Discovery of Cosmic Fractals

The following sections are included:

• Contents

• Foreword by Benoit Mandelbrot

• Preface

• Prologue

The following sections are included:

• The seeds were sown – the myth explains the world

• Celestial writing on the Babylonian sky

• The Ionian Revolution

• Anaximander solves the paradox of unfalling Earth

• The sky becomes a sphere

• Atomists see a glimpse of the microcosm

• Plato's mathematical heaven

• Aristotle's scientific method

• The principle of circular motion

• But what is actually rotating?

• Towards the principle of no center

• The Wisdom of Antiquity was kept alive

• The world edifice of the Middle Ages

The following sections are included:

• Roots of De Revolutionibus

• New understanding on matters celestial

• The young Rheticus visits the old Copernicus

• Bruno breaks the stellar sphere

• …and Galileo opens the gate

• The blurred new view through the magnifying tube

• Kepler's laws of cosmic order

• Nicholas of Cusa: the center is everywhere

• Digges, Bruno, and the Copernican Principle

• The first steps on the cosmic distance ladder

• Stars are remote suns

• Understanding the new cosmic order

• The triumph of Newton's universal gravity

• Just add one particle more …

The following sections are included:

• Structure of the Heavens

• Newton's Cosmology in a Nutshell

• Cosmological paradox

• Why do we not feel an infinite gravity force?

• How to tame the infinite gravity?

• If, however, a uniform infinite cloud of stars exists, why has it not collapsed?

• Why is the night sky so dark?

• The riddle of the shining stars

• What has saved us from the ultimate heat death?

The following sections are included:

• Stars and nebulae

• Emanuel Swedenborg

• Cartesian physics

• The Swedenborg self-similar universe

• Towards the origin of the Solar System

• Hierarchies of Kant and Lambert

• Finite or infinite?

• Emerging protofractals

• Inwards and outwards

The following sections are included:

• The Principle of Relativity

• The relativistic physics of Poincaré and Einstein

• Velocity of light

• From classical space and time…

• …to relativistic space-time

• Time travel into the future with a one-way ticket

• Rest mass energy: E = mc²

• Light, electricity, and magnetism

• Least action, symmetry, conservation laws

• Quantum physics of the microworld

• Heisenberg's Uncertainty Principle: nebulous particle

• The search for genuine atoms

• Quarks hide inside protons

• The quantum nature of fundamental forces

• “Fur coat” of virtual particles and the boiling vacuum

• Spiraling down into the microcosm?

• From terrestrial to cosmic laboratory

The following sections are included:

• The nature of gravity

• Newton's law and the gravitational constant

• The riddle of inertial and gravitating masses

• Relativistic gravity emerges in our Solar System

• Geometry of curved spaces

• General relativity as geometrical gravity theory

• What causes gravity according to general relativity?

• Big Bang, Black Hole, Time Machine…

• …but riddles still exist

• Feynman's quantum field approach to gravity

• Relativistic effects in quantum field gravity

• Gravity as a builder of celestial structures

• Energy flows and order from chaos

• A star is a self-gravitating nuclear reactor

• Exploding stars – the end of the fight?

The following sections are included:

• The Island Universes

• The distance to the “Little Cloud” is measured

• Pulsating stars light up the way to Andromeda

• The diversity of galactic geometries

• Our home galaxy – the Milky Way

• Spectra – fingerprints of stellar matter

• Spectral line shift – a celestial message

• Discovery of extragalactic redshifts

• The search for a relation between redshift and distance

• The law of redshifts: a new cosmic phenomenon

• Galaxies live in swarms

• Super energies in the galaxy universe

• Anomalous redshifts – the exception to the rule?

• Redshift quantization?

The following sections are included:

• Friedmann's discovery of expanding universes

• Cosmological redshift in expanding space

• Uniformity gives rise to the Hubble law

• The Hubble constant measures the age of the universe

• The oldest stars – almost as ancient as the universe

• The geometries of Friedmann's world models

• The cosmic density of matter in the universe

• George Gamow's hot beginning

• Discovery of the cosmic thermal radiation

• The 3 Kelvin glow – the cool relic of the hot bang

• Cooking the light elements

• After solving Newton's paradoxes of infinity…

• …new enigmas of Friedmann's uniform world appear

• Inflation comes and resolves the paradoxes

• The age of the inflationary universe

• When were the galaxies and their clusters born?

• The big bang triumph – its logic and components

The following sections are included:

• A uniform matter distribution leads to a singularity

• What is a black hole singularity?

• Einstein objects to the physical reality of the singularity

• Are there alternatives to singularity?

• Gravastars, eternally collapsing objects, dark stars…

• Relativistic astrophysics probes strong gravity

• A binary pulsar – an ideal gravity laboratory

• The search for gravity waves from collapsing stars

• Two closest supernovae – signs of gravity waves?

• X-rays betray black holes in binary stars

• The best candidate sits at the center of the Milky Way

• Supermassive objects in the nuclei of other galaxies

• Approaching the horizon…

• …may offer unexpected surprises

• The rapid variability of quasars as probe of gravity

• Cosmology requires relativistic and quantum gravity

The following sections are included:

• Early signs of dark matter

• Invisible matter makes galaxies revolve rapidly

• Gravity lenses probe the dark matter

• MACHOs in the halo of the Milky Way

• Do Arp's quasars reveal dark matter in galaxy haloes?

• Much more in a cluster of galaxies than the eye sees

• The total amount of dark matter in the universe

• An ocean of massive neutrinos?

• The search for dark matter goes on

The following sections are included:

• Revolution in cosmology – Einstein's lambda returns!

• A short course in the physics of “nothing”

• Dark energy, quintessence, spintessence…

• A bit of history: redshift and de Sitter's effect

• The age of an accelerating universe

• The fifth element may rule in your backyard

The following sections are included:

• The nature of redshift – Allan Sandage's 15th problem

• Understanding the expansion of space

• The Lemaître phenomenon versus the Doppler effect

• What is the fate of energy in expanding space?

• Superluminal recession of remote galaxies

• Geometry and physics: views of Poincaré and Einstein

• Absolutely soft and hard meter sticks

• Geometry of space in the local galaxy universe

• The classical cosmological tests of space geometry

• The patchy microwave sky brings Euclid back

• The enigmatic unity of space, matter, and energy

The following sections are included:

• Searching the heavens for nebulae

• John Herschel's principle of subordinate grouping

• Fournier d'Albe's brave new worlds

• Gravity within Fournier's hierarchy

• Carl Charlier wrestles with infinities

• Charlier's criteria for infinite worlds

• Towards hierarchic worlds without a middle point

• Knut Lundmark's great plan

The following sections are included:

• The “fractal orbit” of Mandelbrot

• The concept of the fractal

• Koch's curve or snow flake

• The simple measure of complex structures

• The fractal dimension of Fournier-Charlier worlds

• Creativity of fractals

• Random fractals and Brownian motion

• Percolation – a process leading to fractals

• Fractal structures versus smooth distributions

• Fractal view of Nature

• The fractal dimension of abstract art

The following sections are included:

• Order and chaos revealed by the Solar System

• Chaos, strange attractors, and fractals

• How a pendulum connects chaos and fractals

• “Protochaos” in Swedenborg's vision of evolution

• To the microcosmos – and back to the planets again: Nottale's fractal space-time

• Rugged planetary landscapes

• Dense dust clouds – cocoons of stars

• A case study of natural fractals: interstellar clouds

• Dark clouds, molecular complexes, cirrus filaments…

• Galaxy haloes – dark mass hiding in fractals?

• Fractal gas clouds between galaxies

The following sections are included:

• Hubble's law of redshifts is a distance indicator

• The Hubble constant measured before Hubble…

• The Hubble constant: 100 or 72 or 50?

• Distances to galaxies – a mission impossible?

• The notorious Malmquist bias

• What, after all, is the value of the Hubble constant?

• Galaxy clusters painted on the celestial sphere

• The origin of the debate on superclusters

• Abell's rich clusters of galaxies

• Looking through the dusty window

• 3-D astronomy from the vertex of a space cone

• Excursions into the local galaxy universe and beyond

• The mysterious quietness of the Hubble flow

• The redshift of quasars as a distance indicator

The following sections are included:

• Einstein's Cosmological Principle

• Many faces of the Cosmological Principle

• The derivation of uniformity from local isotropy

• The galaxy universe may seem rather smooth…

• …but the uniformity is elusive

• Carpenter – de Vaucouleurs's law of galaxy clustering

• Mandelbrot's fractal view of galaxy clustering

• Does isotropy always imply uniformity?

• Do we live on the peak of a mountain?

• Modern redshift surveys of galaxies

• Pietronero and the five megaparsec mystery

• The Great Fractal Debate

• The correlation function points at 5 Mpc

• The conditional density comes and finds fractality

• To search for or to count on uniformity?

• Towards Einstein–Mandelbrot concordance

• Everything we know about the cosmos?

• Opening the millenium: the race to a fair sample

The following sections are included:

• The ladder of key discoveries

• The three whales of cosmology

• The art of making universes

• Art is long, life is short

• Growth of large scale structures in big bang cosmology

• The smooth Hubble law ignores local roughness

• Gravitational redshift inside a fractal structure

• A Friedmann universe with fractal galaxy distribution

• Dark energy drives the remote and the local universe

• Early work around fractal dimension one…

• …and intriguing aspects of fractal dimension two

• The fractal state of many gravitating particles

• Cosmological questions within quantum field gravity

• The cosmic architecture of complexity

• What is the message of the megafractals?

• Through deeper observations to novel perspectives

The following sections are included:

• Appendix A
  • Definition of the astronomical magnitude
  • The mass of the Milky Way
  • A standard candle in the Hubble diagram
  • The classical electron and gravitational radiuses
  • The cosmological gravitational redshift

• Suggestions for Reading
  • General astronomy and history
  • Fractals
  • Large scale structure and cosmology
  • Gravitation and general physics

• Index