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Fractalary: Fractals from Planets 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 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: e-mail:




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

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

-          Unparticles

Fractal Geometry


-          Fractal Geometry is the iteration of complex functions like (inverted) polynomials (z2, z3, 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 i2 =-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:



-          Download free trial version:








The fractal nature of nature,  from planet to atoms


The Universe as a Fractal Space



-          Is the Universe a Fractal Space ?

-          Article in New Scientist d.d. 9 March 2007:



-          For fractals in general see:



Julia: zn+1 = zn2 + c

(iteration = 200)




-          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 = zn2 + c at a point with x>0 and small y=0,04

Milky Way and the Sun

Julia: zn+1 = zn2 + c

(iterations = 300)






-          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




-          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



Julia: zn+1 = 1/(cos(zn)2 -sin(zn)2)^(5) + c




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









-          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




-          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





Animals and repeating patterns









-          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 = zn2 + c

for c=(0,0)




-          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.


Julia: zn+1 = zn2 + c

for c = (-1,0)




-          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 = zn2 + c




-          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 = zn2 + c


See for complete Julius Ruis Gallery:




-          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.


Mandelbrot: zn+1 = exp(zn) * c




-          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.



Julia: zn+1 = c / ( (cos(zn)2)- (sin(zn)2))^5 (power = 10)





-          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).



Julia: if (x>=0) then zn+1 = ((zn -1)2) / c, else zn+1 = ((zn +1)2) / conj.c^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.


Julia: if x>0 then zn+1 = (zn +1)/c else zn+1 = (zn -1)/conj.c




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:




-          Polynomials are functions of the form zn2, zn3, etc. and combinations of these.

-          Inverted function of  zn2 is 1/ zn2 or 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 / (zn3 + zn2 zn)

(inverted polynomial)




Human organs


25 * Julia: if x>0 then zn+1 = (zn +1)/c else zn+1 = (zn -1)/conj.c




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\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




-          Healthy brain (left) compared to brain tumor (shown in blue, right).

-          There are many types of brain tumors.

-          Fractal formula is zn-2 without brackets around ‘-2’.

Brain and malignant brain tumor

Julia: zn+1 = zn-2+ c




-          Polynomials near tot the point c=0 look like a mandala.

-          Each mandala possesses three repeating characteristics:


* symmetry

* essential points

- The eye retina looks like a mandala fractal structure.

Eye Retina

Julia: zn+1 = zn6 + c




-          Several objects in nature (shells etc.) have a spiral form.

-          Also human ears possess a spiral ‘cochlea’ inside the head.

-          A spiral is related to the figure phi; phi = 1,618 i.e. the Golden Ratio.

-          This ratio is derived from the Fibonaccio set of numbers: 1, 2, 3, 5, 8, 13, 21, 34, 55, ……….

-          Each next number is the addition of the two former numbers.

-          The division of a number by its former number delivers always the result of phi.

-          The showed 3D fractal is constructed with formula of sin(z) and subtraction of sinh(z).

Ear and its inner structure

3D Juliaquat: zn+1 = (sin(zn) - sinh(zn)+c   and

Julia: zn+1 = zn2 + c^2+c.fim




-          Complex sine and cosine are the most suitable functions for the design and construction of  3D artificial blood vessels.

-          A special ‘superformula’ has been developed.

-          Scaffolds are printed with biodegradable polymers.

-          Methods of ‘Direct Write’ are under development.

Heart and its blood vessels

3D Juliaquat: zn+1 = c / (cos(zn)2 -  (sin(zn)2)^(2)

Print me a heart and a set of arteries (publication d.d. April 2006)

Sitting in a culture dish, a layer of chicken heart cells beats in synchrony. But this muscle layer was not sliced from an intact heart, nor even grown laboriously in the lab. Instead, it was "printed", using a technology that could be the future of tissue engineering.

Gabor Forgacs, a biophysicist at the University of Missouri in Columbia, described his "bioprinting" technique last week at the Experimental Biology 2006 meeting in San Francisco. It relies on droplets of "bioink", clumps of cells a few hundred micrometres in diameter, which Forgacs has found behave just like a liquid.

This means that droplets placed next to one another will flow together and fuse, forming layers, rings or other shapes, depending on how they were deposited. To print 3D structures, Forgacs and his colleagues alternate layers of supporting gel, dubbed "biopaper", with the bioink droplets. To build tubes that could serve as blood vessels, for instance, they lay down successive rings containing muscle and endothelial cells, which line our arteries and veins. "We can print any desired structure, in principle," Forgacs told the meeting.

Other tissue engineers have tried printing 3D structures, using modified ink-jet printers which spray cells suspended in liquid (New Scientist, 25 January 2003, p 16). Now Forgacs and a company called Sciperio have developed a device with printing heads that extrude clumps of cells mechanically so that they emerge one by one from a micropipette. This results in a higher density of cells in the final printed structure, meaning that an authentic tissue structure can be created faster.

Cells seem to survive the printing process well. When layers of chicken heart cells were printed they quickly begin behaving as they would in a real organ. "After 19 hours or so, the whole structure starts to beat in a synchronous manner," says Forgacs.

Most tissue engineers trying to build 3D structures start with a scaffold of the desired shape, which they seed with cells and grow for weeks in the lab. This is how Anthony Atala of Wake Forest University in Winston-Salem, North Carolina, and his colleagues grew the bladders which he successfully implanted into seven people (New Scientist, 8 April 2006, p 10). But if tissue engineering goes mainstream, faster and cheaper methods will be a boon. "Bioprinting is the way to go," says Vladimir Mironov, a tissue engineer at the Medical University of South Carolina in Charleston.





-          The left photo shows the internal (and external) anatomy of a blood vessel and has been artificial designed (fractal in the middle).

-          The used function is an inverted polynomial.

-          The number of furrows depends on the power of the function.

-          Walls thickness is determined by the number of iterations.

Blood vessel anatomy

Julia: zn+1 = zn-2 + c

 (iterations = 8)^-2-it=8.fim




-          The lung is  the best example of a tree-structured organ in human body.

-          The middle image is a stem with branches.

-          The tree is constructed with the so called ‘if-then-else’ function.

-          The number of branches depends on the quantity of iterations.

Lungs and its branches [1]

Julia: if x>0 then zn+1 = (zn +1)/c else zn+1 = (zn -1)/conj.c

(it. = 7; without orbits)




-          The lung in more detail still shows a fractal form.

-          The color is created by setting on  the orbits-parameter.

-          Each branch has its own color following the color-spectrum.

-          The number of branch-bifurcations has been increased by changing the number of  iterations.

Lungs and its branches [2]

Julia: if x>0 then zn+1 = (zn +1)/c else zn+1 = (zn -1)/conj.c

 (it. = 10; with orbits)





-          The spleen is similar to a lymph node in shape and structure but is much larger. In fact, it is the largest  lymphatic organ in the body.  It is responsible for the destruction of old red blood cells.  It is a major site for mounting the immune response.


Spleen and lymphatic system









-          Digestion and intestines from mouth to anus.


-          The so called ‘Superformula’ has been developed for the creation of different kind of tubes, usable for gullet and colons.

Digestion and intestines

3D Juliaquat: zn+1 = c * (cos(zn)2 - sin(zn)2)^(2)




-          The stomach is the first stop for the food after the esophagus.

-          Once the food gets to the stomach the stomach uses chemicals to try to make the food tinier.

-          The showed fractal is a special so called ‘if-then-else’ formula for x#>x.

-          The fractal is a 3D construction (i.e. .obj or .stl file)



3D Juliaquat: if (x#>x) then zn+1 = (zn -1) * c, else zn+1 = (zn +1) * conj.c




-          The liver is used for the creation of chemicals for breaking off the food.

-          The used formula is the standard ‘if-then-else’ formula for a tree.

-          We distinguish two kinds of bloodvessels: arteries (coming from the heart) and  veins (going to the heart).


Julia: if x>0 then zn+1 = (zn +1)/c else zn+1 = (zn -1)/conj.c

(without orbits)






-          The pancreas is located behind the liver and is where the hormone insulin is produced.

-          Insulin is used by the body to store and utilize glucose.









-          The liver produces bile which aids in the digestion of fats. The bile travels through tiny canals which eventually drain through the common bile duct into the small intestine.

-          The gallbladder stores excess bile that is not immediately needed for digestion.







-          The small intestine is the part of the gastrointestinal tract between the stomach and the large intestine and comprise the duodenum,  jejunum and ileum. It is where the vast majority of digestion takes place.

-          The fractal formula is a Julia set of a standard polynomial with orbits.

Small intestine structure

Julia: zn+1 = zn12 + c^12.fim




-          The large intestine or colon, is a muscular tube that begins at the end of the small intestine and runs to the rectum. The colon absorbs water from liquid stool that is delivered to it from the small intestine.

-          The fractal formula for the 3D Julia set is the ‘if-then-else’ formula for (z+1) or (z-1).



3D Juliaquat: if x>0 then zn+1 = (zn +1)/c else zn+1 = (zn -1)/conj.c




-          The colon is a long hollow organ lined with mucous membrane (mucosa). Muscle layers wrap around the entire length and help move food material through to the rectum.

-          The fractal formula used is the so called inversed superformula.

-          The showed image is a 3D presentation.

Colon structure

3D Juliaquat: zn+1 = c / (cos(zn)2 -  (sin(zn)2)^( 2)




-          The kidneys are ‘bean-shaped’ organs.

-          The kidneys filter wastes from the blood and excrete them, along with water as urine.

-          The showed images are two  turned fractal trees with the ‘if-then-else’ structure.


2 * Julia: if x>0 then zn+1 = (zn +1)/c else zn+1 = (zn -1)/conj.c (with orbits)^2-z-quat.fim







-          The bladder has an outer wall of muscle and connective tissue

-          Nerves in the wall monitor bladder filling and signal the body to urinate.

-          Like other organs the bladder's inner surface is lined with epithelial cells.

-          The formula for the fractal image is an subtraction of two polynomials, one for the circle and one for the inverted z^2.

-          Also the internal structure of the fractal has a beautiful shape.



3D Juliaquat: zn+1 = c * zn - (1/zn2 * c)







-          The overall architecture of bone is divided into inside cancellous bone (also referred to as trabecular bone) and cortical bone (shell).

-          Trabecular bone has a branching pattern, as seen in the vertebral specimen (left image).

-          For more details see:

Trabecular bone


3D Juliaquat: if (x>=0) then zn+1 = (zn -1)/c, else zn+1 = (zn +1)/conj.c




-          An ovum is a female reproductive cell. This cell is the largest cell in human body.

-          The middle image is a fractal made of tan(z) on the position a=0 and b=3,298575

-          The exp(z) gives also a repeated pattern on the y-axis on multiplied positions of  1/10 pi (0,31415)

Ovum (egg cell)

Julia: zn+1 = c * tan (zn)




-          The human cell consists of a nucleus embedded in cytoplasm and surrounded by a membrane.

-          Different  organelles are driving in the cytoplasm. e.g. the mitochondria that are the ‘power factories’ of the cell.

-          The fractal formula for the middle image is a polynomial divided by another polynomial.

Animal/human cell

Julia: zn+1 = (3 * (zn3 + 3)/(2 *(zn5 –5 ) * c




-          Neurons (also known as nerve cells) are electrically excitable cells in the nervous system that process and transmit information.

-          Neurons are typically composed of a soma, or cell body, a dendritic tree and an axon.

-          The fractal formula is of the form ‘if-then-else’ composed with sin(n)+1 and sin(n)-1.

Neuron (dendrites and axon)

Julia: if x>0 then zn+1 = (sin(zn)+1)/c else zn+1 = (sin(zn)-1)/conj.c




-          Mitochondria are complex organelles that convert energy from food into a form that the cell can use. They have their own genetic material, separate from the DNA in the nucleus.

-          Analysis of the mtDNA (about 16.000 base-pairs of nucleotides) let us see that the mtDNA is fractal structured.


Julia: zn+1 = 1 / (zn6+ c)




-          Bacteria are prokaryotes. Unlike animals and other eukaryotes, bacterial cells do not contain a nucleus or other membrane-bound organelles.

-          It has many shapes including spheres, rods, and spirals.

-          The fractal formula is a special composition of cos(n) multiplied with a polynomial.



Julia: zn+1 = cos(zn) * zn2.5 + c (zoomed in)




-          A virus can infect the cells of a biological organism. They cannot reproduce on their own.

-          Their surface carries specific tools designed to cross the barriers of their host cells.

-          The fractal formula for the virus and the bacterium is the same.

-          Zooming in and out creates the difference.



Julia: zn+1 = cos(zn) * zn2.5 + c

Andras Pellionisz (International PostGenetics Society):

“My knowledge of the universe is almost nil; 40+ years of research and 3 Ph.D.-s in Computer Engineering, Biology, and Physics don't measure up to the universe. However, having devoted the most recent 7 years entirely to the deep study of the DNA, I have evidence that the DNA is fractal. Perhaps not surprisingly, since it is a rather conspicuous tiny part of the universe.”

-          DNA is a double helix formed by base pairs attached to a sugar-phosphate backbone.

-          DNA can also be found in the mitochondria. This mtDNA is fractal structured. 


Julia: zn+1 = c * (cos(zn)2 - sin(zn)2)^(2)




-          Dendrimers are macro-molecules that are made up of branching molecules joined together.

-          The fractal structure is realised with polynomials with the c-value near to zero.



Julia: zn+1 = zn2 + c




-          A molecule is a group of atoms in a definite arrangement held together by chemical bonds.

-          Scientists  have created and captured a man-made molecule at the nanoscale.

-          It shows a fractal structure.

-          The geometric fractal was made with a formula for the inverted polynomial z^-6

Man-made molecule

Julia: zn+1 = zn-6+ c^-6.fim




-          Atoms are the building blocks of the molecules.

-          Each atom consists out of a nucleus with protons and neutrons.

-          Electrons are moving in 8 orbits around the nucleus.

-          See Periodic Chart of Elements:



Julia: zn+1 = zn6 + c




-          There are 2 types of  subatomic elements: fermions which make matter and bosons which make forces.

-          Each element is as well particle as wave, depending of the eye of the beholder.

-          The fractal orbits of z^6 can be influenced by the external ‘flightnumber’ going to infinity.

Subatomic particles

Julia: zn+1 = zn6 + c







-          Unparticles, but not particles, can fit in a theory that has the property of continuous scale-invariance, which is to visualize by fractal images (iterating complex functions).

-          A fractal like this Koch Curve is an example of discrete scale-invariance because it looks the same if multiplied by a fixed number.


Julia: if (x>=0) then zn+1 = (zn -1)*c, else zn+1 = (zn +1)*conj.c




The quadratic iteration of the Mandelbrot set [King] compared with the interactive effects of inverse quadratic charge interaction in tissues [Campbell].


© Chris King, 2007

The quadratic iteration of the Mandelbrot set [King] compared with the interactive effects of inverse quadratic charge interaction in tissues [Campbell]. Although their genesis arises from differing non-linear process, iteration on the one hand and interaction on the other, the multi-fractal structures of tissues have features similar to the Mandelbrot set on changes of scale. These fractal effects reach from the molecular (a) in which individual proteins are illustrated embedded in the lipid membrane, through cell organelles (b) to the intercellular structure of whole organs as illustrated by skin (c). Such scale-dependent coherence of structure is possible only because of the highly nonlinear nature of the electromagnetic force in quantum charge interactions of fermionic matter.




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Version d.d. 11 September  2007