The Fractal Geometry of the Universe

By Lori Gardi

Copyright © Lori Gardi,



A simple fractal pattern is studied that seems to reproduce the seemingly complex geometric patterns that appear in our Universe including supernova, planetary nebula and galaxy formations. The Mandelbrot set fractal construct is also shown to reproduce quite accurately the many shapes of galaxies and galaxy clusters. Some of the more complicated shapes like the Sting Ray Nebula and Cartwheel Galaxy can be easily reproduced using the simplest of fractal formulas.



Whether the Universe is a fractal or is fractal in nature has been an ongoing debate for the last decade or so. The fractal distribution of galaxies up to certain scales is well known (1,2), however, most cosmologists believe that the Universe transitions into homogeneity at some large scale (3) and so the debate continues. Galaxy distribution is only one aspect of the fractal nature of the Universe. What about the shapes of galaxies themselves, which also appear to be fractal in nature? The mechanism that generates these beautiful shapes is still not well known.


In this work, I describe a simple fractal construct that generates a geometry that is similar to many patterns that we find in our Universe, from spiral galaxies to planetary nebula formed from supernova explosions. I also show how another very simple yet complex fractal construct, The Mandelbrot Set, can be used to generate star field images and some of the more complicated structures in our universe like galaxy clusters, The Cartwheel Galaxy and the Stingray Nebula for instance.


As complicated as these patterns appear to be, there could be very simple yet fundamental rules that govern the formation and evolution of these celestial objects. What this says about the current state of cosmology is uncertain. Even Stephen Hawking is not sure how to fit fractals into his theory of the Universe (4). This work attempts to show how fractals fit into the study of cosmology and how our universe may be fractal “at all scales”.




Part I – A Simple Fractal Template


Generating the Fractal Template

The fractal structure that I will be describing here is one of the simplest fractal algorithms that I have ever developed. Being a computer scientist, I have explored all the fractals, in particular the famous Mandelbrot Set fractal construct, which I have studied intensely since the mid 1980’s. I will be getting into that a bit later. For now, I will describe how to generate very simple fractal pattern based on two intersecting circles.



Start with a unit circle at a point in space (ie. radius = 1.0). Add another circle offset by x. Find the two intersection points of the circles.









Figure 1: Unit circle on left.

Two unit circles on right, separated by x.

Two intersection points highlighted in red.



In general, to generate a fractal pattern, all you need to do is scale, rotate and translate an object recursively. Most if not all fractals are generated in this manner.


To generate the fractal structure we must now recursively scale, rotate and translate.

                 First we double the radius (scaling).

                 Draw a circle with new radius at both intersection points (translate).

                 Find the two intersection points of these new circles (rotate 90).

                          Note: rotating by 90 is the same as multiplying by i.



Here is what this pattern looks like after several iterations:


Figure 2: Fractal Template based on above algorithm


When you study this fractal structure, you can see it has a lot of interesting features. The first thing you might notice is that the spheres on the inside, are completely enclosed by the intersection region of the two larger spheres. In other words, the smaller scaled spheres do not intersect with the larger scales spheres. This allows different scales behave independent of the other scales. I think this is an important feature of this particular fractal. The other thing that becomes apparent when you study this structure is that it allows one to JUMP from one scale to the next very easily at the points where the differently scaled circles are touching. This structure appears to be related to the famous phi spiral as seen below which should come as no surprise to the audience.



Figure 3: Spiralling IN using this Fractal Template.


Figure 4: The famous Phi spiral.


So, what has this got to do with the formation and evolution of the Universe? I will now demonstrate through a series of images that this particular fractal pattern can be seen over and over again in our universe and that each object is just a variation on the theme.



Supernova 1987A


Here is an image of Supernova 1987A. Notice the similarity between the dynamic of this exploding star and the fractal pattern described above.



Figure 5:  Supernova 1987A (left), Fractal Template (right)

Hubble Space Telescope

Wide Field Planetary Camera 2

Image courtesy Space Telescope Science Institute



SN 1987A is a supernova in the outskirts of the Tarantula Nebula in the Large Magellanic Cloud a nearby dwarf galaxy. It was visible to the naked eye from the Southern Hemisphere on Earth. The light from the SN reached Earth on February 23, 1987. Current understanding is that the progenitor was a binary star system, which merged around 20,000 years before the explosion, producing a blue supergiant. Difficulties persist with this interpretation”. (Wikipedia).



The Hourglass Nebula (MyCn18)


Here is the image of the Hourglass Nebula next to the fractal template pattern, oriented slightly different to match the orientation of the nebula.




Figure 6: Fractal Template (left), Hour Glass Nebula (right)

Credit: Raghvendra Sahai and John Trauger (JPL), the WFPC2 team, and NASA.

Photo No.: STScI-PRC96-07 and JPL-P-46535


The Hourglass Nebula, also known as MyCN18, is a young planetary nebula situated in the southern constellation of Musca around 8,000 light-years away from Earth. It is conjectured the MyCn18’s hourglass shape is produced by the expansion of a fast stellar wind within a slowly expanding cloud which is denser near its equator than its poles”. (Wikipedia).


Octave Circle Added


Optionally we can add a circle of the same radius to the origin at each scale:








Figure 7: Octave Circle Added



Generating this fractal pattern:



Figure 8: Fractal Template with Extra Octave Circle

This slightly augmented fractal template (with an additional circle added to the origin at each scale), can now be matched to many more celestial objects as I will demonstrate.



Helix Nebula (NGC 7293) Visible Light


Here is an image of the Helix Nebula along with a modified version of the fractal template pattern. Notice the region near the top of the Helix Nebula, which corresponds nicely to the arch near the top of the fractal template on the right. The “eye” shape in the nebula matches accurately with the “eye” shape in the fractal template as well.


Figure 9: Helix Nebula visible light image (left), Fractal Template (right).

Credit: NASA, ESA, and C.R. O’Dell (Vanderbilt University)


The Helix Nebula, also known as NGC 7293, is a large planetary nebula (PN) located in the constellation of Aquarius. It is similar in appearance to the Ring Nebula, whose size, age, and physical characteristics are similar to the Dumbbell Nebula. The Helix Nebula has often been referred to as the Eye of God”. (Wikipedia).



Helix Nebula (NGC 7293) Infrared Light


Here is another image of the Helix Nebula taken in the infrared spectrum. The arch near the top is clearly visible in this image.



Figure 10: Helix Nebula  (left), Fractal Template (right).

Spitzer Space Telescope, IRAC, MIPS, ssc 2007-03a

Nasa/ JPL-Caltech/K.Su (University of Arizona)


The arc shape is emphasised by the arrows in both images. The aspect ratio of the eye shape (width/height) in the nebula exactly corresponds with the fractal template image on the right at 1.618 or phi.




The Little Ghost Nebula (NGC 6369)


Notice the “eye” shape again in this nebula image (left) along with the “tail” in the lower left region that matches well with my fractal template as shown on the right.



Figure 11: Little Ghost Nebula (left). Fractal Template (right)

Credit: NASA and The Hubble Heritage Team


The Little Ghost Nebula (NGC 6369) is a planetary nebula in the constellation Ophiuchus. Planetary nebulae in general are created at the end of a sun-like star’s life as its outer layers expand into space while the star’s core shrinks to become a white dwarf. The transformed white dwarf star, near the centre, radiates strongly at ultraviolet wavelengths and powers the expanding nebula’s glow”. (Wikipedia).




Barred Spiral NGC 1300


Here’s an example of a barred spiral galaxy. Notice how the spiral arm geometry matches exactly with the geometry of the fractal geometric figure.


Figure 12: Barred Spiral NGC 1300 (top) and my Fractal Template (bottom).



The Coiled Galaxy NGC 1097


Here again, the eye shape and orientation and the spiral arms geometry matches quite well with the fractal template geometry.



Figure 13: The galaxy, called NGC 1097


NGC 1097 is located 50 million light-years away. It is spiral-shaped like our Milky Way, with long, spindly arms of stars. The "eye" at the center of the galaxy is actually a monstrous black hole surrounded by a ring of stars. In this color-coded infrared view from Spitzer, the area around the invisible black hole is blue and the ring of stars, white.






Barred Spiral NGC1365


Again, the “eye” shape matches nicely with the fractal pattern and the bifurcation in the upper right spiral arm is predicted by the fractal template or pattern.


Figure 14, NGC 1365 (left), Fractal Template (right).




Whirlpool Galaxy (M51)



Notice the similarity in shapes of these two objects.


Figure 15: Whirlpool Galaxy (M51), Spitzer Infrared


Spitzer Space Telescope shows that the M51 spiral galaxy (left) is rich in dust, and actively forming new stars, while its blue companion galaxy hosts an older stellar population.





Three-Ring Configuration

When you start with a seed of three rings instead of two in the configuration shown below, you end up with a slightly different fractal pattern that is also seen in some obscure celestial objects.












Figure 16, Three-Ring Configuration



The three-ring seed generates the fractal pattern as seen in the lower right image, which is reminiscent of the image on the left, the Flower Nebula.


Figure 17, The Flower Nebula (left), Fractal Template using three-ring configuration.





The Electron


                    Figure 18:  Image of a single electron (left), Fractal Template (right).


The first movie of a single electron’s motion was made at Lund University, Sweden. It shows how an electron rides on a light wave after being pulled away from an atom. Notice the similarity between this shape and the shape of my fractal template diagram. Unfortunately, this is the only “real” photograph that I could find of an atomic element, so an extensive comparison cannot be made, however, the fact that this matches so nicely to the fractal template pattern is suggestive of the idea that the universe is fractal at all scales.



Conclusion Part I

A simple algorithm was developed that generates fractal patterns whose geometry is very similar to that which is generated by supernova explosions and galaxy formations. This simple fractal pattern may also be found at the atomic level, which suggests that the universe is fractal “at all scales” which is the premise of this work.



Discussion Part I:

The fractal geometry of the universe has been a debated subject since the mid 1980’s when Benoit Mandelbrot and others speculated about the fractal distribution of galaxies in the universe. However, in order for the Universe to be truly and intrinsically fractal, it must be fractal at all scales.  Physicist, Dario Benedetti from the Perimeter Institute for Theoretical Physics in Waterloo believes that the geometry (of space-time) may have fractal qualities.  Tim Palmer’s Invariant Set Postulate points to a fractal-geometry at the quantum level. Interestingly, fractal patterns have recently been discovered at the quantum level (6) by several scientists, as reported in Science Daily in January, 2010. Researchers from the Helmholtz-Zentrum Berlin für Materialien und Energie (HZB), in cooperation with colleagues from Oxford and Bristol Universities, as well as the Rutherford Appleton Laboratory, UK, have measured the signatures of a symmetry hidden in solid state matter, showing the same attributes as the golden ratio (phi). 


When applying a magnetic field at right angles to an aligned spin the magnetic chain will transform into a new state called quantum critical, which can be thought of as a quantum version of a fractal pattern.”


Colin Hill’s Electro-Fractal Universe ties a lot of things together (7). He argues that, since the electromagnetic force is enormously more powerful than gravity, then we should conclude that the universe is dominated by it and therefore is formed primarily because of it. Also, since electromagnetic plasmas are scalable, they are well suited for generating self-similar fractal patterns. He also contends that, if even one part of our Universe is shown to be fractal, then the rest must be also fractal, a notion I happen to agree with.


A scientist by the name of Nassim Haramein from has developed a fractal pattern called the 64-etrahedron grid which is a scalable fractal that he contends is related to the structure of the vacuum. His theory also suggests that black holes exist at many scales including the quantum scale, implying that they are fractal in nature or at the very least are fractal generators. In his recent paper, The Schwarzschild Proton, he demonstrates how atoms might also be black holes (5), a concept that is far from the currently accepted model of the universe.


It's interesting to note that the two celestial object types we have been discussing so far, supernova and galaxy formations, are both associated with black holes. Supernova are said to create black holes during the alleged "death of a star" and all galaxies that we know of have supermassive black holes at their center. Could it be that black holes are responsible for generating these fractal patterns? In a recent paper entitled “Quasar Induced Galaxy Formation: A New Paradigm” (8) it is argued that quasars (black holes) play a key role in the formation of their host galaxies. Using quasar HE04050-2958 as their test case, they demonstrate that this particular quasar is actually “forming” its future host galaxy through what they call jet-induced galaxy formation. There are other studies indicating that jet-induced star formation in galaxies is universal (9, 10) and that black holes are creators as well as destroyers.


Another interesting connection is that when you map the orbits around the Schwarzschild black hole using Mathematica, you end up with the following image. Coincidence?


Schwarzschild Black Hole Orbits

Figure 19: Orbits around Schwarzschild Black Holes, source Mathematica Player



There are copious other theories out there that show how fractals play a pivotal role in the formation and transformation of our Universe. In short, the fractal template pattern described in this paper seems to match the geometry of celestial objects at various scales which implies that the same "dynamics" is going on “at all scales”. This points to the intrinsic fractal nature of the universe as a whole. It was also shown how black holes might fit into the fractal nature of the universe being potentially the main source of structure formation in our universe.


In Part II, I will demonstrate how the Mandelbrot set construct can be used to generate galaxy shapes, galaxy cluster patterns, and planetary nebula structures which are very similar to the structures we find in our universe.






Part II: Mandelbrot Cosmology




Star Field Simulation


     Figure ???: Fractal Dynamic Field Generated using Mandelbrot Set



Spiral Galaxy M101

          Figure ???: Fractal Dynamic Field (lower left), Spiral Galaxy M101 (upper right)


Galaxy Clusters (Abel 370)


Figure ??? Fractal Dynamic Field (left), Abel 370 Galaxy Cluster (right)






Cartwheel Galaxy


Figure ???: Cartwheel Galaxy (upper left), Fractal Dynamic Fields (the rest)




Planetary Nebula NGC 6210



Figure 18: Planetary Nebula NGC 6210 (left), Mandelbrot Fractal Dynamic Field (right).





Sting Ray Nebula



Figure 19: Sting Ray Nebula (upper left), Mandelbrot Fractal Dynamic Fields (the rest)


Conclusion Part II



Discussion Part II





1-Peebles, P.J.E., The Large-Scale Structure of the Universe,

Princeton University Press (1980).


2-Mandelbrot, B.B., The Fractal Geometry of Nature, Freeman,

San Francisco (1982).


3- Jose´ Gaite, Alvaro Domı´nguez, and Juan Pe´rez-Mercader


The Astrophysical Journal, 522:L5–L8, 1999 September 1


4-Hawking, Virtual Black Holes, arXiv:hep-th/9510029v1 6 Oct 1995


5-Golden Ratio Discovered in Quantum World: Hidden Symmetry Observed for the First Time in Solid State Matter. Science Daily, Science News, Jan. 7, 2010.


6-Nassim Haramein, The Schwarzschild Proton


7-Colin Hill’s Electro-Fractal Universe


8- Quasar induced galaxy formation.


9- Molecular Gas at High Redshif: Jet-induced Star Formation?


10-Jet-induced start formation: Good news from the big, bad black holes.;jsessionid=133870C51BC00447AC9BFD1DC0372AF2.tomcat1?fromPage=online&aid=260837