Irisneyda Almodóvar

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De izquierda a derecha la derivada de Sec(x)

De derecha a izquierda la integral de Tan(x)*Sec(x)From left to right the derivative of Sec (x)

From right to left the integral of Tan (x) * Sec (x)

Math emotions.

Spotted in the Stanford Math Department (photo cred: Jessica Su)

No attractive as in fixed points? =(

I don’t mean to “steal” this post from Mathispun, but here’s a quick caption for this cool gif. (Caption made available to you by Wikipedia and my copy and pasting skills).

A pair of parabolas face each other symmetrically: one on top and one on the bottom. Then the top parabola is rolled without slipping along the bottom one, and its successive positions are shown in the animation. Then the path traced by the vertex of the top parabola as it rolls is a roulette shown in red, which happens to be a cissoid of Diocles. In geometry, the cissoid of Diocles is a cubic plane curve notable for the property that it can be used to construct two mean proportionals to a given ratio. In particular, it can be used to double a cube. Doubling the cube (also known as the Delian problem) is one of the three most famous geometric problems unsolvable by compass and straightedge construction. It was known to the Egyptians, Greeks, and Indians. To “double the cube” means to be given a cube of some side length s and volume V = s^3, and to construct the side of a new cube, larger than the first, with volume 2V and therefore side length s * cube root 2. The problem is known to be impossible to solve with only compass and straightedge, because cube root 2 (≈ 1.25992105…) is not a constructible number.

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of a formal mathematical science emerging in the West as early as 6th Century BC. By the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow. Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia.

For nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, basic understanding of space remained essentially the same. Immanuel Kant argued that there is only one absolutegeometry. This dominant view was overturned by the revolutionary discovery of non-Euclidean geometry in the works of Gauss (who never published his theory), Bolyai, and Lobachevsky, who demonstrated that ordinary Euclidean space is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by Riemann in his 1867 inauguration lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (On the hypotheses on which geometry is based),

^{}published only after his death. Riemann’s new idea of space proved crucial in Einstein's general relativity theory and Riemannian geometry, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.Where the traditional geometry allowed dimensions 1 (a line), 2 (a plane) and 3 (our world conceived of as three-dimensional space), mathematicians have used higher dimensions for nearly two centuries. Dimension has gone through stages of being any natural number

n, possibly infinite with the introduction of Hilbert space, and any positive real number in fractal geometry.The above image contains examples of simple 2D geometric shapes.