The Fractal Geometry of the Universe
By Lori Gardi
Copyright © Lori Gardi, ButterflyEffect.ca
Abstract
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.
Introduction
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.
Setup:
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.
Repeat.
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 TheResonanceProject.org 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?
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.
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
Introduction
TBD
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
TBD
Discussion Part II
TBD
References
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 FRACTAL DISTRIBUTION OF GALAXIES AND
THE TRANSITION TO HOMOGENEITY
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.
http://arxiv.org/PS_cache/arxiv/pdf/0907/0907.2923v2.pdf
9- Molecular Gas at High Redshif: Jet-induced Star Formation?
http://www.iop.org/EJ/article/1538-4357/612/2/L97/18669.text.html
10-Jet-induced start formation: Good news from the big, bad black holes.
Wikipedia