Max Planck Institute of Molecular Physiology
The Max Planck Institute of Molecular Physiology is located in Dortmund, Germany next to the Dortmund University of Technology. It is one of 80 institutes in the Max Planck Society; the Department of Mechanistic Cell Biology aims to better understand the molecular mechanisms of cell division and their regulation. The main focus is on the key proteins that control the division of chromosomes during mitosis, a process that separates sister chromatids into two identical daughter cells, thereby maintaining chromosome stability; the Department of Systemic Cell Biology studies the regulation of signal transduction processes in cells. These processes control significant cellular functions such as tissue growth or the differentiation of cells into specialized cell types and, determine the fate of each cell; the Department of Structural Biochemistry focuses on structural and functional analyses of biologically and medically relevant membrane proteins and macromolecular complexes. Special attention is given to the investigation into the molecular mechanisms of muscle contraction and the infection with bacterial toxins.
Furthermore, membrane proteins that play an important role in the synthesis and homeostasis of cholesterol in the body are examined. Research in the Department of Chemical Biology concentrates on the interface between organic chemistry and biology. By using biochemical and chemical techniques, researchers identify and develop new tools for the investigation of biologically relevant processes and phenomena. Structural Biology Physical Biochemistry Compound Management and Screening Center Dortmund Protein Facility Biotechnology Electron Microscopy Chemical Genomics Centre Official site Federation European Physiological Societies Official site European Federation for Pharmaceutical Sciences Official site European Federation for Medicinal Chemistry Official site Federation of European Biochemical Societies
In mathematics, the Menger sponge is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet, it was first described by Karl Menger in 1926, in his studies of the concept of topological dimension. The construction of a Menger sponge can be described. Divide every face of the cube into 9 squares, like a Rubik's Cube; this will sub-divide the cube into 27 smaller cubes. Remove the smaller cube in the middle of each face, remove the smaller cube in the center of the larger cube, leaving 20 smaller cubes; this is a level-1 Menger sponge. Repeat steps 2 and 3 for each of the remaining smaller cubes, continue to iterate ad infinitum; the second iteration gives a level-2 sponge, the third iteration gives a level-3 sponge, so on. The Menger sponge; the nth stage of the Menger sponge, Mn, is made up of 20n smaller cubes, each with a side length of n. The total volume of Mn is thus n; the total surface area of Mn is given by the expression 2n + 4n.
Therefore the construction's volume approaches zero. Yet any chosen surface in the construction will be punctured as the construction continues, so that the limit is neither a solid nor a surface; each face of the construction becomes a Sierpinski carpet, the intersection of the sponge with any diagonal of the cube or any midline of the faces is a Cantor set. The cross section of the sponge through its centroid and perpendicular to a space diagonal is a regular hexagon punctured with hexagrams arranged in six-fold symmetry; the number of these hexagrams, in descending size, is given by a n = 9 a n − 1 − 12 a n − 2, with a 0 = 1, a 1 = 6. The sponge's Hausdorff dimension is log 20/log 3 ≅ 2.727. The Lebesgue covering dimension of the Menger sponge is the same as any curve. Menger showed, in the 1926 construction, that the sponge is a universal curve, in that every curve is homeomorphic to a subset of the Menger sponge, where a curve means any compact metric space of Lebesgue covering dimension one.
In a similar way, the Sierpinski carpet is a universal curve for all curves that can be drawn on the two-dimensional plane. The Menger sponge constructed in three dimensions extends this idea to graphs that are not planar, might be embedded in any number of dimensions; the Menger sponge is a closed set. It has Lebesgue measure 0; because it contains continuous paths, it is an uncountable set. Formally, a Menger sponge can be defined as follows: M:= ⋂ n ∈ N M n where M0 is the unit cube and M n + 1:=. MegaMenger is a project aiming to build the largest fractal model, pioneered by Matt Parker of Queen Mary University of London and Laura Taalman of James Madison University; each small cube is made from 6 interlocking folded business cards, giving a total of 960 000 for a level-four sponge. The outer surfaces are covered with paper or cardboard panels printed with a Sierpinski carpet design to be more aesthetically pleasing. In 2014, twenty level-three Menger sponges were constructed, which
An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into some larger string of symbols, an initial "axiom" string from which to begin construction, a mechanism for translating the generated strings into geometric structures. L-systems were introduced and developed in 1968 by Aristid Lindenmayer, a Hungarian theoretical biologist and botanist at the University of Utrecht. Lindenmayer used L-systems to describe the behaviour of plant cells and to model the growth processes of plant development. L-systems have been used to model the morphology of a variety of organisms and can be used to generate self-similar fractals; as a biologist, Lindenmayer worked with yeast and filamentous fungi and studied the growth patterns of various types of bacteria, such as the cyanobacteria Anabaena catenula. The L-systems were devised to provide a formal description of the development of such simple multicellular organisms, to illustrate the neighbourhood relationships between plant cells.
On, this system was extended to describe higher plants and complex branching structures. The recursive nature of the L-system rules leads to self-similarity and thereby, fractal-like forms are easy to describe with an L-system. Plant models and natural-looking organic forms are easy to define, as by increasing the recursion level the form slowly'grows' and becomes more complex. Lindenmayer systems are popular in the generation of artificial life. L-system grammars are similar to the semi-Thue grammar. L-systems are now known as parametric L systems, defined as a tuple G =,where V is a set of symbols containing both elements that can be replaced and those which cannot be replaced ω is a string of symbols from V defining the initial state of the system P is a set of production rules or productions defining the way variables can be replaced with combinations of constants and other variables. A production consists of the predecessor and the successor. For any symbol A, a member of the set V which does not appear on the left hand side of a production in P, the identity production A → A is assumed.
The rules of the L-system grammar are applied iteratively starting from the initial state. As many rules as possible are applied per iteration; the fact that each iteration employs as many rules as possible differentiates an L-system from a formal language generated by a formal grammar, which applies only one rule per iteration. If the production rules were to be applied only one at a time, one would quite generate a language, rather than an L-system. Thus, L-systems are strict subsets of languages. An L-system is context-free if each production rule refers only to an individual symbol and not to its neighbours. Context-free L-systems are thus specified by a context-free grammar. If a rule depends not only on a single symbol but on its neighbours, it is termed a context-sensitive L-system. If there is one production for each symbol the L-system is said to be deterministic. If there are several, each is chosen with a certain probability during each iteration it is a stochastic L-system. Using L-systems for generating graphical images requires that the symbols in the model refer to elements of a drawing on the computer screen.
For example, the program Fractint uses turtle graphics to produce screen images. It interprets each constant in an L-system model as a turtle command. Lindenmayer's original L-system for modelling the growth of algae. Variables: A B constants: none axiom: A rules:, which produces: n = 0: A n = 1: AB n = 2: ABA n = 3: ABAAB n = 4: ABAABABA n = 5: ABAABABAABAAB n = 6: ABAABABAABAABABAABABA n = 7: ABAABABAABAABABAABABAABAABABAABAAB n=0: A start / \ n=1: A B the initial single A spawned into AB by rule, rule couldn't be applied /| \ n=2: A B A former string AB with all rules applied, A spawned into AB again, former B turned into A / | | | \ n=3: A B A A B note all A's producing a copy of themselves in the first place a B, which turns... / | | | \ | \ \ n=4: A B A A B A B A... into an A one generation starting to spawn/repeat/recurse The result is the sequence of Fibonacci words. If we count the length of each string, we obtain the famous Fibonacci sequence of numbers: 1 2 3 5 8 13 21 34 55 89...
For each string, if we count the k-th position from the left end of the string, the value is determined by whether a multiple of the golden ratio falls within the interval. The ratio of A to B converges to the golden mean; this example yields the same result if the rule is replaced with, except that the strings are mirrored. This sequence is a locally catenative sequence because G = G G, where
Filled Julia set
The filled-in Julia set K of a polynomial f is: a Julia set and its interior, non-escaping set The filled-in Julia set K of a polynomial f is defined as the set of all points z of the dynamical plane that have bounded orbit with respect to f K = d e f where: C is the set of complex numbers f is the k -fold composition of f with itself = iteration of function f The filled-in Julia set is the complement of the attractive basin of infinity. K = C ∖ A f The attractive basin of infinity is one of the components of the Fatou set. A f = F ∞ In other words, the filled-in Julia set is the complement of the unbounded Fatou component: K = F ∞ C; the Julia set is the common boundary of the filled-in Julia set and the attractive basin of infinity J = ∂ K = ∂ A f where: A f denotes the attractive basin of infinity = exterior of filled-in Julia set = set of escaping points for f A f = d e f. If the filled-in Julia set has no interior the Julia set coincides with the filled-in Julia set; this happens. Such critical points are called Misiurewicz points.
The most studied polynomials are those of the form f = z 2 + c, which are denoted by f c, where c is any complex number. In this case, the spine S c of the filled Julia set K is defined as arc between β -fixed point and − β, S c = with such properties: spine lies inside K; this makes sense when K is connected and full spine is invariant under 180 degree rotation, spine is a finite topological tree, Critical point z c r = 0 always belongs to the spine. Β -fixed point is a landing point of external ray of angle zero R 0 K, − β is landing point of external ray R 1 / 2 K. Algorithms for constructing the spine: detailed version is described by A. Douady Simplified version of algorithm: connect − β and β within K by an arc, when K has empty interior arc is unique, otherwise take the shortest way that contains 0. Curve R: R = d e f R 1 / 2 ∪ S c ∪ R 0 divides dynamical plane into two components. Airplane Douady rabbit dragon basilica or San Marco fractal cauliflower dendrite Siegel disc Peitgen Heinz-Otto, Ri
The Mandelbulb is a three-dimensional fractal, constructed by Daniel White and Paul Nylander using spherical coordinates in 2009. A canonical 3-dimensional Mandelbrot set does not exist, since there is no 3-dimensional analogue of the 2-dimensional space of complex numbers, it is possible to construct Mandelbrot sets in 4 dimensions using quaternions and bicomplex numbers. White and Nylander's formula for the "nth power" of the vector v = ⟨ x, y, z ⟩ in ℝ3 is v n:= r n ⟨ sin cos , sin sin , cos ⟩ where r = x 2 + y 2 + z 2, ϕ = arctan = arg , θ = arctan = arccos ; the Mandelbulb is defined as the set of those c in ℝ3 for which the orbit of ⟨ 0, 0, 0 ⟩ under the iteration v ↦ v n + c is bounded. For n > 3, the result is a 3-dimensional bulb-like structure with fractal surface detail and a number of "lobes" depending on n. Many of their graphic renderings use n = 8. However, the equations can be simplified into rational polynomials. For example, in the case n = 3, the third power can be simplified into the more elegant form: ⟨ x, y, z ⟩ 3 = ⟨ x x 2 + y 2, y x 2 + y 2, z ⟩.
The Mandelbulb given by the formula above is one in a family of fractals given by parameters given by: v n:= r n ⟨ sin cos , sin sin , cos ⟩ Since p and q do not have to equal n for the identity |vn|=|v|n to hold. More general fractals can be found by setting v n:= r n ⟨ sin cos , sin sin , cos ⟩ for functions f and g. Ot
In mathematics, a fractal is a subset of a Euclidean space for which the Hausdorff dimension exceeds the topological dimension. Fractals tend to appear nearly the same at different levels, as is illustrated here in the successively small magnifications of the Mandelbrot set. Fractals exhibit similar patterns at small scales called self similarity known as expanding symmetry or unfolding symmetry. One way that fractals are different from finite geometric figures is the way. Doubling the edge lengths of a polygon multiplies its area by four, two raised to the power of two. If the radius of a sphere is doubled, its volume scales by eight, two to the power of three. However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power, not an integer; this power is called the fractal dimension of the fractal, it exceeds the fractal's topological dimension. Analytically, fractals are nowhere differentiable. An infinite fractal curve can be conceived of as winding through space differently from an ordinary line – although it is still 1-dimensional, its fractal dimension indicates that it resembles a surface.
Starting in the 17th century with notions of recursion, fractals have moved through rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano, Bernhard Riemann, Karl Weierstrass, on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century. The term "fractal" was first used by mathematician Benoit Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus, meaning "broken" or "fractured", used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature. There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard useful. That's fractals." More formally, in 1982 Mandelbrot stated that "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension exceeds the topological dimension."
Seeing this as too restrictive, he simplified and expanded the definition to: "A fractal is a shape made of parts similar to the whole in some way." Still Mandelbrot settled on this use of the language: "...to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants". The consensus is that theoretical fractals are infinitely self-similar and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth. Fractals are not limited to geometric patterns, but can describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images and sounds and found in nature, art and law. Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractals; the word "fractal" has different connotations for laymen as opposed to mathematicians, where the layman is more to be familiar with fractal art than the mathematical concept.
The mathematical concept is difficult to define formally for mathematicians, but key features can be understood with little mathematical background. The feature of "self-similarity", for instance, is understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer invisible, new structure. If this is done on fractals, however, no new detail appears. Self-similarity itself is not counter-intuitive; the difference for fractals is. This idea of being detailed relates to another feature that can be understood without mathematical background: Having a fractal dimension greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric shapes are perceived. A regular line, for instance, is conventionally understood to be one-dimensional. A solid square is understood to be two-dimensional. We see that for ordinary self-similar objects, being n-dimensional means that when it is rep-tiled into pieces each scaled down by a scale-factor of 1/r, there are a total of rn pieces.
Now, consider the Koch curve. It can be rep-tiled into four sub-copies, each scaled down by a scale-factor of 1/3. So by analogy, we can consider the "dimension" of the Koch curve as being the unique real number D that satisfies 3D = 4, which by no means is an integer! This number is; the fact th
The Barnsley fern is a fractal named after the British mathematician Michael Barnsley who first described it in his book Fractals Everywhere. He made it to resemble Asplenium adiantum-nigrum; the fern is one of the basic examples of self-similar sets, i.e. it is a mathematically generated pattern that can be reproducible at any magnification or reduction. Like the Sierpinski triangle, the Barnsley fern shows how graphically beautiful structures can be built from repetitive uses of mathematical formulas with computers. Barnsley's 1988 book Fractals Everywhere is based on the course which he taught for undergraduate and graduate students in the School of Mathematics, Georgia Institute of Technology, called Fractal Geometry. After publishing the book, a second course was called Fractal Measure Theory. Barnsley's work has been a source of inspiration to graphic artists attempting to imitate nature with mathematical models; the fern code developed by Barnsley is an example of an iterated function system to create a fractal.
This follows from the collage theorem. He has used fractals to model a diverse range of phenomena in science and technology, but most plant structures. IFSs provide models for certain plants and ferns, by virtue of the self-similarity which occurs in branching structures in nature, but nature exhibits randomness and variation from one level to the next. V-variable fractals allow for such randomness and variability across scales, while at the same time admitting a continuous dependence on parameters which facilitates geometrical modelling; these factors allow us to make the hybrid biological models......we speculate that when a V -variable geometrical fractal model is found that has a good match to the geometry of a given plant there is a specific relationship between these code trees and the information stored in the genes of the plant. —Michael Barnsley et al. Barnsley's fern uses four affine transformations; the formula for one transformation is the following: f = + Barnsley shows the IFS code for his Black Spleenwort fern fractal as a matrix of values shown in a table.
In the table, the columns "a" through "f" are the coefficients of the equation, "p" represents the probability factor. These correspond to the following transformations: f 1 = f 2 = + f 3 = + f 4 = + Though Barnsley's fern could in theory be plotted by hand with a pen and graph paper, the number of iterations necessary runs into the tens of thousands, which makes use of a computer mandatory. Many different computer models of Barnsley's fern are popular with contemporary mathematicians; as long as the math is programmed using Barnsley's matrix of constants, the same fern shape will be produced. The first point drawn is at the origin and the new points are iteratively computed by randomly applying one of the following four coordinate transformations:ƒ1 xn + 1 = 0yn + 1 = 0.16 yn. This coordinate transformation is chosen 1% of the time and just maps any point to a point in the first line segment at the base of the stem; this part of the figure is the first to be completed in during the course of iterations.
Ƒ2 xn + 1 = 0.85 xn + 0.04 ynyn + 1 = −0.04 xn + 0.85 yn + 1.6. This coordinate transformation is chosen 85% of the time and maps any point inside the leaflet represent