“Golden Ratio”的概念、定义、习题解题方法及翻译-数学辞典

单词

Golden Ratio

释义

Golden Ratio

A number often encountered when taking the ratios of distances in simple geometric figures such as the Decagon andDodecagon. It is denoted , or sometimes (which is an abbreviation of the Greek ``tome,'' meaning ``tocut''). is also known as the Divine Proportion, Golden Mean, and Golden Section and is aPisot-Vijayaraghavan Constant. It has surprising connections with ContinuedFractions and the Euclidean Algorithm for computing the Greatest Common Divisor of twoIntegers.

Given a Rectangle having sides in the ratio , is defined such that partitioning the originalRectangle into a Square and new Rectangle results in a new Rectangle having sides with aratio . Such a Rectangle is called a Golden Rectangle, and successive points dividing aGolden Rectangle into Squares lie on a Logarithmic Spiral. This figure is known as aWhirling Square.

This means that

(1)

(2)

So, by the Quadratic Equation,

			(3)

			(4)

(Sloane's A001622).

A geometric definition can be given in terms of the above figure. Let the ratio . The Numerator andDenominator can then be taken as and without loss of generality. Now define theposition of by

(5)

Plugging in gives

(6)

(7)

which can be solved using the Quadratic Equation to obtain

(8)

is the ``most'' Irrational number because it has a Continued Fraction representation

(9)

(Sloane's A000012). Another infinite representation in terms of a Continued Square Root is

(10)

Ramanujan

gave the curious Continued Fraction identities

(11)

(12)

(Ramanathan 1984).

The legs of a Golden Triangle are in a golden ratio to its base. In fact, this was the method used by Pythagorasto construct . Euclid used the following construction.

Draw the Square , call the Midpoint of , so that . Now draw the segment, which has length

(13)

and construct

with this length. Now construct

, then

(14)

The ratio of the Circumradius to the length of the side of a Decagon is also ,

(15)

Similarly, the legs of a Golden Triangle (an Isosceles Triangle with a Vertex Angleof 36°) are in a Golden Ratio to the base. Bisecting a Gaullist Cross also gives a golden ratio (Gardner1961, p. 102).

In the figure above, three Triangles can be Inscribed in the Rectangle ofarbitrary aspect ratio such that the three Right Triangles have equal areas by dividing and in the golden ratio. Then

			(16)
			(17)
			(18)

which are all equal.

The golden ratio also satisfies the Recurrence Relation

(19)

Taking

gives

(20)

(21)

But this is the definition equation for

(when the root with the plus sign is used). Squaring gives


			(22)
			(23)

and so on.

For the difference equations

(24)

is also given by

(25)

In addition,

(26)

where

is the

th Fibonacci Number.

The Substitution Map

			(27)
			(28)

gives

(29)

giving rise to the sequence

(30)

(Sloane's A003849). Here, the zeros occur at positions 1, 3, 4, 6, 8, 9, 11, 12, ... (Sloane's A000201), and the ones occurat positions 2, 5, 7, 10, 13, 15, 18, ... (Sloane's A001950). These are complementary Beatty Sequences generated by

and

. The sequence also has many connections with theFibonacci Numbers.

Salem showed that the set of Pisot-Vijayaraghavan Constants is closed, with the smallest accumulation pointof the set (Le Lionnais 1983).

See also Beraha Constants, Decagon, Five Disks Problem, Golden Ratio Conjugate, GoldenTriangle, Icosidodecahedron, Noble Number, Pentagon, Pentagram, Phi Number System,Secant Method

References

Golden Ratio

Boyer, C. B. History of Mathematics. New York: Wiley, p. 56, 1968.

Coxeter, H. S. M. ``The Golden Section, Phyllotaxis, and Wythoff's Game.'' Scripta Mathematica 19, 135-143, 1953.

Dixon, R. Mathographics. New York: Dover, pp. 30-31 and 50, 1991.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/cntfrc/cntfrc.html

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/gold/gold.html

Gardner, M. ``Phi: The Golden Ratio.'' Ch. 8 in The Second Scientific American Book of Mathematical Puzzles & Diversions, A New Selection. New York: Simon and Schuster, 1961.

Gardner, M. ``Notes on a Fringe-Watcher: The Cult of the Golden Ratio.'' Skeptical Inquirer 18, 243-247, 1994.

Herz-Fischler, R. A Mathematical History of the Golden Number. New York: Dover, 1998.

Huntley, H. E. The Divine Proportion. New York: Dover, 1970.

Knott, R. ``Fibonacci Numbers and the Golden Section.'' http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 40, 1983.

Markowsky, G. ``Misconceptions About the Golden Ratio.'' College Math. J. 23, 2-19, 1992.

Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 122-134, 1990.

Pappas, T. ``Anatomy & the Golden Section.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 32-33, 1989.

Ramanathan, K. G. ``On Ramanujan's Continued Fraction.'' Acta. Arith. 43, 209-226, 1984.

Sloane, N. J. A. SequencesA003849,A000012/M0003,A000201/M2322,A001622/M4046,A001950/M1332in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

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