Logarithmic spiral

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number x to the base b is the exponent to which b must be raised, to produce x. For example, since 1000 = 103, the logarithm base 10 of 1000 is 3, or log10 (1000) = 3. The logarithm of x to base b is denoted as logb (x), or without parentheses, logb x, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation.

The number e, also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of natural logarithms. It is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series

The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), loge(x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

Latin is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area around present-day Rome, but through the power of the Roman Republic it became the dominant language in the Italian region and subsequently throughout the Roman Empire. Even after the fall of Western Rome, Latin remained the common language of international communication, science, scholarship and academia in Europe until well into the 18th century, when other regional vernaculars supplanted it in common academic and political usage, and it eventually became a dead language in the modern linguistic definition.

Jacob Bernoulli was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Leibniz–Newton calculus controversy. He is known for his numerous contributions to calculus, and along with his brother Johann, was one of the founders of the calculus of variations. He also discovered the fundamental mathematical constant e. However, his most important contribution was in the field of probability, where he derived the first version of the law of large numbers in his work Ars Conjectandi.

In mathematics, a self-similar object is exactly or approximately similar to a part of itself. Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed.

The nautilus is a pelagic marine mollusc of the cephalopod family Nautilidae. The nautilus is the sole extant family of the superfamily Nautilaceae and of its smaller but near equal suborder, Nautilina.

A headstone, tombstone, or gravestone is a stele or marker, usually stone, that is placed over a grave. It is traditional for burials in the Christian, Jewish, and Muslim religions, among others. In most cases, it has the deceased's name, date of birth, and date of death inscribed on it, along with a personal message, or prayer, but may contain pieces of funerary art, especially details in stone relief. In many parts of Europe, insetting a photograph of the deceased in a frame is very common.

Eadem mutata resurgo is a Latin phrase that literally translates to "Although changed, I arise the same".

The Archimedean spiral is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates (r, θ) it can be described by the equation

In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. It is a subtype of whorled patterns, a broad group that also includes concentric objects.

Evangelista Torricelli was an Italian physicist and mathematician, and a student of Galileo. He is best known for his invention of the barometer, but is also known for his advances in optics and work on the method of indivisibles. The Torr is named after him.

In mathematics, a self-similar object is exactly or approximately similar to a part of itself. Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed.

In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.

In mathematics, an involute is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from or wrapped around the curve.

In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that curve. The evolute of a circle is therefore a single point at its center. Equivalently, an evolute is the envelope of the normals to a curve.

In mathematics, a pedal curve of a given curve results from the orthogonal projection of a fixed point on the tangent lines of this curve. More precisely, for a plane curve C and a given fixed pedal point P, the pedal curve of C is the locus of points X so that the line PX is perpendicular to a tangent T to the curve passing through the point X. Conversely, at any point R on the curve C, let T be the tangent line at that point R; then there is a unique point X on the tangent T which forms with the pedal point P a line perpendicular to the tangent T – the pedal curve is the set of such points X, called the foot of the perpendicular to the tangent T from the fixed point P, as the variable point R ranges over the curve C.

The exponential function is a mathematical function denoted by or . Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation, but modern definitions allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to opine that the exponential function is "the most important function in mathematics".

Extratropical cyclones, sometimes called mid-latitude cyclones or wave cyclones, are low-pressure areas which, along with the anticyclones of high-pressure areas, drive the weather over much of the Earth. Extratropical cyclones are capable of producing anything from cloudiness and mild showers to severe gales, thunderstorms, blizzards, and tornadoes. These types of cyclones are defined as large scale (synoptic) low pressure weather systems that occur in the middle latitudes of the Earth. In contrast with tropical cyclones, extratropical cyclones produce rapid changes in temperature and dew point along broad lines, called weather fronts, about the center of the cyclone.

Iceland is a Nordic island country in the North Atlantic Ocean and in the Arctic Ocean. Iceland is the most sparsely populated country in Europe. Iceland's capital and largest city is Reykjavík, which is home to about 36% of the population. Iceland is the largest part of the Mid-Atlantic Ridge that rises above sea level, and its central volcanic plateau is erupting almost constantly. The interior consists of a plateau characterised by sand and lava fields, mountains, and glaciers, and many glacial rivers flow to the sea through the lowlands. Iceland is warmed by the Gulf Stream and has a temperate climate, despite a high latitude just outside the Arctic Circle. Its high latitude and marine influence keep summers chilly, and most of its islands have a polar climate.

The nautilus is a pelagic marine mollusc of the cephalopod family Nautilidae. The nautilus is the sole extant family of the superfamily Nautilaceae and of its smaller but near equal suborder, Nautilina.

Hawks are birds of prey of the family Accipitridae. They are widely distributed and are found on all continents except Antarctica.The subfamily Accipitrinae includes goshawks, sparrowhawks, sharp-shinned hawks and others. This subfamily are mainly woodland birds with long tails and high visual acuity. They hunt by dashing suddenly from a concealed perch. In America, members of the Buteo group are also called hawks; this group is called buzzards in other parts of the world. Generally, buteos have broad wings and sturdy builds. They are relatively larger-winged, shorter-tailed and fly further distances in open areas than accipiters. Buteos descend or pounce on their prey rather than hunting in a fast horizontal pursuit.

A galaxy is a system of stars, stellar remnants, interstellar gas, dust, dark matter, bound together by gravity. The word is derived from the Greek galaxias (γαλαξίας), literally 'milky', a reference to the Milky Way galaxy that contains the Solar System. Galaxies, averaging an estimated 100 million stars, range in size from dwarfs with less than a hundred million stars, to the largest galaxies known – supergiants with one hundred trillion stars, each orbiting its galaxy's center of mass. Most of the mass in a typical galaxy is in the form of dark matter, with only a few percent of that mass visible in the form of stars and nebulae. Supermassive black holes are a common feature at the centres of galaxies.

The Milky Way is the galaxy that includes the Solar System, with the name describing the galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars that cannot be individually distinguished by the naked eye. The term Milky Way is a translation of the Latin via lactea, from the Greek γαλακτικὸς κύκλος, meaning "milky circle". From Earth, the Milky Way appears as a band because its disk-shaped structure is viewed from within. Galileo Galilei first resolved the band of light into individual stars with his telescope in 1610. Until the early 1920s, most astronomers thought that the Milky Way contained all the stars in the Universe. Following the 1920 Great Debate between the astronomers Harlow Shapley and Heber Doust Curtis, observations by Edwin Hubble showed that the Milky Way is just one of many galaxies.

The Archimedean spiral is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates (r, θ) it can be described by the equation

A hyperbolic spiral is a plane curve, which can be described in polar coordinates by the equation

The cornea is the transparent front part of the eye that covers the iris, pupil, and anterior chamber. Along with the anterior chamber and lens, the cornea refracts light, accounting for approximately two-thirds of the eye's total optical power. In humans, the refractive power of the cornea is approximately 43 dioptres. The cornea can be reshaped by surgical procedures such as LASIK.

Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary information encoded in genes, which can be transmitted to future generations. Another major theme is evolution, which explains the unity and diversity of life. Energy processing is also important to life as it allows organisms to move, grow, and reproduce. Finally, all organisms are able to regulate their own internal environments.

Mollusca is the second-largest phylum of invertebrate animals after the Arthropoda, the members of which are known as molluscs or mollusks. Around 85,000 extant species of molluscs are recognized. The number of fossil species is estimated between 60,000 and 100,000 additional species. The proportion of undescribed species is very high. Many taxa remain poorly studied.

Half Moon Bay is a bay of the Pacific Ocean on the coast of San Mateo County, California. The bay is approximately semi-circular, hence the name half moon, with sea access to the south. Coastal towns located there are Princeton-by-the-Sea, Miramar, El Granada, and the city of Half Moon Bay.