Math

Penrose Triangle

Penrose triangle is a triangular impossible object, an optical illusion consisting of an object which can be depicted in a perspective drawing, but cannot exist as a solid object. It was first created by the Swedish artist Oscar Reutersvärd in 1934.

Independently from Reutersvärd, the triangle was devised and popularized in the 1950s by psychiatrist Lionel Penrose and his son, prominent Nobel Prize-winning mathematician Sir Roger Penrose, who described it as “impossibility in its purest form”.

Penrose Triangle

Recaman Sequence

Recamán’s sequence is a well known sequence defined by a recurrence relation, because its elements are related to the previous elements in a straightforward way, they are often defined using recursion.

Values of the sequence can be associated with musical notes, in such that case the running of the sequence can be associated with an execution of a musical tune.

The calculation of terms of the sequence can be programmed.

Recaman Sequence

Prime Knot

Prime knot is a knot (embedding of a topological circle in 3-dimensional Euclidean space, considered up to continuous deformations) that is, in a certain sense, indecomposable.

Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be composite knots or composite links.

It can be a nontrivial problem to determine whether a given knot is prime or not.

Prime Knot

Poncelet Porism

Poncelet’s porism states that whenever a polygon is inscribed in one conic section and circumscribes another one, the polygon must be part of an infinite family of polygons that are all inscribed in and circumscribe the same two conics.

Poncelet’s porism can be proved by an argument using an elliptic curve, whose points represent a combination of a line tangent to one conic and a crossing point of that line with the other conic.

Poncelet Porism

Borromean Rings

Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the three is cut or removed.

Most commonly, these rings are drawn as three circles in the plane, in the pattern of a Venn diagram, alternatingly crossing over and under each other at the points where they cross.

Other triples of curves are said to form the Borromean rings as long as they are topologically equivalent to the curves depicted in this drawing.

Borromean Rings

Golden Ratio

Golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio.

That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes.

Approximate logarithmic spirals can occur in nature, for example the arms of spiral galaxies – golden spirals are one special case of these logarithmic spirals, although there is no evidence that there is any general tendency towards this case appearing.

Golden Ratio

Mandelbrot Set

The Mandelbrot set is the set of complex numbers c for which the function fc(z) = z2 + c does not diverge when iterated from z = 0, i.e., for which the sequence fc(0), fc(fc(0)), etc., remains bounded in absolute value.

Images of the Mandelbrot set exhibit an elaborate and infinitely complicated boundary that reveals progressively ever-finer recursive detail at increasing magnifications, making the boundary of the Mandelbrot set a fractal curve.

The “style” of this repeating detail depends on the region of the set being examined. Mandelbrot set images may be created by sampling the complex numbers and testing, for each sample point c, whether the sequence fc(0), fc(fc(0))  goes to infinity.

Mandelbrot Set

Tesseract

Tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square.

Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells.

The tesseract is one of the six convex regular 4-polytopes. It is the four-dimensional hypercube, or 4-cube as a part of the dimensional family of hypercubes or measure polytopes.

Tesseract

Schroeder´s stairs

Schroeder´s stairs is an optical illusion which is a two-dimensional drawing which may be perceived either as a drawing of a staircase leading from left to right downwards or the same staircase only turned upside down, a classical example of perspective reversal in psychology of perception.

Drawing of Schroeder stairs may be variously described as an “ambiguous figure”, “reversible figure” or “bistable figure”.

Schroeder´s stairs

Calculus

Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.

It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while integral calculus concerns accumulation of quantities, and areas under or between curves.

Calculus

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