Gta San Andreas Extreme Edition 2011 Golden Pen !LINK! Download
Trish Irelan
Andrea Marie Breiling's large-scale paintings employ swooping bands of spray paint built up in fevered layers. Bristling with an evangelical fervor and a fierce commitment to improvisation and freedom, each painting acts as a document of extreme effort-both physical and emotional. Churned up like a storm, the compositions tend to pour forth from the relatively ordered logic of an axis or horizon, drenching the picture plane and creating a sublime depth of space and feeling. Though based primarily in Los Angeles, Breiling made this cohesive body of work here in New York and the paintings reflect the influence of our weather, architecture (bridges in particular), and a bruised and moody aura that feels directly related to her deeply personal sensitivities to location and daily life.
Golden Gate contains a wide array of geologic features, from the Franciscan Complex, which represents subduction-related tectonic events that took place over a hundred million years ago, to the Merced Formation, which records glacier-driven sea level changes over the last 3 million years, to modern dune sands formed from bits of the Sierra Nevada Mountains transported to the coast by the Sacramento River. Whether formed from deep within the ocean crust or mantle under extreme heat and pressure, like serpentinite, or built by millions of microscopic sea creatures over eons, like radiolarian chert, the Recreation Area is a rock lover's paradise. These rocks also provide the substrate for the plants and animals that make the park their home.
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a \displaystyle a and b \displaystyle b with a > b > 0 \displaystyle a>b>0 ,
Some 20th-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing it to be aesthetically pleasing. These uses often appear in the form of a golden rectangle.
Ancient Greek mathematicians first studied the golden ratio because of its frequent appearance in geometry;[12] the division of a line into "extreme and mean ratio" (the golden section) is important in the geometry of regular pentagrams and pentagons.[13] According to one story, 5th-century BC mathematician Hippasus discovered that the golden ratio was neither a whole number nor a fraction (it is irrational), surprising Pythagoreans.[14] Euclid's Elements (c. 300 BC) provides several propositions and their proofs employing the golden ratio,[15][c] and contains its first known definition which proceeds as follows:[16]
Luca Pacioli named his book Divina proportione (1509) after the ratio; the book, largely plagiarized from Piero della Francesca, explored its properties including its appearance in some of the Platonic solids.[20][21] Leonardo da Vinci, who illustrated Pacioli's book, called the ratio the sectio aurea ('golden section').[22] Though it is often said that Pacioli advocated the golden ratio's application to yield pleasing, harmonious proportions, Livio points out that the interpretation has been traced to an error in 1799, and that Pacioli actually advocated the Vitruvian system of rational proportions.[23] Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. 16th-century mathematicians such as Rafael Bombelli solved geometric problems using the ratio.[24]
German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio;[25] this was rediscovered by Johannes Kepler in 1608.[26] The first known decimal approximation of the (inverse) golden ratio was stated as "about 0.6180340 \displaystyle 0.6180340 " in 1597 by Michael Maestlin of the University of Tübingen in a letter to Kepler, his former student.[27] The same year, Kepler wrote to Maestlin of the Kepler triangle, which combines the golden ratio with the Pythagorean theorem. Kepler said of these:
Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio. The first we may compare to a mass of gold, the second we may call a precious jewel.[28]
Eighteenth-century mathematicians Abraham de Moivre, Nicolaus I Bernoulli, and Leonhard Euler used a golden ratio-based formula which finds the value of a Fibonacci number based on its placement in the sequence; in 1843, this was rediscovered by Jacques Philippe Marie Binet, for whom it was named "Binet's formula".[29] Martin Ohm first used the German term goldener Schnitt ('golden section') to describe the ratio in 1835.[30] James Sully used the equivalent English term in 1875.[31]
By 1910, inventor Mark Barr began using the Greek letter phi ( φ \displaystyle \varphi ) as a symbol for the golden ratio.[32][e] It has also been represented by tau ( τ \displaystyle \tau ), the first letter of the ancient Greek τομή ('cut' or 'section').[35]
The zome construction system, developed by Steve Baer in the late 1960s, is based on the symmetry system of the icosahedron/dodecahedron, and uses the golden ratio ubiquitously. Between 1973 and 1974, Roger Penrose developed Penrose tiling, a pattern related to the golden ratio both in the ratio of areas of its two rhombic tiles and in their relative frequency within the pattern.[36] This gained in interest after Dan Shechtman's Nobel-winning 1982 discovery of quasicrystals with icosahedral symmetry, which were soon afterward explained through analogies to the Penrose tiling.[37]
This means that the constant 5 \displaystyle \sqrt 5 cannot be improved without excluding the golden ratio. It is, in fact, the smallest number that must be excluded to generate closer approximations of such Lagrange numbers.[40]
Fibonacci numbers and Lucas numbers have an intricate relationship with the golden ratio. In the Fibonacci sequence, each number is equal to the sum of the preceding two, starting with the base sequence 0 , 1 \displaystyle 0,1 :
Both the Fibonacci sequence and the sequence of Lucas numbers can be used to generate approximate forms of the golden spiral (which is a special form of a logarithmic spiral) using quarter-circles with radii from these sequences, differing only slightly from the true golden logarithmic spiral. Fibonacci spiral is generally the term used for spirals that approximate golden spirals using Fibonacci number-sequenced squares and quarter-circles.
The golden ratio features prominently in geometry. For example, it is intrinsically involved in the internal symmetry of the pentagon, and extends to form part of the coordinates of the vertices of a regular dodecahedron, as well as those of a 5-cell. It features in the Kepler triangle and Penrose tilings too, as well as in various other polytopes.
In a regular pentagon the ratio of a diagonal to a side is the golden ratio, while intersecting diagonals section each other in the golden ratio. The golden ratio properties of a regular pentagon can be confirmed by applying Ptolemy's theorem to the quadrilateral formed by removing one of its vertices. If the quadrilateral's long edge and diagonals are a , \displaystyle a, and short edges are b , \displaystyle b, then Ptolemy's theorem gives a 2 = b 2 + a b . \displaystyle a^2=b^2+ab. Dividing both sides by a b \displaystyle ab yields (see Calculation above),
The diagonal segments of a pentagon form a pentagram, or five-pointed star polygon, whose geometry is quintessentially described by φ \displaystyle \varphi . Primarily, each intersection of edges sections other edges in the golden ratio. The ratio of the length of the shorter segment to the segment bounded by the two intersecting edges (that is, a side of the inverted pentagon in the pentagram's center) is φ , \displaystyle \varphi , as the four-color illustration shows.
The triangle formed by two diagonals and a side of a regular pentagon is called a golden triangle or sublime triangle. It is an acute isosceles triangle with apex angle 36 and base angles 72.[45] Its two equal sides are in the golden ratio to its base.[46] The triangle formed by two sides and a diagonal of a regular pentagon is called a golden gnomon. It is an obtuse isosceles triangle with apex angle 108 and base angle 36. Its base is in the golden ratio to its two equal sides.[46] The pentagon can thus be subdivided into two golden gnomons and a central golden triangle. The five points of a regular pentagram are golden triangles,[46] as are the ten triangles formed by connecting the vertices of a regular decagon to its center point.[47]
Bisecting one of the base angles of the golden triangle subdivides it into a smaller golden triangle and a golden gnomon. Analogously, any acute isosceles triangle can be subdivided into a similar triangle and an obtuse isosceles triangle, but the golden triangle is the only one for which this subdivision is made by the angle bisector, because it is the only isosceles triangle whose base angle is twice its apex angle. The angle bisector of the golden triangle subdivides the side that it meets in the golden ratio, and the areas of the two subdivided pieces are also in the golden ratio.[46]
If the apex angle of the golden gnomon is trisected, the trisector again subdivides it into a smaller golden gnomon and a golden triangle. The trisector subdivides the base in the golden ratio, and the two pieces have areas in the golden ratio. Analogously, any obtuse triangle can be subdivided into a similar triangle and an acute isosceles triangle, but the golden gnomon is the only one for which this subdivision is made by the angle trisector, because it is the only isosceles triangle whose apex angle is three times its base angle.[46]
The golden ratio appears prominently in the Penrose tiling, a family of aperiodic tilings of the plane developed by Roger Penrose, inspired by Johannes Kepler's remark that pentagrams, decagons, and other shapes could fill gaps that pentagonal shapes alone leave when tiled together.[48] Several variations of this tiling have been studied, all of whose prototiles exhibit the golden ratio:
df19127ead