Введение
Соотношение чисел, примерно 1:2.4 – это соотношение, также известное как серебряное соотношение.
Ratio of numbers, approximately 1:2.4
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
В математике две величины находятся в серебряном соотношении (или серебряном среднем), если отношение меньшей из этих двух величин к большей величине равно отношению большей величины к сумме меньшей величины и удвоенной большей величины (см. ниже). Это определяет серебряное отношение как иррациональную математическую постоянную, значение которой, равное одному плюс квадратный корень из двух, приблизительно равно 2.4142135623. Его название является аллюзией на золотое сечение; аналогично тому, как золотое сечение является предельным отношением последовательных чисел Фибоначчи, серебряное соотношение является предельным отношением последовательных чисел Пелла. Серебряное отношение иногда обозначается δS, но может варьироваться от λ до σ.
Ratio of numbers, approximately 1:2.4
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
Математики изучали серебряное отношение со времен древних греков (хотя, возможно, и не давали ему специального названия до недавнего времени) из-за его связи с квадратным корнем из двух, его сходящимися, квадратными треугольными числами, числами Пелла, восьмиугольниками и тому подобным. Описанное выше отношение можно выразить алгебраически, при a > b:
Ratio of numbers, approximately 1:2.4
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
или, эквивалентно,
Ratio of numbers, approximately 1:2.4
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
Серебряное отношение также можно определить с помощью простой непрерывной дроби [2; 2, 2, 2, …]:
Ratio of numbers, approximately 1:2.4
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
Сходящиеся этой непрерывной дроби (, , , , , …) являются отношениями последовательных чисел Пелла. Эти дроби дают точные рациональные приближения серебряного отношения, аналогичные приближениям золотого сечения отношениями последовательных чисел Фибоначчи. Серебряный прямоугольник связан с правильным восьмиугольником. Если правильный восьмиугольник разделен на два равнобедренных трапеции и прямоугольник, то прямоугольник является серебряным прямоугольником с отношением сторон 1:δS, а 4 стороны трапеций находятся в соотношении 1:1:1:δS. Если длина стороны правильного восьмиугольника равна t, то расстояние между противоположными сторонами восьмиугольника (его протяженность) равно δSt, а площадь восьмиугольника равна 2δSt².
Ratio of numbers, approximately 1:2.4
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
a ratio also known as the silver ratio
In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity (see below). This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is sometimes denoted by δS but it can vary from λ to σ.
Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like. The relation described above can be expressed algebraically, for a > b:
or equivalently,
The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ]:
The convergents of this continued fraction (, , , , , ) are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers. The silver rectangle is connected to the regular octagon. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δS, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δS. If the edge length of a regular octagon is t, then the span of the octagon (the distance between opposite sides) is δSt, and the area of the octagon is 2δSt^(2).
Теоретические свойства чисел
Серебряное соотношение является числом Писот-Виджаярагхаван (PV number), так как абсолютная величина его сопряженного меньше 1. Фактически, это второе наименьшее квадратичное PV-число после золотого сечения. Это означает, что расстояние от серебряного соотношения до ближайшего целого числа равно… Таким образом, последовательность дробных частей серебряного соотношения, рассматриваемая как элементы тора, сходится. В частности, эта последовательность не является равномерно распределенной по модулю 1.
The silver ratio is a Pisot–Vijayaraghavan number (PV number), as its conjugate has absolute value less than 1. In fact it is the second smallest quadratic PV number after the golden ratio. This means the distance from to the nearest integer is Thus, the sequence of fractional parts of , (taken as elements of the torus) converges. In particular, this sequence is not equidistributed mod 1.