Link found between golden ratio and atomic symmetry

Link found between golden ratio and atomic symmetry

Researchers
in Germany and the UK have discovered symmetry hidden in solid state
matter at very small scales. The findings, published in the journalScience,
indicate that symmetry involves the golden ratio famous from art and
architecture. The research was supported in part by the NMI3
('Integrated infrastructure initiative for neutron scattering and muon
spectroscopy') project, which was funded under the 'Coordination of
research activities' Thematic area of the EU's Sixth Framework
Programme (FP6) to the tune of EUR 21 million.

Scientists say unexpected and even irrational behaviour goes hand in
hand with particles at the atomic, or quantum, scale. The emergence of
new properties is triggered by what experts call 'Heisenberg's
uncertainty principle', which basically states that it is impossible to
know both the exact position and exact velocity of an object at the
same time. The effect, however, is just noticeable on a subatomic scale.

For this latest study, the researchers investigated the magnetic
material cobalt niobate, which is made up of linked magnetic atoms that
form chains just one atom wide. According to them, cobalt niobate comes
in handy when someone wants to describe ferromagnetism at tiny scales
in solid state matter.

The magnetic chain changes into a new state called 'quantum critical'
when a magnetic field is applied at right angles to an aligned spin,
the team said. The quantum critical, experts say, can be considered as
being a quantum version of a fractal pattern.

'The system reaches a quantum uncertain — or a Schrödinger cat state
[composed of two diametrically opposed conditions at the same time],'
explained Professor Alan Tennant of Helmholtz-Zentrum Berlin für
Materialien und Energie (HZB) in Germany, who co-authored the study.
'This is what we did in our experiments with cobalt niobate. We have
tuned the system exactly in order to turn it quantum critical.'

研究人员发现,当他们the system and
artificially introduced more quantum uncertainty, the chain of atoms
acted like a nanoscale guitar string. They used a special probe,
'neutron scattering', which enabled them to see the actual atomic scale
vibrations of a system.

'Here the tension comes from the interaction between spins causing them
to magnetically resonate,' said lead author Dr Radu Coldea of Oxford
University in the UK. 'For these interactions we found a series (scale)
of resonant notes: the first 2 notes show a perfect relationship with
each other. Their frequencies (pitch) are in the ratio of 1.618...,
which is the golden ratio famous from art and architecture.'

Scientists say that in mathematics and the arts, two quantities are in
the golden ratio if the ratio of the sum of the quantities to the
larger quantity is the same as the ratio of the larger quantity to the
smaller one.

Dr Coldea underlined that there is nothing coincidental about that. 'It
reflects a beautiful property of the quantum system — a hidden
symmetry. Actually quite a special one called E8 by mathematicians, and
this is its first observation in a material.'

针对粒子physi开发的数学理论cs could find a niche
in nanoscale science, and could fuel technology in future, according to
the researchers.

'Such discoveries are leading physicists to speculate that the quantum,
atomic scale world may have its own underlying order,' Professor
Tennant, who led the HZB team, said, adding that 'similar surprises may
await researchers in other materials in the quantum critical state.'

Participating in the NMI3 project were researchers from the Czech
Republic, Denmark, Germany, Hungary, the Netherlands, Poland, Russia,
Sweden and the UK.

Journal Reference:

1. R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, K. Kiefer.Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry.Science, Jan. 8, 2010

Golden ratio
From Wikipedia, the free encyclopedia

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Thegolden sectionis a line segment divided according to the golden ratio: The total lengtha + bis to the longer segmentaasais to the shorter segmentb.

Inmathematicsand thearts, two quantities are in thegolden ratioif theratioof the sum of the quantities to the larger quantity is equal to (=) theratioof the larger quantity to the smaller one. The golden ratio is anirrationalmathematical constant, approximately 1.6180339887.[1]Other names frequently used for the golden ratio are thegolden section(Latin:sectio aurea) andgolden mean.[2][3][4]Other terms encountered includeextreme and mean ratio,[5]medial section,divine proportion,divine section(Latin:sectio divina),golden proportion,golden cut,[6]golden number, andmean ofPhidias.[7][8][9]The golden ratio is often denoted by theGreekletterphi, usually lower case (φ).

The figure on the right illustrates the geometric relationship that defines this constant. Expressed algebraically:

This equation has as its unique positive solution thealgebraicirrational number

[1]

At least since theRenaissance, manyartistsandarchitectshave proportioned their works to approximate the golden ratio—especially in the form of thegolden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to beaestheticallypleasing.Mathematicianshave studied the golden ratio because of its unique and interesting properties.

Construction of agolden rectangle:
1. Construct a unit square (red).
2. Draw a line from the midpoint of one side to an opposite corner.
3. Use that line as the radius to draw an arc that defines the long dimension of the rectangle.

Contents
[hide]

• 1Calculation
• 2History
  • 2.1Timeline
• 3Aesthetics
  • 3.1Architecture
  • 3.2Painting
  • 3.3Book design
  • 3.4Perceptual studies
  • 3.5Music
• 4Nature
• 5Mathematics
  • 5.1Golden ratio conjugate
  • 5.2Short proofs of irrationality
    • 5.2.1Contradiction from an expression in lowest terms
    • 5.2.2Derivation from irrationality of √5
  • 5.3Alternate forms
  • 5.4Geometry
    • 5.4.1Golden triangle, pentagon and pentagram
      • 5.4.1.1Golden triangle
      • 5.4.1.2Pentagram
      • 5.4.1.3Ptolemy's theorem
    • 5.4.2Scalenity of triangles
  • 5.5Relationship to Fibonacci sequence
  • 5.6Other properties
  • 5.7Decimal expansion
• 6Pyramids
  • 6.1Mathematical pyramids and triangles
  • 6.2Egyptian pyramids
• 7Disputed sightings
• 8See also
• 9References and footnotes
• 10Further reading
• 11External links

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