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 zeta-function 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.



Particle tracks


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.


Cosmic Microwave Background

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 Robertson-Walker metric. A metric is an advanced version of Pythagoras´ formula. It calculates distances in space-time from coordinates. As the scale factor of space grows so does the wavelength of radiation and other particles. The R-W 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.


Diagram over CMB fluctuations at different scales

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 1st 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/cm3. 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.


Computer simulation of structure formation

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.



Sideview of spiral galaxy


The estimated number of galaxies in the observable universe is 1010. This spiral galaxy contains approximately 3·1011 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 All-Sky Survey, 1997-2001). Visible light from the central parts is obscured by interstellar dust.


The sun


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.


Sunspot


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.


The earth

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 2-5 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 4.6·109 years old. Life began 0.6·109 years later and it has evolved ever since. Intelligent creatures appeared 7 million years ago. They have now reached a self-conscious level. To learn more about the geology follow these links 1 2. For information on evolution, try these 1 2 3.


Illustration of -Exp(i*Pi)=1


Mathematics has a long and rich history. You can read about it in the MacTutor History of Mathematics archive.

The 1st 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 = 1-1/3+1/5-1/7+… . 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 2nd 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 7th century, especially Brahmagupta 628. It took a millennium before they were generally accepted among mathematicians.

The 3rd 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 16th century by Italians in their efforts to solve polynomial equations of higher orders.

The 4th 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=limn→∞(1+1/n)n. The prime importance of e is as as a function y=ex, the unique solution to y’=y with y(0)=1. y=ex is not confined to real numbers. All traditional functions can be extended to complex numbers via polynomial series.  ez=1+z+z2/2!+z3/3!+... ez is closely tied to the trigonometric functions via Euler’s formula: eix=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 5th square by using Euler’s formula.

0 - eiπ = 1


Snowflake in von Koch curve


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 shapesn. A point belongs to the von Koch Curve if its distance to these shapesn 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


The Mandelbrot Set

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: zn+1=zn2+c. For a given value of the complex number c you get a specific iteration. Setting c=0 gives zn+1=zn2. 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 c-values 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.


Part of the Mandelbrot image


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, self-similarity. 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.


Another part of the Mandelbrot image


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


The Riemann zeta function


On top of the most wanted list of unsolved mathematical problems is the Riemann Hypothesis: all non-trivial roots of the Riemann zeta function (1, 2) defined by ζ(z) = 1 + 1/2z + 1/3z + 1/4z+ … 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 20th 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).


Starry sky