Back to Main Index Fractalary: Fractals from Planet to Atoms Patent Application (The Netherlands) by Julius J.C.M. Ruis, number 10 33 147 d.d. 29 December 2006 The use of Fractal Geometry in Rapid Prototyping and Tissue-Engineering of artificial human and/or animal organs, more specifically human blood vessels Click here for prototype artificial blood vessel
Julia Set		Mandelbrot Set	Julius Ruis Set
Introduction and summary What is a Fractal? A fractal is a geometric object (like a line or a circle) which is however rough or irregular on all scales of length, and so which appears to be 'broken up' in a radical way. Some of the best examples can be divided into parts, each of which is similar to the original object. Fractals are said to possess infinite detail, and they may actually have a self-similar structure that occurs at different levels of magnification.  The most well-known fractals are the Mandelbrot Set and Julia Sets. Jules Ruis developed the so called Julius Ruis Set. This is a smart presentation of 400 Julia sets, showing that the Mandelbrot Set is the parameter basin of all closed Julia Sets. Why Fractal Geometry? Fractals provide scientists with a new vocabulary to read the book of nature. Galileo's circles and triangles are insufficient to describe nature in all its rugged complexity. In addition, the fact that natural objects are commonly self-similar, makes fractals ideal models for many of those objects. Fractal geometry also provides scientists with a new way of looking and experimenting with old problems using a different perspective. What is most exciting about fractals is that they successfully bring geometry to where it did not appear to belong, an idea reminiscent of general relativity, which is based on the introduction of geometry to understand the cosmos. What is Fractal Geometry good for? Fractal geometry is a compact way of encoding the enormous complexity of many natural objects. By iterating a relatively simple construction rule, we see how an original simple object can be transformed into an enormously complex one by adding ever increasing detail to it, at the same time preserving affinity between the whole and the parts, or scale invariance. Just think of a big oak tree in winter. Its branches are naked so it is easy to distinguish the way in which a twig splits and becomes two which then split again, to become four; in much the same way in which the trunk first split into slender branches which split again and then again, and again. The self-similarity is evident, the whole looks just like its parts, yet not exactly. Nature has slightly altered the construction rule, introducing some degree of randomness which will make one oak slightly different from any other oak tree in the world. Now, imagine packing all the information required by the tree to become a beautiful large oak tree into the smallest possible space, with the greatest economy of means. It would appear logical that rather than encoding all the unique, intricate complex branching of a mature oak in its seed (an acorn), all the details of its evolving shape, nature simply encodes the splitting rule, and the urge to repeat it, to iterate. This, plus a little randomness during growth that changes the number of splits or their place in a branch is enough to create a unique oak tree. In fact a whole computer data compression industry is based on similar ideas that permit coding and compressing large files to be quickly transmitted through the Internet. Patent application Jules Ruis, managing director of Fractal Consultancy (The Netherlands), filed end 2006 a patent application on a procedure using fractal geometry for the design and manufacturing of artificial human and/or animal organs, more specifically human blood vessels. The designed structures can be presented in a two-dimensional as well as a three-dimensional way. The patent pending also emphasizes the use of fractal geometry for the direction of print- and injectionheads in equipment used for the application of materials (inkjet printing and methods of direct writing), and equipment that directs laserbeams and electronic beams (electron microscopes). More information Jules Ruis, Fractal Consultancy, Son-Eindhoven, The Netherlands, tel. +31 499 47 10 55;  internet: www.fractal.org e-mail: Jules.Ruis@fractal.org
Content 1. Fractal Design Cycle 2. The Fractal Nature of Nature (from Planet to Atoms) 3. Fractal Images of Human Organs 4. Print me a Heart and a Set of Arteries 5. Fractal Structure of DNA and smaller particles
Fractal Design Cycle and Fractalary of fractal data sets
Main Index of examples Nature/Body - Universe - Milky Way and the sun - Sun and its planets - Earth - Clouds/mountains/rivers - Butterfly/zebra/peacock - Pyramid/Sagrada familia - Dicentra spectabilis/cactus - Fungus/flower/cauliflower - Tree - Human body and organs - Brain/eye/ear/ - Heart/blood vessel/lungs - Digestion/intestines/liver - Stomach/colon/kidneys - Bladder/trabecular bone - Cell/neuron/mitochondrium - Bacterium/virus - DNA/RNA/protein - Dendrimer - Molecule/atom/particle	Fractal Geometry - Fractal Geometry is the iteration of complex functions like (inverted) polynomials (z^2, z^3, etc) and complex (inverted) transcendental functions  (sin(z), cos(z), tan(z), exp(z) ). - A complex number has the form z=x + i*y or c=a + i*b with i^2 =-1 - In fractal formulas zn+1 means z(new) and zn means z(old). - Formula in  function is iterated from 1 to maximal ‘k’ times. - Iteration goes on until predeter-mined small/great value has been reached (function is going to zero or infinity). - Quantity of real done iterations is called ‘f’ (‘flightnumber’). - Instruction at the end of the procedure, coupled on reached ‘f’, is : pset color, position machine printhead (ink, matter, cell or molecule, etc), manipulate beam/bundle or position motor. - Calculate next computer-pixel and manipulate next machine-voxel, layer for layer.		Fractal Imaginator - The Fractal Imaginator is a software program to create fractals. - Using the program Fi you can input your own mathematical formulas and other relevant data. - The created images are saved as bmp/jpg/png files or obj/stl/pov files. - The parameters of the image are stored in separated data-files (.fim files). - This way of storing saves much computer capacity. - After installation of Fi on your own computer the Fi program will automatically start by clicking the .fim files (just like Adobe pdf files). - Buy program: - http://www.mysticfractal.com/FractalImaginator.html - Download free trial version: - http://www.mysticfractal.com/fractalimaginator_trial.htm
The fractal nature of nature,  from planet to atoms The Universe as a Fractal Space
http://www.fractal.org/Bewustzijns-Besturings-Model/Fractal-Nature.pdf
			- Is the Universe a Fractal Space ? - Article in New Scientist d.d. 9 March 2007: - http://www.fractal.org/Bewustzijns-Besturings-Model/Is-the-universe-a-fractal-space.htm - For fractals in general see: - http://classes.yale.edu/fractals/worldOfFractals.html
Univers	Julia: zn+1 = zn^2 + c (iteration = 200)		http://www.fractal.org/Julius-Ruis-Gallery/Fractalary/Univers.fim
			- The Milky Way system is a spiral galaxy consisting of over 400 billion stars , plus gas and dust arranged into the halo, the nuclear bulge and the disk, which contains the majority of the stars, including the sun. - The best fractal formula for a spiral is zn+1 = zn^2+ c at a point with x>0 and small y=0,04
Milky Way and the Sun	Julia: zn+1 = zn^2 + c (iterations = 300)		Http://www.fractal.org/Julius-Ruis-Gallery/Fractalary/Spiral.fim
			- The Sun possesses 9 planets: Pluto, Neptune, Uranus, Saturn, Jupiter, Mars, Earth, Venus and Mercury. - On earth we see much more celestial bodies at the starry sky. - Only the tan-functions give the fractal structure with balls.
Sun and its planets	Julia: zn+1 = tan(zn) * c		http://www.fractal.org/Julius-Ruis-Gallery/Fractalary/sun.fim
			- The Earth is the only planet of the sun on which Life is possible. - The inner side of the earth consists out of 4 different layers. - See also Google Earth - The showed fractal is the ‘superformula’ near c=0 with iterations=215
Earth	Julia: zn+1 = 1/(cos(zn)^2 -sin(zn)^2)^(5) + c		http://www.fractal.org/Julius-Ruis-Gallery/Fractalary/earth.fim
Benoit Mandelbrot: "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."
Clouds and  mountains	River		Trees
			- The Pyramids in Egypt were constructed as images of the stars at the sky. - So a connection was realised between heaven and earth. - Maybe the pyramids were the first sign of the fractal structure of space. - The so called Sierpinski fractal looks like a pyramid.
Pyramid of Cheops	Julia: if (x>=0) then zn+1=( zn-1) * c, else zn+1 =  (zn +1) * conj.c		http://www.fractal.org/Julius-Ruis-Gallery/Fractalary/Pyramid.fim
			- La Sagrada Familia (Cathedral in Barcelona) was developed by Gaudi. - The architectural design (inside and outside) is completely nature-inspired (=fractal). - The complex transcendental functions,  especially the sine and cosine, give the best impression of  this man-made construction.
Sagrada Familia (Barcelona)	3D Juliaquat: zn+1 = sin(zn) * c		http://www.fractal.org/Julius-Ruis-Gallery/Fractalary/Sagrada.fim
Animals and repeating patterns
Butterfly	Zebra		Peacock
			- The middle fractal  is the image of the most popular fractal: the Mandelbrot Set (after the Polish Frenchman Benoit Mandelbrot ). - Zooming in at the border of the fractal gives new mini-M-sets. - The M-set looks like the left showed flower, called the dicentra spectabilis (in Dutch: broken heart).
Dicentra spectabilis (gebroken hartje)	Mandelbrot: zn+1 = zn^2+ c for c=(0,0)		http://www.fractal.org/Julius-Ruis-Gallery/Fractalary/Mandel-z2.fim
			- The middle fractal  is called the  Julia set (after the Frenchman Gaston Julia). - The Julia set has been constructed in the complex mathematical plane. - We distinguish ‘closed’ and ‘open’ Julia sets. - The cactus plant in nature has a Julia fractal structure.
Cactus	Julia: zn+1 = zn^2+ c for c = (-1,0)		http://www.fractal.org/Julius-Ruis-Gallery/Fractalary/Julia-set.fim
			- The middle fractal is the so called Julius Set: a smart presentation of 400 Julia sets, showing that the Mandelbrot set is the parameter basin of all closed Julia sets. - The whole is reflected in each part. Each part is projected in the whole. See showed set of cacti at the left.
Set of cacti	Julius: zn+1 = zn^2+ c		http://www.fractal.org/Julius-Ruis-Gallery/Fractalary/Julius-z2.fim
			- Julia sets can be showed with and without external orbits. - Julia sets with external orbits get the addition ‘Ruis’ (i.e. ‘noise’).   - A Julius set can be shown as 400 Julia sets with orbits.  This set is called a Julius Ruis Set (after Jules Ruis from The Netherlands).
Plant dicentra spectabilis	Julius Ruis: zn+1 = zn^2+ c		See for complete Julius Ruis Gallery: http://www.fractal.org/Julius-Ruis-Gallery/Index-Gallery.htm
			- The fungus is an example of the form that belongs to the exponential functions. - A repeating pattern can be found along the y-axis, so x=0, especially on y-values that are  multiplications of pi (=3,1415). - In many cases mathematical fractals show an attractor. This are special points, with moving orbits around it, like a rainbow.
Fungus	Mandelbrot: zn+1 = exp(zn) * c		http://www.fractal.org/Julius-Ruis-Gallery/Fractalary/Mandel-exp.fim
			- Formulas for inverted sine and cosine are very suitable for the creation of fractal flowers. - A special ‘superformula’ has been developed by Jules Ruis.. - The number of leaves is equal to the power of the sine/cosine. - The colours can be changed by a special parameter.
Flower	Julia: zn+1 = c / ( (cos(zn)^2)- (sin(zn)^2))^5 (power = 10)		http://www.fractal.org/Julius-Ruis-Gallery/Fractalary/Flower.fim
			- Cauliflowers, broccoli and romansco are examples of food with a typical fractal tree-like structure. - The fractal image is constructed with a so called ‘if-then-else’ formula. - The branches on the stem originate from the power of two for  (zn -1) or (zn+1).
Cauliflower	Julia: if (x>=0) then zn+1 = ((zn -1)^2) / c, else zn+1 = ((zn +1)^2) / conj.c		http://www.fractal.org/Julius-Ruis-Gallery/Fractalary/r8^2.fim
			- A tree is the most typical example of fractals in living nature (biology). - A very simple mathematical formula zn+1 = (zn +1)/c or zn+1  = (zn –1)/c for resp. x>0 and x<0 leads to this beautiful fractal form. - The number of iterations is related to the number of branches.
Tree	Julia: if x>0 then zn+1 = (zn +1)/c else zn+1 = (zn -1)/conj.c		http://www.fractal.org/Julius-Ruis-Gallery/Fractalary/Tree-it=10.fim
Fractals support growing organs Today scientists can regenerate tissue such as skin, but they are still figuring out how to grow replacement organs. The challenge is in coaxing cells from organs to grow into new organs rather than unstructured clusters of cells.	Researchers found a way to impart structure to growing cells that may eventually allow for growth of entire organs. If the method proves successful, "we can use [a] patient's own cells to create a living organ and this will remove the problems with organ rejections" and a shortage of donor organs, said Mohammed Kaazempur-Mofrad, a researcher at MIT Key to the method is supporting the growing cells with something akin to the circulatory system, which provides cells with oxygen and nutrients. "In order to make living replacements for large vital organs such as the liver and kidney, it is essential to integrate the creation of vasculature with the tissue engineering," said Kaazempur-Mofrad. And the growth of these vascular networks has to be highly controlled and precise, he said.		The researchers used computer-generated fractal patterns to fabricate a network of branching, microscopic tubes. Fractals are patterns that repeat at different scales. If, for instance, one portion of a fractal looks like a tree, zooming in on its branches and twigs will show that they also look like trees, and zooming further will show that their branches and twigs follow the same pattern. These self-similar patterns are common in nature, including natural blood vessel networks, and can scale up or down in size. "Using [the] fractal concept will make it easier to mimic... nature and also to scale up our designs from one animal to another," said Kaazempur-Mofrad. See article: http://www.fractal.org/Life-Science-Technology/Publications/Fractals-support-growing-organs.htm
			- Polynomials are functions of the form zn^2, zn^3, etc. and combinations of these. - Inverted function of  zn^2 is 1/ zn^2or written as zn^-2 - It is important to know that the inversion of a function means that all points inside the object go outside and recursively.
Human body within a body	Julia Ruis: zn+1 = c / (zn^3+ zn^2– zn) (inverted polynomial)		http://www.fractal.org/Julius-Ruis-Gallery/Fractalary/cdiv-poly.fim
Human organs	25 * Julia: if x>0 then zn+1 = (zn +1)/c else zn+1 = (zn -1)/conj.c		http://www.fractal.org/Julius-Ruis-Gallery/Fractalary/if-then-JR5.fim
			Using Fractal Geometry for Rapid Prototyping and Tissue Engineering - Fractal geometry is used for rapid prototyping of so called scaffolds. - Middle photo shows tree-like scaffolds (negative and positive) used for the creation of artificial blood vessels. - Biodegradable polymer scaffolds are seeded with endothelial cells and conditioned ‘in vitro’ in a bioreactor (Tissue Engineering). - After some weeks the artificial organ will be placed in human body and natural growth starts.
Brain blood vessels	3D scaffold in wax; Juliaquat: if x>0 then zn+1 = (zn +1)/c else zn+1 =  (zn -1)/conj.c		http://www.fractal.org\Fractal-tree-scaffold.htm
			Fractal Dimension - The dimensions of objects in euclidean mathematics are 1, 2 or 3 (resp. line, surface and content). - The fractal dimension is a non-integer e.g. 2,79 for the surface of the brain. - Due to the patented invention no CAD-files are needed any longer for printing 3D forms.
Brain surface	3D Juliaquat: if (x>=0) then zn+1 = (zn -3)/c, else zn+1 = (zn+3)/conj.c		http://www.fractal.org/Julius-Ruis-Gallery/Fractalary/Tree-quat.fim
			- Healthy brain (left) compared to brain tumor (shown in blue, right). - There are many types of brain tumors. - Fractal formula is zn^-2without brackets around ‘-2’.
Brain and malignant brain tumor	Julia: zn+1 = zn^-2+ c		http://www.fractal.org/Julius-Ruis-Gallery/Fractalary/malignant-tumor.fim