"13, 2008. Lewis performed the song live on the March 6, 2008 results show of the seventh season of ""American Idol"". The single has sold 94,000 copies to date. How Many Words ""How Many Words"" is the official second single by the singer-songwriter Blake Lewis, from his debut album ""A.D.D. (Audio Day Dream)"". The song was expected to be followed by ""Know My Name"", and then ""Without You"". However, due to being dropped by Arista Records, ""How Many Words"" is the final single from his debut album. He is expected to release a single from his second album at the"
"snowflake. After each iteration, the number of sides of the Koch snowflake increases by a factor of 4, so the number of sides after iterations is given by: If the original equilateral triangle has sides of length , the length of each side of the snowflake after iterations is: the perimeter of the snowflake after iterations is: The Koch curve has an infinite length because the total length of the curve increases by a factor of four thirds with each iteration. Each iteration creates four times as many line segments as in the previous iteration, with the length of each"
"in the same micro-environment does not guarantee that each arm grow the same; indeed, for some crystal forms it does not because the underlying crystal growth mechanism also affects how fast each surface region of a crystal grows. Empirical studies suggest less than 0.1% of snowflakes exhibit the ideal six-fold symmetric shape. Very occasionally twelve branched snowflakes are observed; they maintain the six-fold symmetry. Snowflakes form in a wide variety of intricate shapes, leading to the notion that ""no two are alike"". Although nearly-identical snowflakes have been made in laboratory, they are very unlikely to be found in nature. Initial"
"of ice. At that stage, the snowflake has the shape of a minute hexagon. The six ""arms"" of the snowflake, or dendrites, then grow independently from each of the corners of the hexagon, while either side of each arm grows independently. The microenvironment in which the snowflake grows changes dynamically as the snowflake falls through the cloud and tiny changes in temperature and humidity affect the way in which water molecules attach to the snowflake. Since the micro-environment (and its changes) are very nearly identical around the snowflake, each arm tends to grow in nearly the same way. However, being"
"a stylized charge, often used to represent winter or winter sports. Three different snowflake symbols are encoded in Unicode: ""snowflake"" at U+2744 (❄); ""tight trifoliate snowflake"" at U+2745 (❅); and ""heavy chevron snowflake"" at U+2746 (❆). A selection of photographs taken by Wilson Bentley (1865–1931): Snowflake A snowflake is a single ice crystal that has achieved a sufficient size, and may have amalgamated with others, then falls through the Earth's atmosphere as snow. Each flake nucleates around a dust particle in supersaturated air masses by attracting supercooled cloud water droplets, which freeze and accrete in crystal form. Complex shapes emerge"
"of infinitely many equilateral triangles (see figure). Each side of the green triangle is exactly 1/3 the size of a side of the large blue triangle, and therefore has exactly 1/9 the area. Similarly, each yellow triangle has 1/9 the area of a green triangle, and so forth. Taking the blue triangle as a unit of area, the total area of the snowflake is The first term of this series represents the area of the blue triangle, the second term the total area of the three green triangles, the third term the total area of the twelve yellow triangles, and"
"are snowflakes, and are usually the type of ice particle that falls to the ground. Guinness World Records list the world's largest snowflakes as those of January 1887 at Fort Keogh, Montana; allegedly one measured 38 cm (15 inches) wide. The exact details of the sticking mechanism remain a subject of research. Although the ice is clear, scattering of light by the crystal facets and hollows/imperfections mean that the crystals often appear white in color due to diffuse reflection of the whole spectrum of light by the small ice particles. The shape of the snowflake is determined broadly by the"
"unlikely that any two snowflakes are alike due to the estimated 10 (10 quintillion) water molecules which make up a typical snowflake, which grow at different rates and in different patterns depending on the changing temperature and humidity within the atmosphere that the snowflake falls through on its way to the ground. Snowflakes that look identical, but may vary at the molecular level, have been grown under controlled conditions. Although snowflakes are never completely symmetrical, a non-aggregated snowflake often grows so as to exhibit an approximation of six-fold radial symmetry. The symmetry gets started due to the hexagonal crystalline structure"
"1.7712. The formula_12 is obtained because each iteration has 7 hexagons that are scaled by 1/3. The boundary of a hexaflake is the standard Koch curve of 60 degrees and infinitely many Koch snowflakes are contained within. Also, the projection of the cantor cube onto the plane orthogonal to its main diagonal is a hexaflake. Like the pentaflake, there is also a variation of the hexaflake, called the Sierpinski hexagon, that has no central hexagon. Its Hausdorff dimension equals formula_14 ≈ 1.6309. This variation still contains infinitely many Koch curves of 60 degrees. ""n""-flakes of higher polygons also exist, though"
"Hexaflake A hexaflake is a fractal constructed by iteratively exchanging each hexagon by a flake of seven hexagons; it is a special case of the ""n""-flake. A hexaflake has 7 hexagons in its ""n""th iteration, each smaller by 1/3 than the hexagons in the previous iteration. Its exterior boundary is the von Koch flake, and the full boundary contains an infinite number of Koch snowflakes. The Hausdorff dimension of the hexaflake is equal to ln(7)/ln(3), approximately 1.7712. It may also be constructed by projecting the Cantor cube onto the plane orthogonal to its main diagonal. A closely related fractal, the"
"crystal faces from forming properly or at all. However, when snowflakes crystallize, they do not touch each other. Thus, snowflakes form euhedral, six-sided twinned crystals. In rocks, the presence of euhedral crystals may signify that they formed early in the crystallization of magma or perhaps crystallized in a cavity or vug, without hindrance from other crystals. ""Euhedral"" is derived from the Greek ""eu"" meaning ""well, good"" and ""hedron"" meaning ""shape"". Euhedral crystals have flat faces with sharp angles. The flat faces (also called facets) are oriented in a specific way relative to the underlying atomic arrangement of the crystal: They"
"N-flake An ""n""-flake, polyflake, or Sierpinski ""n""-gon, is a fractal constructed starting from an ""n""-gon. This ""n""-gon is replaced by a flake of smaller ""n""-gons, such that the scaled polygons are placed at the vertices, and sometimes in the center. This process is repeated recursively to result in the fractal. Typically, there is also the restriction that the ""n""-gons must touch yet not overlap. The most common variety of ""n""-flake is two-dimensional (in terms of its topological dimension) and is formed of polygons. The four most common special cases are formed with triangles, squares, pentagons, and hexagons, but it can"
"rhombus, or triangle is rep-4. The sphinx hexiamond (illustrated above) is rep-4 and rep-9, and is one of few known self-replicating pentagons. The Gosper island is rep-7. The Koch snowflake is irrep-7: six small snowflakes of the same size, together with another snowflake with three times the area of the smaller ones, can combine to form a single larger snowflake. A right triangle with side lengths in the ratio 1:2 is rep-5, and its rep-5 dissection forms the basis of the aperiodic pinwheel tiling. By Pythagoras' theorem, the hypotenuse, or sloping side of the rep-5 triangle, has a length of"
"paper snowflakes by folding a piece of paper several times, cutting out a pattern with scissors and then unfolding it. Snowflakes are also often used as symbols representing winter or cold conditions. For example, snow tires which enhance traction during harsh winter driving conditions are labelled with a snowflake on the mountain symbol. A stylized snowflake has been part of the emblem of the 1968 Winter Olympics, 1972 Winter Olympics, 1988 Winter Olympics, 1998 Winter Olympics and 2002 Winter Olympics. Snowflakes are also seen as a symbol of uniqueness as no two are perfectly identical. In heraldry, the snowflake is"
"points on the Koch snowflake is infinite. No small piece of it is line-like, but rather it is composed of an infinite number of segments joined at different angles. The fractal dimension of a curve can be explained intuitively thinking of a fractal line as an object too detailed to be one-dimensional, but too simple to be two-dimensional. Therefore its dimension might best be described not by its usual topological dimension of 1 but by its fractal dimension, which is often a number between one and two; in the case of the Koch snowflake, it is about 1.262. A fractal"
"Snowflake A snowflake is a single ice crystal that has achieved a sufficient size, and may have amalgamated with others, then falls through the Earth's atmosphere as snow. Each flake nucleates around a dust particle in supersaturated air masses by attracting supercooled cloud water droplets, which freeze and accrete in crystal form. Complex shapes emerge as the flake moves through differing temperature and humidity zones in the atmosphere, such that individual snowflakes differ in detail from one another, but may be categorized in eight broad classifications and at least 80 individual variants. The main constituent shapes for ice crystals, from"
"Teragon A teragon is a polygon with an infinite number of sides, the most famous example being the Koch snowflake. Typically, a teragon will be bounded by one or more self-similar fractal curves, which are created by replacing each line segment in an initial figure with multiple connected segments, then replacing each of those segments with the same pattern of segments, then repeating the process an infinite number of times for every line segment in the figure. The word ""teragon"" comes from Greek (""teras""), ""monster, marvel"", + γωνία (""gōnía"") ""corner, angle"" (as in poly-gon). By the standards of classical geometry,"
"graupel. Although ice by itself is clear, snow usually appears white in color due to diffuse reflection of the whole spectrum of light by the scattering of light by the small crystal facets of the snowflakes of which it is comprised. The shape of the snowflake is determined broadly by the temperature and humidity at which it is formed. Rarely, at a temperature of around , snowflakes can form in threefold symmetry — triangular snowflakes. The most common snow particles are visibly irregular, although near-perfect snowflakes may be more common in pictures because they are more visually appealing. It is"
"so forth. Excluding the initial 1, this series is geometric with constant ratio ""r"" = 4/9. The first term of the geometric series is ""a"" = 3(1/9) = 1/3, so the sum is Thus the Koch snowflake has 8/5 of the area of the base triangle. The convergence of a geometric series reveals that a sum involving an infinite number of summands can indeed be finite, and so allows one to resolve many of Zeno's paradoxes. For example, Zeno's dichotomy paradox maintains that movement is impossible, as one can divide any finite path into an infinite number of steps wherein"
"could/should have different colours than the plane you're solving. If considering the pentagonal regions as faces, like in the great dodecahedron represented by Schläfli symbol {5,5/2}, then the requirement is for all faces to be monochrome and for opposite faces to share the same color. The puzzle does not turn smoothly, due to its unique design. There are 30 edges, each of which can be flipped into two positions, giving a theoretical maximum of 30!×2 permutations. This value is not reached for the following reasons: This gives a total of formula_1 possible combinations. The precise figure is 72 431 714"
"to reach another position; or (≈ 7×10) when mirror images (defined as the same sequence of turns, but from the other end of the snake) are counted as the one position. Rubik's Snake A Rubik's Snake (also Rubik's Twist, Rubik's Transformable Snake, Rubik’s Snake Puzzle) is a toy with 24 wedges that are right isosceles triangular prisms. The wedges are connected by spring bolts, so that they can be twisted, but not separated. By being twisted, the Rubik's Snake can be made to resemble a wide variety of objects, animals, or geometric shapes. Its ""ball"" shape in its packaging is"
"Sierpinski hexagon, is formed by repeatedly replacing each hexagon by six smaller hexagons, omitting the central seventh hexagon. It is named after the work of Wacław Sierpiński and by analogy to the Sierpinski triangle, and has the same Koch snowflake exterior boundary as the hexaflake. The hexaflake has been applied in the design of antennas and optical fibers. Hexaflake A hexaflake is a fractal constructed by iteratively exchanging each hexagon by a flake of seven hexagons; it is a special case of the ""n""-flake. A hexaflake has 7 hexagons in its ""n""th iteration, each smaller by 1/3 than the hexagons"
"The Koch snowflake is self-replicating with six copies around a central point and one larger copy at the center. Hence it is an irreptile which is irrep-7. The fractal dimension of the Koch curve is ≈ 1.26186. This is greater than the dimension of a line (1) but less than Peano's space-filling curve (2). The Koch curve is continuous everywhere but differentiable nowhere. It is possible to tessellate the plane by copies of Koch snowflakes in two different sizes. However, such a tessellation is not possible using only snowflakes of one size. Since each Koch snowflake in the tessellation can"
"perimeter diverges to infinity. Consequently, the snowflake has a finite area bounded by an infinitely long line. The Koch snowflake can be constructed by starting with an equilateral triangle, then recursively altering each line segment as follows: After one iteration of this process, the resulting shape is the outline of a hexagram. The Koch snowflake is the limit approached as the above steps are followed over and over again. The Koch curve originally described by Helge von Koch is constructed with only one of the three sides of the original triangle. In other words, three Koch curves make a Koch"
"of twenty regular dodecahedrons. Each flake is formed by placing a dodecahedron scaled by formula_19 in each corner. Its Hausdorff dimension is equal to formula_20 ≈ 2.3296. An icosahedron flake, or sierpinski icosahedron, is formed by successive flakes of twelve regular icosahedrons. Each flake is formed by placing an icosahedron scaled by formula_10 in each corner. Its Hausdorff dimension is equal to formula_22 ≈ 2.5819. N-flake An ""n""-flake, polyflake, or Sierpinski ""n""-gon, is a fractal constructed starting from an ""n""-gon. This ""n""-gon is replaced by a flake of smaller ""n""-gons, such that the scaled polygons are placed at the vertices,"
"3 31 honeycomb In 7-dimensional geometry, the 3 honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,3} and is composed of 3 and 7-simplex facets, with 56 and 576 of them respectively around each vertex. It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram. Removing the node on the short branch leaves the 6-simplex facet: Removing the node on the end of the 3-length branch leaves the 3 facet: The vertex figure is determined by removing the ringed node and"
"out of a turn had to be symmetrical. The simple ""figure eight"" shape was executed by connecting two circles ""about three times the height of the skater with one circle skated on each foot"". The figure eight has four variations: inside edges, outside edges, backward, and forward. A turn added at the halfway point of each circle increased the level of complexity. Other figures included three-lobed figures with a counter turn or a rocker turn, which were completed at the points where the lobes touched. Counters and rockers had to be executed symmetrically, with no change of edge, with the"
"Snowflake schema In computing, a snowflake schema is a logical arrangement of tables in a multidimensional database such that the entity relationship diagram resembles a snowflake shape. The snowflake schema is represented by centralized fact tables which are connected to multiple dimensions.. ""Snowflaking"" is a method of normalizing the dimension tables in a star schema. When it is completely normalized along all the dimension tables, the resultant structure resembles a snowflake with the fact table in the middle. The principle behind snowflaking is normalization of the dimension tables by removing low cardinality attributes and forming separate tables. The snowflake schema"
"to 2 while the progression for the perimeter diverges to infinity, so as in the case of the Koch snowflake, we have a finite area bounded by an infinite fractal curve. The resulting area fills a square with the same center as the original, but twice the area, and rotated by radians, the perimeter touching but never overlapping itself. The total area covered at the th iteration is: while the total length of the perimeter is: which approaches infinity as increases. Koch snowflake The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a mathematical"
"they are less common and don't usually have a central polygon. Some examples are shown below; the 7-flake through 12-flake. While it may not be obvious, these higher polyflakes still contain infinitely many Koch curves, but the angle of the Koch curves decreases as ""n"" increases. Their Hausdorff dimensions are slightly more difficult to calculate than lower ""n""-flakes because their scale factor is less obvious. However, the Hausdorff dimension is always less than two but no less than one. An interesting ""n""-flake is the ∞-flake, because as the value of ""n"" increases, an ""n""-flake's Hausdorff dimension approaches 1, ""n""-flakes can"
"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 is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the ""regular enneacontahexagon"", ""m""=48, and it can be divided into 1128: 24 squares and 23 sets of 48 rhombs. This decomposition is based on a Petrie polygon projection of a 48-cube. An enneacontahexagram is a 96-sided star polygon. There are 15 regular forms given by Schläfli symbols {96/5}, {96/7}, {96/11}, {96/13}, {96/17}, {96/19}, {96/23}, {96/25}, {96/29}, {96/31}, {96/35},"
"formula_8 (golden ratio). The formula_7 is obtained because each iteration has 6 pentagons that are scaled by formula_10. The boundary of a pentaflake is the Koch curve of 72 degrees. There is also a variation of the pentaflake that has no central pentagon. Its Hausdorff dimension equals formula_11 ≈ 1.6723. This variation still contains infinitely many Koch curves, but they are somewhat more visible. A hexaflake, is formed by successive flakes of seven regular hexagons. Each flake is formed by placing a scaled hexagon in each corner and one in the center. Its Hausdorff dimension is equal to formula_12 ≈"
"has facets and , and vertex figure . 1 33 honeycomb In 7-dimensional geometry, 1 is a uniform honeycomb, also given by Schläfli symbol {3,3}, and is composed of 1 facets. It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram. Removing a node on the end of one of the 3-length branch leaves the 1, its only facet type. The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the trirectified 7-simplex, 0. The edge"