GOLDEN THOUGHTS a whistlestop tour of some beautiful mathematical places

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GOLDEN THOUGHTSa whistle-stop tour of some
beautiful mathematical places

• Toni Beardon
• AIMS
• 30 September 2004

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Ruler and compass construction for ?

• Draw a square of side 1 unit.
• Bisect the base
• Draw the diagonal to form a triangle with sides 1
  unit and 1/2 unit so that the diagonal has length
  ?5/2
• Draw the arc of the circle with radius ?5/2 and
  find the point where this arc cuts the extended
  base to give one vertex of the golden rectangle
  with sides (1?5)/2 and 1.

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The Golden Ratio ? (Phi)

• The golden rectangle has sides in the ratio
  x1 such that, when a square of side 1 unit is
  removed, the new rectangle has the same
  proportions. That is
• x 1 1 x-1
• x2 x 1 0
• x (1 ?5)/2.
• Thus ? (1?5)/2 1.618033
• and
• 1/ ? ? - 1 0.618033

• Successive points splitting a golden
  rectangle into squares lie on a logarithmic
  spiral, also known as a whirling square.

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Golden Rectangles and the Fibonacci Sequence

• The Fibonacci sequence is defined by the
  recurrence relation
• Fn Fn-1 Fn-2
• F01, F11
• 1, 1, 2, 3, 5, 8,13, 21, 34, 55,

• The ratios of the terms
• (shown by the rectangles)
• converges to ?

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Continued fraction for ?

• Values of the finite continued fractions

• 
  
  

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The Golden Ratio

• The golden ratio features in Euclids
  Elements (c.300 BC) as the extreme and mean
  ratio. It was studied earlier by Plato and the
  Pythagoreans and it appears even earlier in
  Egyptian architecture in the Great Pyramid. The
  term golden and the use of ? as the notation
  were introduced in the 19th and 20th centuries.

• Taking b1 gives the quadratic equation
• a2 a 1 0, with solution
• ?(1 ?5) /2 1.618033
• So ? is an algebraic number and not
  transcendental. More

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The Great Golden Pyramid of Gizac. 2560 BC

• The structure consists of approximately 2
  million blocks of stone, each weighing more than
  two tons. It has been suggested that there are
  enough blocks in the three pyramids to build a 3
  m (10 ft) high, 0.3 m (1 ft) thick wall around
  France. The area covered by the Great pyramid can
  accommodate St Peter's in Rome, the cathedrals of
  Florence and Milan, and Westminster and St Paul's
  in London combined.

• Man fears Time, yet Time fears the Pyramids
• Arab proverb

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The Divine Proportion in Art and Nature

• The Parthenon in Athens, built in 447 BC, is
  an example of the use of the golden section in
  architecture.
• http//academic.reed.edu/humanities/110Tech/Parth
  enon.html

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Golden Ratio or Mean or Section or Number also
called The Divine Proportion

• The Greeks left no record of any work on
  algebra. The Arabs introduced algebra and
  preserved and built on the legacy of Greek
  mathematics.
• Al-Khwarizmi, (780-850), from Iraq,
  introduced problems on dividing a line of length
  10 units in two parts using a quadratic equation.
  Abu-Kamil (850-930), from Egypt, developed
  similar equations arising from dividing a line in
  various ways.
• Leonardo of Pisa (Fibonacci) used many
  Arabic sources and he specifically referred to
  the golden ratio in Liber Abaci, published in
  1202.
• Pacioli in Divina Proportione (1509) gives
  an account of what was known about the golden
  ratio. He explains his title as follows
• it seems to me that the proper title for this
  treatise must be Divine Proportion. This is
  because there are very many similar attributes
  which I find in our proportion all befitting
  God himself which is the subject of our very
  useful discourse.

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Leonardo da Vinci 1452-1519 The Last Supper
Convent of Santa Maria delle Grazie (Refectory),
Milan

• The Golden Ruler

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The Divine Proportion in Art and Nature

• Compare these two Leonardo da Vinci sketches.
  This one is based on the work of the Roman
  architect Vitruvius. It shows the human body with
  the parts in prescribed proportions based on a
  circle and a square. Leonardo studied this
  configuration in relation to the classical
  squaring the circle problem of trying to find a
  square with the same area as a circle.

• This drawing illustrates the use of the
  golden section. The height of a person was
  divided into two sections, the dividing point
  being the navel. Leonardo drew his portraits so
  that the distance from the soles of the feet to
  the navel, then divided by the distance from the
  navel to the top of the head was equal to 0.618,
  the golden mean.

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Salvador Dali The Last Supper

• Salvador Dali. 1904-1989. "The Sacrament of
  the Last Supper". National Gallery of Art,
  Washington DC. 1955
• The painting incorporates part of a
  dodecahedron which involves the golden ratio.

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Pentakite ABCDE is a regular pentagon. Find the
length BE

• The angles of a regular pentagon are 108 so
  all the angles can be found.
• ?s CDF, EDB are congruent isosceles
  triangles. If the sides of the regular pentagon
  are 1 unit and FD x units then BE x units.
• ?s EDB, BEF are similar, hence
• 
  More
• 
  

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Exact values of trig ratios for special angles

• cos36osin54o?/2 (1?5)/4
• sec72ocosec18o2 ? (1?5)
• cos72osin18o1/2 ? (?5 - 1)/4

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Pent Show ABFE is a rhombus and find the ratio
ECEF

• If EBECx then FCx-1
• ?s EBF, DCF are similar.
• Hence EBEFCDCF
• So we have
• x2 x 1 0
• From which we get x ?
• and EC EF EF FC ?
• So F divides EC and BD
• in the golden ratio. More

• ABCDE is a regular pentagon with sides of 1
  unit.
• So ?s AEB, BFE are congruent isosceles
  triangles with angles 108o, 36o, 36o and ABFE is
  a rhombus.

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Golden TriangleTriangles ABC, CBD, ABD are
isosceles. Find the ratio ABBC

• Hence ?s ABC, BCD are similar with angles
  36o,72o, 72o.
• Let ABp and BCBDDAq then, from these
  similar triangles,
• p q q p-q so writing the ratio p/qr,
• r 1 1 r 1
• Thus r ?, the golden ratio.
• Also AD DC BD DC, so D divides
  the side AC in the golden ratio.
  More

• Let ?BAC?ABD x.
• Then ?BDC ?BCD2x,
• ?CBD180-4x and
• So 180-4xx2x and
• X36o.

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Automata generating Fibonacci numbers

• Variables A and B
• Constants none
• Start A
• Rules A?B, B?AB
• A, B, AB, BAB, ABBAB, BABABBAB,
• ABBABBABABBAB,
• BABABBABABBABBABABBAB,
• 1, 1, 2, 3, 5, 8, 13, 21,

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Gnomons carpenters tools used in Babylonian and
Greek mathematics

• Each Fibonacci number has a gnomon with the
  area of the gnomon equal to the number.

• Each gnomon is a hexagon. You can fit two
  together like a jigsaw to make the next one. The
  diagram shows the lengths of the sides, how F2n-1
  and F2n make up F2n1, and F2n and F2n1 make up
  F2n2 .

• Using gnomons you can find the recurrence
  relations
• More More

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Golden PowersFind a formula for ?n in terms of ?.

• We know ?2 ?1 so ?n2 ?n1 ?n
• Example ?7 ?6 ?5 2?5 ?4 3 ? 4 2?3
  5?3 3?2
• 8?2 5?
  13?8
• F7 ? F6
• Conjecture ?n Fn ? Fn-1
• Proof Let ?n an?bn then, multiplying by ?,
• ?n1 an?2 bn ?
• an(?1) bn ?
• (anbn) ? an
• So an1 an bn and bn1an .
• Then, substituting for bn , we get an1
  an an-1
• which is the recurrence relation defining
  the terms of the Fibonacci sequence,
  hence ?n Fn?Fn-1 More

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Golden FibsWhen is a Fibonacci sequence also a
geometric sequence?

• The Fibonacci sequence Fn is defined by the
  relation Fn2 Fn1 Fn where
  Fo0 and F11.
• Now suppose that we take the same relation
  and more general sequences Xn with any two
  starting values X0 and X1 .For a geometric
  sequence the terms must be
  X0 , r X0 , r2 X0 , rn X0 where r is the
  common ratio, so for a geometric-Fibonacci
  sequence we have
• rn2 X0 rn1Xo rnXo .
• As Xo ?0 and r?0 we can divide by rnXo , so
• r2 r 1, hence r is the golden ratio ?.
• We have shown that a geometric sequence is a
  Fibonacci type when the ratio is golden. It can
  be shown that a Fibonacci sequence with the first
  two terms Fo and ?F0 is a geometric sequence
  with common ratio ? . More

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Golden ThoughtsThe areas A1, A2 and A3 are
equal. What are the ratios RXXS and RYYQ?

• Let PSx, SXa and RXb.
• Knowing that areas A1,A2,A3 are equal the
  other lengths can be found. So
• which simplifies to
• So writing the ratio RXSX as b/ar this
  gives r2-r-10 hence this is the golden ratio.
• The ratio RYYQ is then 1
  ? ? ?2 ? ? 1, golden again!
• 
  More

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Pythagorean Golden Means

• Take any two numbers a and b where 0 lt b lt a.
  The arithmetic mean is A(ab)/2, the geometric
  mean is G?ab and the harmonic mean is

• The means A, G and H can be the lengths of the
  sides of a right angled triangle if and only if
• a b?3
• More

• The proof uses only elementary algebra

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How many solutions?

• One solution is x 2. Substitute and check!
• There is another solution which is easy to
  miss! It is approximately 2.3305 . Use a
  numerical method rather than an analytic method
  to find it.
  
• 
  More

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Golden Ellipse

• An ellipse with semi axes a and b fits between
  two circles of radii a and b.
• If the area of the ellipse is equal to the
  area of the annulus what is the ratio ba?
• The areas are and
• so for he areas to be equal
• so that

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Darts, kites and aperiodic tiling

• The lengths of the sides of this rhombus are
  equal to the golden ratio and POOR ?

• Roger Penrose used this rhombus in his famous
  tilings filling the plane. An aperiodic tiling
  never repeats itself, unlike a tessellation. John
  Conway has asked if Penrose tilings are three
  colourable in such a way that adjacent tiles have
  different colours. Sibley and Wagon (2000) proved
  that tilings by rhombi are three-colourable, and
  Babilon (2001) proved that tilings by kites and
  darts are three-colourable. McClure then found an
  algorithm that appears to three-color tilings by
  kites and darts, rhombi, and pentacles.
  More

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Recent research by Dov Aharonov, Alan Beardon and
Kathy Driver

• It has been shown recently that many of the
  known identities for Fibonacci numbers have
  counterparts for solutions of other recurrence
  relations with constant coefficients.
• The Chebyshev polynomials satisfy a
  recurrence relation with a variable x in the
  coefficients. By taking suitable values for x
  these polynomials reduce to the solutions of
  second order recurrence relations with constant
  coefficients. Thus by proving identities for
  Chebyshev polynomials the identities from
  Fibonacci numbers are recaptured and also, at the
  same time, the corresponding identities for the
  general constant coefficients recurrence relation.

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• Thank you
• Toni Beardon
• LAB11_at_cam.ac.uk