The only reference that historians rely on of Euclid having written the Elements was from Proclus, who briefly in his Commentary on the Elements ascribes Euclid as its author.Although best known for its geometric results, the Elements also includes number theory.…

The answer is because the mean does not provide as much information as the geometric mean.

Rules for quadratic equations were given in the The Nine Chapters on the Mathematical Art, a Chinese treatise on mathematics.These early geometric methods do not appear to have had a general formula.Euclid, the Greek mathematician, produced a more abstract geometrical method around 300 BC.…

Nonetheless, KK remains an important touchstone in theoretical physics and is often embedded in more sophisticated theories.It is studied in its own right as an object of geometric interest in K-theory.Even in the absence of a completely satisfying theoretical physics framework, the idea of exploring extra, compactified, dimensions is of considerable interest in the experimental physics and astrophysics communities.…

A projective geometry of dimension 1 consists of a single line containing at least 3 points.The geometric construction of arithmetic operations cannot be carried out in either of these cases.For dimension 2, there is a rich structure in virtue of the absence of Desargues' Theorem.…

The function (where the list elements are positive numbers) provides the geometric mean.The function (where the list elements are positive numbers) provides the harmonic mean.…

Instead, the geometric mean is simply where n is the number of steps from the initial to final state.

Given a geometric object X, one can consider functions on such objects.

Theoretically, if the resultant of the vertical load on the tire and the lateral force generated by it points directly into the instant center, the suspension links will not move.In this case, all weight transfer at that end of the vehicle will be geometric in nature.This is key information used in finding the force-based roll center as well.…

In geometric units, every time interval is interpreted as the distance travelled by light during that given time interval.That is, one second is interpreted as one light-second, so time has the geometric units of length.This is dimensionally consistent with the notion that, according to the kinematical laws of special relativity, time and distance are on an equal footing.…

It was during this time that Riley began to paint the black and white works for which she is best known.They present a great variety of geometric forms that produce sensations of movement or colour.In the early 1960s, her works were said to induce sensation in viewers as varied as seasick and sky diving.…

The style is traditional in the folk art of many countries.Patterns can be free form style or based on geometric figures.In America it is mostly used with basswood, butternut, pine, or mahogany.…

Geometric patterns of the region are seen in his later works.

The shepherdess, standing at the left, is posed in sexually suggestive fashion, very different from her austere counterpart in the later version.The later version has a far more geometric composition and the figures are much more contemplative.The mask-like face of the shepherdess conforms to the conventions of the Classical "Greek profile".…

The most recent painting, consisting of geometric figures, date to the medieval period.Executed mainly in red and white with the occasional use of green and yellow, the paintings depict the lives and times of the people who lived in the caves, including scenes of childbirth, communal dancing and drinking, religious rites and burials, as well as indigenous animals.…

Replacing the arithmetic and harmonic mean by a pair of generalized means of opposite, finite exponents yields the same result.You can get the same result every time for the geometric mean using a method that involves logarithms.…

For example the Kronecker-Weber theorem can be deduced easily from the analogous local statement.The philosophy behind the study of local fields is largely motivated by geometric methods.In algebraic geometry, it is common to study varieties locally at a point by localizing to a maximal ideal.…

Fractals also includes the idea of a detailed pattern that repeats itself.Fractals are different from other geometric figures because of the way in which they scale.Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in).…

The belief that God created the universe according to a geometric plan has ancient origins.

A rational map is to a regular map as a rational function is to a polynomial: it may be indeterminate at some points.In geometric terms, one has to work with functions defined on some open, dense set of a given variety.The description of the behaviour on the complement may require infinitely near points to be introduced to account for limiting behaviour along different directions.…