In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, two quantities are in the supergolden ratio if the
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of the larger number divided by the smaller one is equal to
:
which is the only
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
solution
Solution may refer to:
* Solution (chemistry), a mixture where one substance is dissolved in another
* Solution (equation), in mathematics
** Numerical solution, in numerical analysis, approximate solutions within specified error bounds
* Soluti ...
to the
equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
. It can also be represented using the
hyperbolic cosine
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
as:
:
The decimal expansion of this number begins 1.465571231876768026656731…, and the ratio is commonly represented by the Greek letter
(psi). Its
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
is:
:
The supergolden ratio is also the fourth smallest
Pisot number
Charles Pisot (2 March 1910 – 7 March 1984) was a French mathematician. He is chiefly recognized as one of the primary investigators of the numerical set associated with his name, the Pisot–Vijayaraghavan numbers.
He followed the classical p ...
.
Supergolden sequence
The supergolden sequence, also known as the
Narayana's cows
Nārāyaṇa Paṇḍita ( sa, नारायण पण्डित) (1340–1400) was an Indian mathematician. Plofker writes that his texts were the most significant Sanskrit mathematics treatises after those of Bhaskara II, other than the K ...
sequence, is a
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
where the ratio between consecutive terms approaches the supergolden ratio.
The first three terms are each one, and each term after that is calculated by adding the previous term and the term two places before that; that is,
, with
. The first values are 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595…
(
OEIS:A000930).
Properties
![Supergolden triangle](https://upload.wikimedia.org/wikipedia/commons/5/5a/Supergolden_triangle.png)
Many of the properties of the supergolden ratio are related to those of the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
''φ''. For example, the ''n''th item of Narayana's sequence is the number of ways to tile a 1 × ''n''
rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containi ...
with 1 × 1 and 1 × 3 tiles, while the ''n''th term of the
Fibonacci sequence
In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start ...
is the number of ways to tile a 1 × ''n'' rectangle with 1 × 1 and 1 × 2 tiles. The supergolden ratio satisfies
, while the golden ratio satisfies
. In
Fibonacci's rabbit problem, each pair breeds each cycle starting after two cycles, while in
Narayana's cow problem, each pair breeds each cycle starting after three cycles.
There is a supergolden rectangle that has the property that if a
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
is removed from one side, the remaining rectangle can be divided into two supergolden rectangles of opposite orientations.
Another example is that both the golden ratio and the supergolden ratio are
Pisot numbers.
The supergolden ratio's
algebraic conjugate
In mathematics, in particular field theory, the conjugate elements or algebraic conjugates of an algebraic element , over a field extension , are the roots of the minimal polynomial of over . Conjugate elements are commonly called conju ...
s are
and have a
magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...
of
, as the product of the
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
of
is 1.
Supergolden rectangle
![Supergolden Ratio](https://upload.wikimedia.org/wikipedia/commons/9/9c/Supergolden_Ratio.png)
A supergolden rectangle is a rectangle whose side lengths are in the supergolden ratio, i.e. the length of the longer side divided by the length of the shorter side is equal to
, the supergolden ratio ''ψ''. When a square with the same side length as the shorter side of the rectangle is removed from one side of the rectangle, the sides resulting rectangle will be in a ratio. This rectangle can be divided into rectangles with side-length ratios of and , two supergolden ratios of perpendicular orientations,
and their
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape
A shape or figure is a graphics, graphical representation of an obje ...
s will be in a ratio.
In addition, if the
line that separates the two supergolden rectangles from each other is extended across the rest of the original rectangle such that it – along with the side of the square that was removed from the original rectangle – divides the original rectangle into quadrants, then the larger supergolden rectangle has the same area as the opposite quadrant,
its diagonal length is the length of the short side of the original rectangle divided by
, the fourth quadrant is also a supergolden rectangle, and its diagonal length is
times the length of the short side of the original rectangle.
See also
* Solutions to equations similar to
:
**
Golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
– the only positive solution to the equation
**
Plastic number
In mathematics, the plastic number (also known as the plastic constant, the plastic ratio, the minimal Pisot number, the platin number, Siegel's number or, in French, ) is a mathematical constant which is the unique real solution of the cubic ...
– the only
real solution to the equation
Notes
References
{{Irrational number
Golden ratio
History of geometry
Cubic irrational numbers
Mathematical constants