The picture above illustrates the changing universe and our place in cosmos.
The lower half shows some snapshots from the world of mathematics.
Magnified views of parts of the image can be downloaded from the Info&Downloads section.
The picture starts with some particle tracks taken from a bubble chamber.
They represent the first phase, a universe of interacting particles. After the tracks comes the
cosmic microwave background. The universe is about 400 000 years old and has just become
transparent for photons. They have travelled for 13 billion years to reach us.
The next part is taken from a computer simulation. It shows the evolution of large scale
structure in the universe. A galaxy appears, it's the Milky Way Galaxy.
Below is the Sun, one among 100 billion stars in our galaxy. Planet Earth hovers
above the sunspot.
The bottom left end is a history of mathematics. Discoveries from different areas and
different epochs are brought together in an elegant formula. The next part represents
the close relation between mathematics and physics. The von Koch Curve is of infinite
length but surrounds a finite area. It is continous everywhere but differentiable nowhere.
It has the same symmetry group as the snowflake. All snowflakes differ but they have the same symmetry,
a consequence of the geometry of the water molecule. In the center is the Mandelbrot set.
A mathematical work of art. The picture ends
with an unsolved mathematical problem, the Riemann zetafunction for z=½+it.
The text below has further explanations of the different parts. They can be reached by clicking in the picture.
The small headline pictures can be enlarged by clicking. Some links in the text are
not meant for a framed webpage. If a link appears broken, try opening it as a separate page.


This picture shows particle tracks from a bubble chamber in
CERN.
When a charged particle enters the chamber containing superheated
liquid it ionizes other particles on its way. The energized ions will start
local boiling with bubbles that can be seen in pictures like the one above.
Today other types of particle detectors are used. Discovery of the Higgs boson
was announced on July 4th 2012. It was discoverd by the large hadron collider
LHC
in two gigantic detectors,
CMS
and
ATLAS.
Now the search is on for supersymmetric particles beyond the
Standard Model
but so far none has been seen.
If you click on a small picture you will get a big picture. To return from a link, use the
right pointer above. The left pointer will take you back to PM projects.
Clicking on a picture element in the top picture takes you to the chosen part.


The universe was once very dense and very hot. There were no neutral atoms
and the photons of light could only travel short distances between
interactions with charged particles. As the universe expanded and cooled
neutral atoms could form. The photons became free to travel without
interruptions. The universe became transparent, nothing we see can be older
than this.
The speckled globe is a snapshot of the photons in the observable
universe 13.7 billion years ago, 380 000 years after big bang. The
unobservable parts are those from which the light has not yet reached us.
At the beginning the radiation had a temperature of 3000K. Space has
expanded by a factor 1090 since then so the wavelength of the radiation
is now in the microwave region. The spectrum is the same as heat radiation
from a body at T=2.7K. The radiation is called
CMB,
cosmic microwave background.
The expansion of space is measured by the scale factor R(t) in the
RobertsonWalker metric. A metric is an advanced version of
Pythagoras´ formula. It calculates distances in spacetime from coordinates.
As the scale factor of space grows so does the wavelength of radiation and other
particles. The RW metric only applies to the universe at large. The metric is
modified by the local mass distribution on scales smaller than local galaxy
clusters. Galaxies are gravitationally bound, they do not expand.
The globe is a map of small variations in temperature
(~100mK)
across the sky. The map comes from a NASA explorer mission, the Wilkinson Microwave
Anisotropy Probe.
WMAP
is a satellite orbiting the sun four lunar distances further out from the earth.
CMB is actually hotter in one direction and cooler in the opposite direction
because of a Doppler effect. We are moving through the observable universe with
an absolute velocity of 370 km/s in the direction of the constellation Leo.
This dipole effect has been removed from the anisotropy map.
The work of WMAP from 2001 to 2010 has been continued and improved by the
Planck mission.


Fluctuations in temperature of the CMB is a rich source
of information if you want to understand the early universe and how the
primordial plasma evolved into large scale structures, galaxies and stars.
The diagram shows the amplitude of temperature fluctuations at different
angular scales, don’t forget to click it.
Disturbances in a plasma travel at a fixed speed so there will be something
resembling sound waves. There will be tones and overtones of certain wavelengths
just as in a musical instrument. If you calculate the wavelength of the 1^{st}
harmonic for the plasma when the CMB was released you will get the length of the
far end of a triangle. The photons have been on their way for 14 billion years
but they have actually travelled 45 billion light years. Their speed is relative
to space but space is expanding. You could compare it to a plane flying in windy
weather. The speed is measured from the surrounding air.
With three known sides the angle of largest temperature fluctuations seems
to be predicted. But if the universe is curved this is not so. Imagine yourself
a member of Flat Earth Society sent to the research station at the South Pole.
He follows two meridians to the equator and measures their lengths and
separation. He calculates the meridians' spread angle only to find that it
doesn't match the measured spread angle. Flat Earth society is about to lose
a member. The angle depend on the curvature of earth given by radius^{1}.
The maximum of temperature variations at an angular scale of about one degree indicates a
universe almost perfectly flat.
Curvature is caused by the energy density. A flat universe corresponds to a density
of 10^{29} g/cm^{3}. This is far more than can be accounted for by
matter made up from known elementary particles. The WMAP data suggests the following
energy composition for cosmos.
4% Known particles, mostly quarks and leptons.
23% Dark matter, consisting of undiscovered particles.
73% Dark energy with negative pressure that accelerates the expansion.


These frames show
formation of large scale structure
in a computer simulation made by NCSA, National Center for Supercomputer Applications by Andrey Kravtsov and Anatoly Klypin.
Observational data from WMAP and other sources to describe the conditions
in the early universe results in a distribution of matter that agrees quite
well with observations in projects like
SDSS.
The CMB reveals fluctuations in the density of
matter in the early universe, variations that caused the evolution of large
scale structure through gravitational attraction. The origin of these
fluctuations is believed to have come from an extreme expansion called
inflation during a very short time in the very early universe.
The timescales could be as short as 10^{36} s.
Small quantum fluctuations occur everywhere all the
time. Virtual pairs of particles and antiparticles are created and destroyed.
Inflation was so rapid that these virtual particles lost contact, quantum
fluctuations were expanded into reality. The expansion caused by inflationary
models is so extreme that it would be almost impossible to know if we are
living in a curved, bounded and very large universe or a flat and spatially infinite
universe. More information on structure formation is available at the center for
Cosmological Physics.


The estimated number of galaxies in the observable universe is 10^{10}.
This spiral galaxy contains approximately 3·10^{11} stars and
at least one of these stars has a planet with intelligent life. That star is located
26 000 light years from the centre of the galaxy which has a radius three
times that distance. This image of the Milky Way Galaxy as seen from the earth is based on the infrared part of the spectrum.
It is made made from star counts based on the
2MASS
star catalog (2 Micron AllSky Survey, 19972001).
Visible light from the central parts is obscured by interstellar dust.


This is the star mentioned above. The picture doesn’t show much
structure, just some sunspots. In the magnified part in the download section you will see some
more details. The left solar image is from the corona. The sun has no real
boundary, it fades out and leaks fast particles to a solar wind. The right solar picture
is taken in ultraviolet during a big eruption. The sun is very active and
constantly changing. These pictures come from the Solar & Heliospheric Observatory
SOHO,
a joint project between ESA and NASA. The satellite is orbiting around the
Lagrangian point
L1 that is situated between the earth and the sun.


This is the same sunspot that can be seen on the sun above. The photo was taken with a telescope on the Canary Island of La Palma by the Swedish
Institute for Solar Physics.
On the collage there is a view of the earth that illustrates the size of the eart in relation to the sun. Sunspots are caused by magnetic fields and they are
not black. The temperature is somewhat cooler in a sunspot 4300K compared to the average 5800K in the photosphere. The spot is not as bright as its
surroundings and the contrast makes the spot appear black.


This is the planet mentioned earlier. It has a lot of water covering 70% of
the surface. The planet is geologically active, the crust is divided into continental
sized plates that are moving around, merging and breaking up. Their speed is
25 cm per year. The movement is caused by convective currents in the
interior driven by heat from radioactive processes in the centre. The planet is
4.6·10^{9} years old.
Life began 0.6·10^{9} years later and it has evolved ever since. Intelligent creatures
appeared 7 million years ago. They have now reached a selfconscious level.
To learn more about the geology follow these links
1
2.
For information on evolution, try these
1
2
3.


Mathematics has a long and rich history. You can read about it in the
MacTutor History of Mathematics archive.
The 1^{st} square illustrates the number
π.
Geometry is the oldest branch of pure mathematics. All the great civilisations used their own fraction for
π.
The Egyptian Rhind papyrus from 1700 b.c. has a good value (16/9)^{2}.
Leibniz used a series in 1674,
π = 11/3+1/51/7+… .
The first proof that
π
is irrational, not expressible as a fraction was made 1770 by Lambert, 3500
years after the Rhind papyrus and 2100 years after the Pythagoreans’ proof that
√ 2
is irrational.
π
is not only irrational it is also transcendental, not a root of any polynomial over the integers.
This was proved 1882 by Lindemann, thereby resolving the ancient problem of
squaring the circle. It is not possible.
The 2^{nd} square illustrates the discovery or invention of negative numbers and zero.
More than 2000 years ago negative numbers were used in Chinese rod numerals.
Positive numbers were red rods and negative numbers were represented by black rods.
They were properly introduced in India in the 7^{th}
century, especially Brahmagupta 628. It took a millennium before they were
generally accepted among mathematicians.
The 3^{rd} square represents the complex number i.
Complex numbers are introduced in the same manner as negative numbers. They met the same resistance
as irrational and negative numbers had met before, they were called imaginary.
Complex numbers were first used in the 16^{th} century by Italians in
their efforts to solve polynomial equations of higher orders.
The 4^{th} square contains another very important number in mathematics.
It is called e and nobody knows why. Maybe Euler named it after himself.
Euler introduced e as the base for natural logarithms in 1728 and he showed that
it was an irrational number. Leonhard Euler is probably the most productive
mathematician of all times. There are several nice formulas for e such as
e=1+1/1!+1/2!+1/3!+... and
e=lim_{n→∞}(1+1/n)^{n}. The prime importance of e is as as a function
y=e^{x}, the unique solution to y’=y with y(0)=1.
y=e^{x} is not confined to real numbers. All traditional functions can be extended to
complex numbers via polynomial series.
e^{z}=1+z+z^{2}/2!+z^{3}/3!+... e^{z} is closely
tied to the trigonometric functions via
Euler’s formula:
e^{ix}=cos(x)+i·sin(x).
All of these discoveries 0, , e, i and
π
from different eras and different parts of mathematics
come together in the 5^{th} square by using Euler’s formula.
0  e^{iπ} = 1


Mathematics is the language of physics. The snowflake represents physics and
the surrounding curve is mathematics. Snowflakes are a form of ice crystals. Their
symmetrical shape is a reflection of the hexagonal arrangement of water
molecules in the solid state. Hydrogen bonds between molecules make the
hexagonal order the most cost effective. Rearranging the water molecules by
breaking the bonds and putting them back in a different pattern would require
a net transfer of energy to the system. Differences in temperature, humidity,
air currents and melting during crystal growth results in a large variety of shapes for
snowflakes.
The surrounding curve is called the von Koch Curve.
You start with an equilateral triangle and add triangular outgrowths of
smaller and smaller sizes in larger and larger numbers. This iterative
procedure gives a series of shapes_{n}. A
point belongs to the von Koch Curve if its distance to these shapes_{n}
goes towards zero as n goes to infinity. The procedure is presented on this
site of the
von Koch Curve. The curve is continuous everywhere but differentiable
nowhere. Its length is infinite but it bounds a finite area.
It has a fractal dimension D=log4/log3≈1.26


Physics and Mathematics have a close connection to aesthetics. The most
successful physical theories are also the most symmetric and elegant ones.
Maxwell’s theory of electromagnetic fields, invariant under the Poincaré group
and General Relativity, invariant under general coordinate transformations are
examples of this.
The beauty of mathematics can be seen in the
Mandelbrot set.
It is based on the simplest of nonlinear mappings: z_{n+1}=z_{n}^{2}+c.
For a given value of the complex number c you get a specific iteration.
Setting c=0 gives z_{n+1}=z_{n}^{2}. Iterations
starting from a point inside the unit circle are driven towards zero and points
outside it are driven to infinity. The boundary between these two domains of
attractions is called a Julia Set. For c=0 the Julia set is the unit circle.
Other choices of c will give you other more complex Julia Sets, some are
connected and some are broken up into disconnected pieces.
The Mandelbrot Set is the black part of the figure above. It is a part of the
complex plane and it represents a classification of Julia Sets. Points in the
black part are cvalues with iterations that lead to connected Julia Sets. All
points outside this set have disconnected Julia Sets. Imagine a path starting
inside M and ending on the outside. As c varies along the path the associated
Julia Sets will transform and when c passes the boundary of the Mandelbrot Set
there will be a dramatic qualitative change. The Julia Set will explode into
a cloud of infinitely many disconnected pieces, a mathematical phase transition.
The colours indicate how close you are to the Mandelbrot Set. They have nothing
to do with mathematics, they are chosen to create beautiful
Mandelbrot and Julia pictures.


The Mandelbrot Set, M has a very complex geometrical form as you can see if
you explore the link given above. It contains mini versions of itself at all
scales, selfsimilarity. Remember that the coloured parts do not belong to M.
It has been proved that M is a connected set so when you find a mini M in a
sea of colours there will always be a black connection leading back to the
main body.


Pictures of the Mandelbrot set with various colorings care plentiful on the Internet and there are many apps to explore their beauty.


On top of the most wanted list of unsolved mathematical problems is the
Riemann Hypothesis:
all nontrivial roots of the Riemann zeta function
(1,
2)
defined by
ζ(z) = 1 + 1/2^{z }+ 1/3^{z} + 1/4^{z}+ … are positioned
on a line given by z = ½ + i·t.
There are infinitely many zeros inside the strip given by 0<Re(z)<1. This was proved by G. H. Hardy, one
of the greatest number theorists of the 20^{th} century. It’s a bit hard
to visualize functions of the complex plane since it requires four dimensions.
The function above is ζ(½+i·t) with
a sign modification. The Riemann Hypothesis is important since the distribution of prime numbers is related to the
location of the zeros of ζ(z).

