Lemniscate of Gerono: solution set of Another lemniscate, the lemniscate of Gerono or lemniscate of Huygens, is the zero set of the quartic polynomial y^2-x^2(a^2-x^2)... Viviani's curve, a three-dimensional curve formed by intersecting a sphere with a cylinder, also has a figure eight shape, and has the lemniscate of Gerono as its planar projection..
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The lemniscate of Bernoulli and its two foci In algebraic geometry, a lemniscate is any of several figure-eight or -shaped curves. The word comes from the Latin "lēmniscātus" meaning "decorated with ribbons", from the Greek λημνίσκος meaning "ribbons",. or which alternatively may refer to the wool from which the ribbons were made. Curves that have been called a lemniscate include three quartic plane curves: the hippopede or lemniscate of Booth, the lemniscate of Bernoulli, and the lemniscate of Gerono.
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The lemniscate of Gerono In algebraic geometry, the lemniscate of Gerono, or lemniscate of Huygens, or figure-eight curve, is a plane algebraic curve of degree four and genus zero and is a lemniscate curve shaped like an \infty symbol, or figure eight. It has equation :x^4-x^2+y^2 = 0. It was studied by Camille-Christophe Gerono.
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Bernoulli's brother Jacob Bernoulli also studied the same curve in the same year, and gave it its name, the lemniscate.. It may also be defined geometrically as the locus of points whose product of distances from two foci equals the square of half the interfocal distance.. It is a special case of the hippopede (lemniscate of Booth), with d=-c, and may be formed as a cross-section of a torus whose inner hole and circular cross-sections have the same diameter as each other. The lemniscatic elliptic functions are analogues of trigonometric functions for the lemniscate of Bernoulli, and the lemniscate constants arise in evaluating the arc length of this lemniscate.
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Erdős lemniscate of degree ten and genus six A conjecture of Erdős which has attracted considerable interest concerns the maximum length of a polynomial lemniscate ƒ(x, y) = 1 of degree 2n when p is monic, which Erdős conjectured was attained when p(z) = zn − 1\. This is still not proved but Fryntov and Nazarov proved that p gives a local maximum. In the case when n = 2, the Erdős lemniscate is the Lemniscate of Bernoulli :(x^2+y^2)^2=2(x^2-y^2)\, and it has been proven that this is indeed the maximal length in degree four. The Erdős lemniscate has three ordinary n-fold points, one of which is at the origin, and a genus of (n − 1)(n − 2)/2.
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When d > 0 the curve has an oval form and is often known as an oval of Booth, and when the curve resembles a sideways figure eight, or lemniscate, and is often known as a lemniscate of Booth, after 19th-century mathematician James Booth who studied them. Hippopedes were also investigated by Proclus (for whom they are sometimes called Hippopedes of Proclus) and Eudoxus. For , the hippopede corresponds to the lemniscate of Bernoulli.
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By inverting the Erdős lemniscate in the unit circle, one obtains a nonsingular curve of degree n.
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The name comes from the shape its central lemniscate takes when graphed. The shape is named after the juggling game diabolo, which involves two sticks, a string, and a spinning prop in the likeness of the lemniscate. The confusion is the result of the Italian word meaning "devil".
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Lemniscate of Bernoulli In 1680, Cassini studied a family of curves, now called the Cassini oval, defined as follows: the locus of all points, the product of whose distances from two fixed points, the curves' foci, is a constant. Under very particular circumstances (when the half-distance between the points is equal to the square root of the constant) this gives rise to a lemniscate. In 1694, Johann Bernoulli studied the lemniscate case of the Cassini oval, now known as the lemniscate of Bernoulli (shown above), in connection with a problem of "isochrones" that had been posed earlier by Leibniz. Like the hippopede, it is an algebraic curve, the zero set of the polynomial (x^2 + y^2)^2 - 2a^2 (x^2 - y^2).
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The "figure 8" immersion of the Klein bottle. Klein bagel cross section employing a figure eight curve (the lemniscate of Gerono).
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A lemniscate of Bernoulli and its two foci F1 and F2 The lemniscate of Bernoulli is the pedal curve of a rectangular hyperbola In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points F1 and F2, known as foci, at distance 2c from each other as the locus of points P so that PF1·PF2 = c2. The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from lemniscatus, which is Latin for "decorated with hanging ribbons". It is a special case of the Cassini oval and is a rational algebraic curve of degree 4.
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The ∞ symbol in several typefaces The infinity symbol (\infty, , or in unicode ∞) is a mathematical symbol representing the concept of infinity. In algebraic geometry, the figure is called a lemniscate.
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Lemniscaat or Lemniscaat Publishers is a Dutch independent publishing house based in Rotterdam, Netherlands. The company publishes both children's literature and non-fiction books for adults. The company is named after the lemniscate symbol.
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The study of lemniscates (and in particular the hippopede) dates to ancient Greek mathematics, but the term "lemniscate" for curves of this type comes from the work of Jacob Bernoulli in the late 17th century.
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Several other curves can be derived from the hyperbola by inversion, the so-called inverse curves of the hyperbola. If the center of inversion is chosen as the hyperbola's own center, the inverse curve is the lemniscate of Bernoulli; the lemniscate is also the envelope of circles centered on a rectangular hyperbola and passing through the origin. If the center of inversion is chosen at a focus or a vertex of the hyperbola, the resulting inverse curves are a limaçon or a strophoid, respectively.
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Inverting the equation of an ellipse or hyperbola :cx^2+dy^2=1 gives :\left(x^2+y^2\right)^2=cx^2+dy^2 which is the hippopede. When this is the lemniscate of Bernoulli.
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Lemniscate of Booth The consideration of curves with a figure-eight shape can be traced back to Proclus, a Greek Neoplatonist philosopher and mathematician who lived in the 5th century AD. Proclus considered the cross-sections of a torus by a plane parallel to the axis of the torus. As he observed, for most such sections the cross section consists of either one or two ovals; however, when the plane is tangent to the inner surface of the torus, the cross-section takes on a figure-eight shape, which Proclus called a horse fetter (a device for holding two feet of a horse together), or "hippopede" in Greek. The name "lemniscate of Booth" for this curve dates to its study by the 19th-century mathematician James Booth.. The lemniscate may be defined as an algebraic curve, the zero set of the quartic polynomial (x^2 + y^2)^2 - cx^2 - dy^2 when the parameter d is negative (or zero for the special case where the lemniscate becomes a pair of externally tangent circles). For positive values of d one instead obtains the oval of Booth.
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503 (German)Franz Kössler, Lothar Kalok: Personenlexikon von Lehrern des 19. Jahrhunderts – Band: Vaders – Vries. Universitätsbibliothek Gießen, preprint 2007, p. 6 (German) In his dissertation De curvis lemniscatis he examined the lemniscate of Bernoulli and discovered a surprising property of certain angles occurring in it.
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The Keyhole Nebula is a dark nebulosity superimposed on the brightest part of the Carina Nebula. The Keyhole, or Keyhole Nebula, is a small dark cloud of cold molecules and dust within the Carina Nebula, containing bright filaments of hot, fluorescing gas, silhouetted against the much brighter background nebula. John Herschel used the term "lemniscate-oval vacuity" when first describing it, and subsequently referred to it simply as the "oval vacuity". The term lemniscate continued to be used to describe this portion of the nebula until popular astronomy writer Emma Converse described the shape of the nebula as "resembling a keyhole" in an 1873 Appleton's Journal article.
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This lemniscate was first described in 1694 by Jakob Bernoulli as a modification of an ellipse, which is the locus of points for which the sum of the distances to each of two fixed focal points is a constant. A Cassini oval, by contrast, is the locus of points for which the product of these distances is constant. In the case where the curve passes through the point midway between the foci, the oval is a lemniscate of Bernoulli. This curve can be obtained as the inverse transform of a hyperbola, with the inversion circle centered at the center of the hyperbola (bisector of its two foci).
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Applying the above transformation to the lemniscate of Bernoulli :\left(x^2 + y^2\right)^2 = a^2 \left(x^2 - y^2\right) gives us :a^2\left(u^2-v^2\right) = 1, the equation of a hyperbola; since inversion is a birational transformation and the hyperbola is a rational curve, this shows the lemniscate is also a rational curve, which is to say a curve of genus zero. If we apply the transformation to the Fermat curve , where is odd, we obtain :\left(u^2+v^2\right)^n = u^n+v^n. Any rational point on the Fermat curve has a corresponding rational point on this curve, giving an equivalent formulation of Fermat's Last Theorem.
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Vinyl Williams' debut studio album, Lemniscate, was released on November 12, 2012 by No Pain in Pop in Europe and Williams' own Salonislam imprint in the US. The Fader premiered videos for "Higher Worlds" and "Harmonious Change" prior to the album's release. The album received a positive reception from critics.
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Camille-Christophe Gerono (1799 in Paris, France – 1891 in Paris) was a French mathematician. He concerned himself above all with geometry. The Lemniscate of Gerono or figure-eight curve was named after him.. With Olry Terquem, he was founding co-editor in 1842 of the scientific journal Nouvelles Annales de Mathématiques..
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When n is an integer, and n points are arranged regularly on a circle of radius a, then the set of points so that the geometric mean of the distances from the point to the n points is a sinusoidal spiral. In this case the sinusoidal spiral is a polynomial lemniscate.
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The normal n of the lemniscate in P intersects the line connecting F1 and F2 in R. Now the interior angle of the triangle OPR at O is one third of the triangle's exterior angle at R. In addition the interior angle at P is twice the interior angle at O.
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The Guardian's New Music critic, Michael Cragg, reviewed the album's track Open Your Mind in October 2012, while "Lemniscate" as a whole was featured as Rough Trade's "Album Of The Week" shortly after its release. Other media outlets reviewing the album included This is fake DIY and James Skinner from BBC Music.
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Produzioni matematiche, 1750 Fagnano is best known for investigations on the length and division of arcs of certain curves, especially the lemniscate; this seems also to have been in his own estimation his most important work, since he had the figure of the lemniscate with the inscription: "Multifariam divisa atque dimensa Deo veritatis gloria", engraved on the title-page of his Produzioni Matematiche, which he published in two volumes (Pesaro, 1750), and dedicated to Pope Benedict XIV. The same figure and words "Deo veritatis gloria" also appear on his tomb. Failing to rectify the ellipse or hyperbola, Fagnano attempted to determine arcs whose difference is rectifiable. The word "rectifiable" meant at that time that the length can be found explicitly, which is different from its modern meaning.
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The function f(u) thus has the period iω' in the imaginary direction while it is 2iω' for F(u). Their zeros and poles will again form a regular lattice reflecting their double periodicity. After Gauss had died it was discovered that he had discovered a corresponding double periodicity in his lemniscate elliptic function.
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Other decks have the woman sitting upon the lion, or merely with one hand upon it. Some decks feature just one of the characters; flowers are often presented on this card. According to Eden Gray, the lemniscate above her represents enlightenment and spiritual powers, whereas the lion represents animal passions and earthly cravings.Gray, Eden.
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The infinity symbol \infty (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode at and in LaTeX as `\infty`. It was introduced in 1655 by John Wallis, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology.
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Examples include the hippopede and the Cassini oval and their relatives, such as the lemniscate of Bernoulli. The Cassini oval has the remarkable property that the product of distances to two foci are constant. For comparison, the sum is constant in ellipses, the difference is constant in hyperbolae and the ratio is constant in circles.
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A special case of a toric section is the spiric section, in which the intersecting plane is parallel to the rotational symmetry axis of the torus. They were discovered by the ancient Greek geometer Perseus in roughly 150 BC.. Well-known examples include the hippopede and the Cassini oval and their relatives, such as the lemniscate of Bernoulli.
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This linkage does not generate a true straight line motion, and indeed Watt did not claim it did so. Rather, it traces out Watt's curve, a lemniscate or figure eight shaped curve; when the lengths of its bars and its base are chosen to form a crossed square, it traces the lemniscate of Bernoulli.. In a letter to Boulton on 11 September 1784 Watt describes the linkage as follows. Although the Peaucellier–Lipkin linkage, Hart's inversor, and other straight line mechanisms generate true straight- line motion, Watt's linkage has the advantage of much greater simplicity than these other linkages. It is similar in this respect to the Chebyshev linkage, a different linkage that produces approximate straight-line motion; however, in the case of Watt's linkage, the motion is perpendicular to the line between its two endpoints, whereas in the Chebyshev linkage the motion is parallel to this line.
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Watt’s linkage: Point P follows a lemniscate while moving almost vertically for some part of its path. page 4. Watt's linkage (also known as the parallel linkage) is a type of mechanical linkage invented by James Watt in which the central moving point of the linkage is constrained to travel on a nearly straight line. It was described in Watt's patent specification of 1784 for the Watt steam engine.
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The first elliptic functions were found by Carl Friedrich Gauss around 1795 in connection with his calculation of the lemniscate arc length, but first published after his death.J. Stillwell, Mathematics and Its History, Springer, New York (2010). . These are special cases of the general, elliptic functions which were first investigated by Abel in 1823 when he still was a student.A. Stubhaug, Niels Henrik Abel and his Times, Springer-Verlag, Berlin (2000). .
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Springer-Verlag, 2013, , 9783322853653, p. 97.K. Strubecker: Vorlesungen der Darstellenden Geometrie. Vandenhoeck & Ruprecht, Göttingen 1967, p. 250. The projection of Viviani's curve onto a plane perpendicular to the line through the crossing point and the sphere center is the lemniscate of Gerono.. In 1692 Viviani tackled the task: Cut out of a half sphere (radius r) two windows, such that the remaining surface (of the half sphere) can be squared, i.e.
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In 1696 Bernoulli solved the equation, now called the Bernoulli differential equation, : y' = p(x)y + q(x)y^n. Jacob Bernoulli also discovered a general method to determine evolutes of a curve as the envelope of its circles of curvature. He also investigated caustic curves and in particular he studied these associated curves of the parabola, the logarithmic spiral and epicycloids around 1692. The lemniscate of Bernoulli was first conceived by Jacob Bernoulli in 1694.
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Vinyl Williams is an American neo-psychedelic band led by Los Angeles-based multimedia artist and musician Lionel Williams, active since 2007. Vinyl Williams has released five studio albums: Lemniscate (2012), Into (2015), Brunei (2016), Opal (2018), and Azure (2020). Williams, who calls his music "celestial pop", has been described as neo-psychedelia,, electronic, dream pop, shoegaze, krautrock, chillwave and hypnagogic pop. Dummy Mag has called Williams a "retro futurist with a penchant for analogue noise and sonic transcendentalism".
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Schematic of Watt's parallel motion: A and G are fixed hinge joints while F is not a joint but merely signifies the point on the linkage which follows a lemniscate. Its motion is magnified in D by the parallelogram BCDE. See the diagram on the right. A is the journal (bearing) of the walking beam KAC, which rocks up and down about A. H is the piston, which is required to move vertically but not horizontally.
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Within 100 years of the appearance of Mendeleev's table in 1869, Edward G. Mazurs had collected an estimated 700 different published versions of the periodic table.Scerri 2007, p. 20 As well as numerous rectangular variations, other periodic table formats have been shaped, for example, like a circle, cube, cylinder, building, spiral, lemniscate, octagonal prism, pyramid, sphere, or triangle. Such alternatives are often developed to highlight or emphasize chemical or physical properties of the elements that are not as apparent in traditional periodic tables.
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The inverse of a sinusoidal spiral with respect to a circle with center at the origin is another sinusoidal spiral whose value of n is the negative of the original curve's value of n. For example, the inverse of the lemniscate of Bernoulli is a rectangular hyperbola. The isoptic, pedal and negative pedal of a sinusoidal spiral are different sinusoidal spirals. One path of a particle moving according to a central force proportional to a power of r is a sinusoidal spiral.
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In mathematics, an elliptic Gauss sum is an analog of a Gauss sum depending on an elliptic curve with complex multiplication. The quadratic residue symbol in a Gauss sum is replaced by a higher residue symbol such as a cubic or quartic residue symbol, and the exponential function in a Gauss sum is replaced by an elliptic function. They were introduced by , at least in the lemniscate case when the elliptic curve has complex multiplication by , but seem to have been forgotten or ignored until the paper .
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Entry 43, dated 1796, October 21, states "Vicimus GEGAN" (We have conquered GEGAN). The meaning of this was a mystery for many years. found a manuscript by Gauss suggesting that GEGAN is a reversal of the acronym NAGEG standing for Nexum medii Arithmetico-Geometricum Expectationibus Generalibus and refers to the connection between the arithmetic geometric mean and elliptic functions. Entry 146, dated 1814 July 9, is the last entry, and records an observation relating biquadratic residues and the lemniscate functions, later proved by Gauss and by .
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If one of the short (uncrossed) edges of an antiparallelogram linkage is fixed in place, and the remaining linkage moves freely, then the crossing point of the antiparallelogram traces out an ellipse that has the fixed edge's endpoints as its foci. The other moving short edge of the antiparallelogram has as its endpoints the foci of another moving ellipse, formed from the first one by reflection across a tangent line through the crossing point.. For both the parallelogram and antiparallelogram linkages, if one of the long (crossed) edges of the linkage is fixed as a base, the free joints move on equal circles, but in a parallelogram they move in the same direction with equal velocities while in the antiparallelogram they move in opposite directions with unequal velocities.. As James Watt discovered, if an antiparallelogram has its long side fixed in this way it forms a variant of Watt's linkage, and the midpoint of the unfixed long edge will trace out a lemniscate or figure eight curve. For the antiparallelogram formed by the sides and diagonals of a square, it is the lemniscate of Bernoulli., pp. 58–59.
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In the systems of this type, an electrical generator, pump, or tasking line is installed on the ground. There are two subtypes, with or without a secondary vehicle. In the subtype without a secondary vehicle,"Yo-Yo" method, the tether slowly unwinds off a drum on the ground, due to the windward pull of the kite system's wing, while the wing travels crosswind, that is, left-right of the wind's ambient direction, along various paths, e.g., a figure-8 flight path, or optimized lemniscate paths, or circular paths (small or large radius).
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From his findings gleaned over 30 years spent observing nature in pristine areas of Austria, Viktor Schauberger sought to implement basic life and movement processes and principles for the production of alternative fuel energy for machines, turbines, engines or to generate heat. Visitors to the Vortex Garden can experience such implosive forces and empathize with the way that life forces can or may emerge, grow and develop. The name “Vortex” derives from the many flowform water features and funnels with spiralling and lemniscate (figure-eight shaped) forms of water movement.
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When light passes through a slice of calcite cut perpendicular to its optic axis, the difference between the propagation times of the ordinary and extraordinary waves has a second-order dependence on the angle of incidence. If the slice is observed in a highly convergent cone of light, that dependence becomes significant, so that a chromatic-polarization experiment will show a pattern of concentric rings. But most minerals, when observed in this manner, show a more complicated pattern of rings involving two foci and a lemniscate curve, as if they had two optic axes.Jenkins & White, 1976, pp. 576–9 (§27.9, esp. Fig. 27M).
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Different forms of symmetry can be deduced from the equation of a polar function r. If the curve will be symmetrical about the horizontal (0°/180°) ray, if it will be symmetric about the vertical (90°/270°) ray, and if it will be rotationally symmetric by α clockwise and counterclockwise about the pole. Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate. Among the best known of these curves are the polar rose, Archimedean spiral, lemniscate, limaçon, and cardioid.
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In the organically-shaped cascading pools by John Wilkes, the water makes flowing lemniscate patterns with various rhythms according to the different basin widths. The trick fountains are supplied with rainwater from the gutters through three tanks; enhanced by the invigorating motion in the flowform basins, the water is used for garden irrigation. In the Vortex Garden, garden soil and plant maintenance is based on principles of the permaculture approach. On the recommendations of Hermann Benjes, a relatively large number of dead wood stacks and woodpiles provide choice breeding grounds for microorganisms in this town garden.
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It was during the regional retreat of the Lake Superior Lobe and glacial meltwater flow from deglaciation and glacial Lake Nemadji and Lake Duluth caused the entrenchment of the St. Croix River and the formation of the deep gorge of the St. Croix River of the St. Croix River valley and its famous potholes occurred. In and surrounding Polk County, Minnesota, geomorphic and stratigraphic relationships evidence exists for at least two drainage events. A strath terrace, known as the Chengwatana surface provides evidence for the occurrence of the first drainage event. The Chengwatana surface is a scoured surface marked by distinct lemniscate landforms; bar-shaped lndforms composed of sand; and a lag layer of cobbles and boulders.
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Biot, assuming that the concentric pattern was the general case, tried to calculate the colors with his theory of chromatic polarization, and succeeded better for some minerals than for others. In 1818, Brewster belatedly explained why: seven of the twelve minerals employed by Biot had the lemniscate pattern, which Brewster had observed as early as 1812; and the minerals with the more complicated rings also had a more complicated law of refraction.Buchwald, 1989, pp. 254–5,402. In a uniform crystal, according to Huygens's theory, the secondary wavefront that expands from the origin in unit time is the ray- velocity surface—that is, the surface whose "distance" from the origin in any direction is the ray velocity in that direction.
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Rider–Waite tarot deck In modern mysticism, the infinity symbol has become identified with a variation of the ouroboros, an ancient image of a snake eating its own tail that has also come to symbolize the infinite, and the ouroboros is sometimes drawn in figure-eight form to reflect this identification—rather than in its more traditional circular form. The book also features this image on its cover. In the works of Vladimir Nabokov, including The Gift and Pale Fire, the figure-eight shape is used symbolically to refer to the Möbius strip and the infinite, as is the case in these books' descriptions of the shapes of bicycle tire tracks and of the outlines of half-remembered people. The poem after which Pale Fire is entitled explicitly refers to "the miracle of the lemniscate".
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The heart of the design is the four- bar linkage consisting of AB, BE and EG and the base link is AG, both joints on the framework of the engine. As the beam rocks, point F (which is drawn to aid this explanation, but is not a marked point on the machine itself) describes an elongated figure-of-eight (more precisely, a lemniscate of Bernoulli) in mid-air. Since the motion of the walking beam is constrained to a small angle, F describes only a short section of the figure-of-eight, which is quite close to a vertical straight line. The figure-of-eight is symmetrical as long as arms AB and EG are equal in length, and straightest when the ratio of BF to FE matches that of AB to EG. If the stroke length (that is, the maximum travel of F) is S, then the straight section is longest when BE is around 2/3 S and AB is 1.5 S.Neil Sclater and Nicholas P. Chironis, Mechanisms and Mechanical Devices Sourcebook Third Edition (2001), page 136.
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