Wu is celebrating his 200th birthday. Despite his age, Louis is in perfect physical condition, he has once again become bored with human society and is thinking about taking one of his periodic sabbaticals, alone in a spaceship far away from other people. He meets a Pierson's puppeteer, who offers him a mysterious job. Intrigued, Louis accepts. Speaker-to-Animals, a Kzin, Teela Brown, a young human woman who becomes Louis' lover join the crew.
They first go to the puppeteer home world, where they learn that the expedition's goal is to investigate the Ringworld, a gigantic artificial ring, to see if it poses any threat. The Ringworld is about one million miles wide and the diameter of Earth's orbit, encircling a sunlike star, it rotates to provide artificial gravity 99.2% as strong as Earth's from centrifugal force. The Ringworld has a habitable, flat inner surface, a breathable atmosphere and a temperature optimal for humans. Night is provided by an inner ring of shadow squares which are connected to each other by thin, ultra-strong wire; when the crew completes their mission, they will be given the starship in which they travelled to the puppeteer home world. When they reach the vicinity of the Ringworld, they are unable to contact anyone, their ship, the Lying Bastard, is disabled by the Ringworld's automated meteoroid-defense system; the damaged vessel collides with a strand of shadow-square wire and crash-lands near a huge mountain, "Fist-of-God".
Although many of the ship's systems survive intact the normal drive is destroyed leaving them unable to launch back into space where they could use the undamaged faster-than-light hyperdrive to return home. They set out to find a way to get the Lying Bastard off of the Ringworld. Using their flycycles, they try to reach the rim of the ring, where they hope to find some technology that will help them, it will take them months to cross the vast distance. When Teela develops "Plateau trance", they are forced to land. On the ground, they encounter primitive human natives who live in the crumbling ruins of a once-advanced city and think that the crew are the engineers who created the ring, whom they revere as gods; the crew is attacked. They continue their journey, during which Nessus reveals some Puppeteer secrets: they have conducted experiments on both humans and Kzinti. Speaker's outrage forces Nessus to follow them from a safe distance. In a floating building over the ruins of a city, they find a map of the Ringworld and videos of its past civilization.
While flying through a giant storm, caused by air escaping through a hole in the Ring floor due to a meteoroid impact, Teela becomes separated from the others. While Louis and Speaker search for her, their flycycles are caught by an automatic police trap designed to catch traffic offenders, they are trapped in the basement of a floating police station. Nessus enters the station to try to help them. In the station, they meet Halrloprillalar Hotrufan, a former crew member of a spaceship used for trade between the Ringworld and other inhabited worlds; when her ship returned to the Ringworld the last time, they found. The crew managed to enter the Ringworld, but some of them were killed and others suffered brain damage when the device that let them pass through the Ringworld floor failed. From her account, Louis surmises that a mold was brought back from one of the other planets by a spaceship like Prill's. Teela reaches the police station, accompanied by her new lover, a native "hero" called Seeker who helped her survive.
Based on an insight gained from studying an ancient Ringworld map, Louis comes up with a plan to get home. Teela, chooses to remain on the Ringworld with Seeker. Louis skeptical about breeding for luck, now wonders if the entire mission was caused by Teela's luck, to unite her with her true love and help her mature; the party collects one end of the shadow-square wire, snapped when the ship crashed. They travel back to their crashed ship in the floating police station, dragging the wire behind them. Louis threads it through the ship to tether it to the police station, he takes the police station up to the summit of "Fist-of-God", the enormous mountain near their crash site. The mountain had not appeared on the Ringworld map, leading Louis to conclude that it is in fact the result of a meteoroid impact with the underside of the ring, which pushed the "mountain" up from the ring's floor
Pentagon
In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be self-intersecting. A self-intersecting regular pentagon is called a pentagram. A regular pentagon has Schläfli symbol and interior angles are 108°. A regular pentagon has five lines of reflectional symmetry, rotational symmetry of order 5; the diagonals of a convex regular pentagon are in the golden ratio to its sides. Its height and width are given by Height = 5 + 2 5 2 ⋅ Side ≈ 1.539 ⋅ Side, Width = Diagonal = 1 + 5 2 ⋅ Side ≈ 1.618 ⋅ Side, Diagonal = R 5 + 5 2 = 2 R cos 18 ∘ = 2 R cos π 10 ≈ 1.902 R, where R is the radius of the circumcircle. The area of a convex regular pentagon with side length t is given by A = t 2 25 + 10 5 4 = 5 t 2 tan 4 ≈ 1.720 t 2. A pentagram or pentangle is a regular star pentagon, its Schläfli symbol is. Its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio.
When a regular pentagon is circumscribed by a circle with radius R, its edge length t is given by the expression t = R 5 − 5 2 = 2 R sin 36 ∘ = 2 R sin π 5 ≈ 1.176 R, its area is A = 5 R 2 4 5 + 5 2. The area of any regular polygon is: A = 1 2 P r where P is the perimeter of the polygon, r is the inradius. Substituting the regular pentagon's values for P and r gives the formula A = 1 2 ⋅ 5 t ⋅ t tan 2 = 5 t 2 tan 4 with side length “f” Like every regular convex polygon, the regular convex pentagon has an inscribed circle; the apothem, the radius r of the inscribed circle, of a regular pentagon is related to the side length t by r = t 2 tan = t 2 5 − 20 ≈ 0.6882 ⋅ t. Like every regular convex polygon, 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 PA + PD = PB + PC + PE; the regular pentagon is constructible with compass and straightedge. 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 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
Equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, an equilateral triangle is equiangular, it is a regular polygon, so it is referred to as a regular triangle. Denoting the common length of the sides of the equilateral triangle as a, we can determine using the Pythagorean theorem that: The area is A = 3 4 a 2 The perimeter is p = 3 a The radius of the circumscribed circle is R = a 3 The radius of the inscribed circle is r = 3 6 a or r = R 2 The geometric center of the triangle is the center of the circumscribed and inscribed circles The altitude from any side is h = 3 2 a Denoting the radius of the circumscribed circle as R, we can determine using trigonometry that: The area of the triangle is A = 3 3 4 R 2 Many of these quantities have simple relationships to the altitude of each vertex from the opposite side: The area is A = h 2 3 The height of the center from each side, or apothem, is h 3 The radius of the circle circumscribing the three vertices is R = 2 h 3 The radius of the inscribed circle is r = h 3 In an equilateral triangle, the altitudes, the angle bisectors, the perpendicular bisectors, the medians to each side coincide.
A triangle ABC that has the sides a, b, c, semiperimeter s, area T, exradii ra, rb, rc, where R and r are the radii of the circumcircle and incircle is equilateral if and only if any one of the statements in the following nine categories is true. Thus these are properties that are unique to equilateral triangles, knowing that any one of them is true directly implies that we have an equilateral triangle. A = b = c 1 a + 1 b + 1 c = 25 R r − 2 r 2 4 R r s = 2 R + r s 2 = 3 r 2 + 12 R r s 2 = 3 3 T s = 3 3 r s = 3 3 2 R A = B = C = 60 ∘ cos A + cos B + cos C = 3 2 sin A 2 sin B 2 sin C 2 = 1 8 T = a 2 + b 2 + c 2 4 3 T = 3 4 2 3 R = 2 r 9 R 2 = a 2 + b 2 + c 2 r = r a +
Polygon
In elementary geometry, a polygon is a plane figure, described by a finite number of straight line segments connected to form a closed polygonal chain or polygonal circuit. The solid plane region, the bounding circuit, or the two together, may be called a polygon; the segments of a polygonal circuit are called its edges or sides, the points where two edges meet are the polygon's vertices or corners. The interior of a solid polygon is sometimes called its body. An n-gon is a polygon with n sides. A simple polygon is one. Mathematicians are concerned only with the bounding polygonal chains of simple polygons and they define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. There are many more generalizations of polygons defined for different purposes; the word polygon derives from the Greek adjective πολύς "much", "many" and γωνία "corner" or "angle".
It has been suggested. Polygons are classified by the number of sides. See the table below. Polygons may be characterized by their convexity or type of non-convexity: Convex: any line drawn through the polygon meets its boundary twice; as a consequence, all its interior angles are less than 180°. Equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. Non-convex: a line may be found which meets its boundary more than twice. Equivalently, there exists a line segment between two boundary points that passes outside the polygon. Simple: the boundary of the polygon does not cross itself. All convex polygons are simple. Concave. Non-convex and simple. There is at least one interior angle greater than 180°. Star-shaped: the whole interior is visible from at least one point, without crossing any edge; the polygon must be simple, may be convex or concave. All convex polygons are star-shaped. Self-intersecting: the boundary of the polygon crosses itself.
The term complex is sometimes used in contrast to simple, but this usage risks confusion with the idea of a complex polygon as one which exists in the complex Hilbert plane consisting of two complex dimensions. Star polygon: a polygon which self-intersects in a regular way. A polygon can not be both star-shaped. Equiangular: all corner angles are equal. Cyclic: all corners lie on a single circle, called the circumcircle. Isogonal or vertex-transitive: all corners lie within the same symmetry orbit; the polygon is cyclic and equiangular. Equilateral: all edges are of the same length; the polygon need not be convex. Tangential: all sides are tangent to an inscribed circle. Isotoxal or edge-transitive: all sides lie within the same symmetry orbit; the polygon is equilateral and tangential. Regular: the polygon is both isogonal and isotoxal. Equivalently, it is both equilateral, or both equilateral and equiangular. A non-convex regular polygon is called a regular star polygon. Rectilinear: the polygon's sides meet at right angles, i.e. all its interior angles are 90 or 270 degrees.
Monotone with respect to a given line L: every line orthogonal to L intersects the polygon not more than twice. Euclidean geometry is assumed throughout. Any polygon has as many corners; each corner has several angles. The two most important ones are: Interior angle – The sum of the interior angles of a simple n-gon is π radians or × 180 degrees; this is because any simple n-gon can be considered to be made up of triangles, each of which has an angle sum of π radians or 180 degrees. The measure of any interior angle of a convex regular n-gon is 180 − 360 n degrees; the interior angles of regular star polygons were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra: for a regular p q -gon, each interior angle is π p radians or 180 p degrees. Exterior angle – The exterior angle is the supplementary angle to the interior angle. Tracing around a convex n-gon, the angle "turned" at a corner is external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360°.
This argument can be generalized to concave simple polygons, if external angles that turn in the opposite direction are subtracted from the total turned. Tracing around an n-gon in general, the sum of the exterior angles can be any integer multiple d of 360°, e.g. 720° for a pentagram and 0° for an angular "eight" or antiparallelogram, where d is the density or starriness of the polygon. See orbit. In this section, the vertices of the polygon under consideration are taken to be, ( x 1
Hexagon
In geometry, a hexagon is a six-sided polygon or 6-gon. The total of the internal angles of any simple hexagon is 720°. A regular hexagon has Schläfli symbol and can be constructed as a truncated equilateral triangle, t, which alternates two types of edges. A regular hexagon is defined as a hexagon, both equilateral and equiangular, it is bicentric, meaning that it is both tangential. The common length of the sides equals the radius of the circumscribed circle, which equals 2 3 times the apothem. All internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, making up the dihedral group D6; the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a triangle with a vertex at the center of the regular hexagon and sharing one side with the hexagon is equilateral, that the regular hexagon can be partitioned into six equilateral triangles. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, so are useful for constructing tessellations.
The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a regular triangular lattice is the honeycomb tessellation of hexagons, it is not considered a triambus, although it is equilateral. The maximal diameter, D, is twice the maximal radius or circumradius, R, which equals the side length, t; the minimal diameter or the diameter of the inscribed circle, d, is twice the minimal radius or inradius, r. The maxima and minima are related by the same factor: 1 2 d = r = cos R = 3 2 R = 3 2 t and d = 3 2 D; the area of a regular hexagon A = 3 3 2 R 2 = 3 R r = 2 3 r 2 = 3 3 8 D 2 = 3 4 D d = 3 2 d 2 ≈ 2.598 R 2 ≈ 3.464 r 2 ≈ 0.6495 D 2 ≈ 0.866 d 2. For any regular polygon, the area can be expressed in terms of the apothem a and the perimeter p. For the regular hexagon these are given by a = r, p = 6 R = 4 r 3, so A = a p 2 = r ⋅ 4 r 3 2 = 2 r 2 3 ≈ 3.464 r 2. The regular hexagon fills the fraction 3 3 2 π ≈ 0.8270 of its circumscribed circle.
If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumscribing circle between B and C PE + PF = PA + PB + PC + PD. The regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups: Dih3, Dih2, Dih1, 4 cyclic subgroups: Z6, Z3, Z2, Z1; these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a group order. R12 is full symmetry, a1 is no symmetry. P6, an isogonal hexagon constructed by three mirrors can alternate long and short edges, d6, an isotoxal hexagon constructed with equal edge lengths, but vertices alternating two different internal angles; these two forms have half the symmetry order of the regular hexagon. The
Larry Niven
Laurence van Cott Niven is an American science fiction writer. His best-known work is Ringworld, which received Hugo, Locus and Nebula awards; the Science Fiction and Fantasy Writers of America named him the 2015 recipient of the Damon Knight Memorial Grand Master Award. His work is hard science fiction, using big science concepts and theoretical physics, it often includes elements of detective fiction and adventure stories. His fantasy includes the series The Magic Goes Away, rational fantasy dealing with magic as a non-renewable resource. Niven was born in Los Angeles, he is a great-grandson of Edward L. Doheny, an oil tycoon who drilled the first successful well in the Los Angeles City Oil Field in 1892 and was subsequently implicated in the Teapot Dome scandal, he attended the California Institute of Technology and graduated with a Bachelor of Arts in mathematics from Washburn University in Topeka, Kansas in 1962. He completed a year of graduate work in mathematics at the University of California, Los Angeles.
On September 6, 1969, he married Marilyn Joyce "Fuzzy Pink" Wisowaty, a science fiction and Regency literature fan. He is an agnostic. Niven is the author of numerous science fiction short stories and novels, beginning with his 1964 story "The Coldest Place". In this story, the coldest place concerned is the dark side of Mercury, which at the time the story was written was thought to be tidally locked with the Sun. Algis Budrys said in 1968 that Niven becoming a top writer despite the New Wave was evidence that "trends are for second-raters". In addition to the Nebula award in 1970 and the Hugo and Locus awards in 1971 for Ringworld, Niven won the Hugo Award for Best Short Story for "Neutron Star" in 1967, he won the same award in 1972, for "Inconstant Moon", in 1975 for "The Hole Man". In 1976, he won the Hugo Award for Best Novelette for "The Borderland of Sol". Niven has written scripts for three science fiction television series: the original Land of the Lost series. Niven has written for the DC Comics character Green Lantern including in his stories hard science fiction concepts such as universal entropy and the redshift effect.
He has included limited psi gifts in some characters in his stories. Several of his stories predicted the black market in transplant organs. Many of Niven's stories—sometimes called the Tales of Known Space—take place in his Known Space universe, in which humanity shares the several habitable star systems nearest to the Sun with over a dozen alien species, including the aggressive feline Kzinti and the intelligent but cowardly Pierson's Puppeteers, which are central characters; the Ringworld series is part of the Tales of Known Space, Niven has shared the setting with other writers since a 1988 anthology, The Man-Kzin Wars. There have been several volumes of short novellas. Niven has written a logical fantasy series The Magic Goes Away, which utilizes an exhaustible resource called mana to power a rule-based "technological" magic; the Draco Tavern series of short stories take place in a more light-hearted science fiction universe, are told from the point of view of the proprietor of an omni-species bar.
The whimsical Svetz series consists of a collection of short stories, The Flight of the Horse, a novel, Rainbow Mars, which involve a nominal time machine sent back to retrieve long-extinct animals, but which travels, in fact, into alternative realities and brings back mythical creatures such as a Roc and a Unicorn. Much of his writing since the 1970s has been in collaboration with Jerry Pournelle and Steven Barnes, but Brenda Cooper and Edward M. Lerner. One of Niven's best known humorous works is "Man of Steel, Woman of Kleenex", in which he uses real-world physics to underline the difficulties of Superman and a human woman mating. Niven appeared in the 1980 science documentary film Target... Earth? Niven's most famous contribution to the SF genre comes from his novel Ringworld, in which he envisions a Ringworld: a band of material a million miles wide, of the same diameter as Earth's orbit, rotating around a star; the idea's genesis came from Niven's attempts to imagine a more efficient version of a Dyson sphere, which could produce the effect of surface gravity through rotation.
Given that spinning a Dyson Sphere would result in the atmosphere pooling around the equator, the Ringworld removes all the extraneous parts of the structure, leaving a spinning band landscaped on the sun-facing side, with the atmosphere and inhabitants kept in place through centrifugal force and 1,000 mi high perimeter walls. After publication of Ringworld, Dan Alderson and Ctein, two fannish friends of Niven, analyzed the structure and told Niven that the Ringworld was dynamically unstable such that if the center of rotation drifts away from the central sun, gravitational forces will not're-center' it, thus allowing the ring to contact the sun and be destroyed. Niven used this as a core plot element in the sequel novel, The Ringworld Engineers
Johannes Kepler
Johannes Kepler was a German astronomer and astrologer. He is a key figure in the 17th-century scientific revolution, best known for his laws of planetary motion, his books Astronomia nova, Harmonices Mundi, Epitome Astronomiae Copernicanae; these works provided one of the foundations for Newton's theory of universal gravitation. Kepler was a mathematics teacher at a seminary school in Graz, where he became an associate of Prince Hans Ulrich von Eggenberg, he became an assistant to the astronomer Tycho Brahe in Prague, the imperial mathematician to Emperor Rudolf II and his two successors Matthias and Ferdinand II. He taught mathematics in Linz, was an adviser to General Wallenstein. Additionally, he did fundamental work in the field of optics, invented an improved version of the refracting telescope, was mentioned in the telescopic discoveries of his contemporary Galileo Galilei, he was a corresponding member of the Accademia dei Lincei in Rome. Kepler lived in an era when there was no clear distinction between astronomy and astrology, but there was a strong division between astronomy and physics.
Kepler incorporated religious arguments and reasoning into his work, motivated by the religious conviction and belief that God had created the world according to an intelligible plan, accessible through the natural light of reason. Kepler described his new astronomy as "celestial physics", as "an excursion into Aristotle's Metaphysics", as "a supplement to Aristotle's On the Heavens", transforming the ancient tradition of physical cosmology by treating astronomy as part of a universal mathematical physics. Kepler was born on December 27, the feast day of St John the Evangelist, 1571, in the Free Imperial City of Weil der Stadt, his grandfather, Sebald Kepler, had been Lord Mayor of the city. By the time Johannes was born, he had two brothers and one sister and the Kepler family fortune was in decline, his father, Heinrich Kepler, earned a precarious living as a mercenary, he left the family when Johannes was five years old. He was believed to have died in the Eighty Years' War in the Netherlands.
His mother, Katharina Guldenmann, an innkeeper's daughter, was a herbalist. Born prematurely, Johannes claimed to have been sickly as a child, he impressed travelers at his grandfather's inn with his phenomenal mathematical faculty. He was introduced to astronomy at an early age, developed a love for it that would span his entire life. At age six, he observed the Great Comet of 1577, writing that he "was taken by mother to a high place to look at it." In 1580, at age nine, he observed another astronomical event, a lunar eclipse, recording that he remembered being "called outdoors" to see it and that the moon "appeared quite red". However, childhood smallpox left him with weak vision and crippled hands, limiting his ability in the observational aspects of astronomy. In 1589, after moving through grammar school, Latin school, seminary at Maulbronn, Kepler attended Tübinger Stift at the University of Tübingen. There, he studied philosophy under Vitus Müller and theology under Jacob Heerbrand, who taught Michael Maestlin while he was a student, until he became Chancellor at Tübingen in 1590.
He proved himself to be a superb mathematician and earned a reputation as a skilful astrologer, casting horoscopes for fellow students. Under the instruction of Michael Maestlin, Tübingen's professor of mathematics from 1583 to 1631, he learned both the Ptolemaic system and the Copernican system of planetary motion, he became a Copernican at that time. In a student disputation, he defended heliocentrism from both a theoretical and theological perspective, maintaining that the Sun was the principal source of motive power in the universe. Despite his desire to become a minister, near the end of his studies, Kepler was recommended for a position as teacher of mathematics and astronomy at the Protestant school in Graz, he accepted the position in April 1594, at the age of 23. Kepler's first major astronomical work, Mysterium Cosmographicum, was the first published defense of the Copernican system. Kepler claimed to have had an epiphany on July 19, 1595, while teaching in Graz, demonstrating the periodic conjunction of Saturn and Jupiter in the zodiac: he realized that regular polygons bound one inscribed and one circumscribed circle at definite ratios, which, he reasoned, might be the geometrical basis of the universe.
After failing to find a unique arrangement of polygons that fit known astronomical observations, Kepler began experimenting with 3-dimensional polyhedra. He found that each of the five Platonic solids could be inscribed and circumscribed by spherical orbs. By ordering the solids selectively—octahedron, dodecahedron, cube—Kepler found that the spheres could be placed at intervals corresponding to the relative sizes of each planet's path, assuming the planets circle the Sun. Kepler found a formula relating the size of each planet's orb to the length of its orbital period: from inner to outer planets, the ratio of increase in orbital period is twice the difference in orb radius. However, Kepler rejected this formula, because it was not precise enough. As
