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The Collected Works and Correspondence of Chauncey Wright

Collected Works of Chauncey Wright, Volume 1

Essays and Reviews
ON THE PHYLLOTAXIS.

ON THE PHYLLOTAXIS.1

By Chauncey Wright,
ASSISTANT IN THE OFFICE OF THE AMERICAN EPIIHMERIS AND NAUTICAL ALMANAC.

§ 1. General Properties of the Phyllotactic Surds.

The fractions 1/2, 1/3, 2/5, 3/5, &c., each of which has a numerator equal to the sum of the two preceding numerators, and a denominator equal to the sum of the two preceding denominators, have been shown to be the successive approximations of the continued fraction A nested continued fraction starting with 1 divided by the sum of 2 plus a fraction, which in turn has 1 divided by the sum of 1 plus another fraction; this pattern continues with each denominator containing 1 plus a subsequent fraction, labeled with "1 + &c." indicating the pattern continues indefinitely.

The value of this fraction is obtained from the expression

Mathematical text presenting a continued fraction expression for x, shown as x = 1 / (1 + x) = 1 / (1 + 1 / (1 + 1 / (1 + ...))). It states successive approximations of x as 1, 1/2, 3/5, 5/8, and so on. The exact value of x is described as the root (√5 − 1) / 2 of the quadratic equation x² + x = 1. The final equality is given as 1 / (2 + x) = 1/2 (3 − √5).

The sum of these two surds is unity, and the latter is equal to the square of the former; hence they divide unity in extreme and mean ratio.

The equation

x2 x = 1

becomes by transposition

x2 = 1 - x

which, in the form of a proportion, is

1 : x = x : 1- x

that is, x and 1 - x divide unity in extreme and mean ratio.

Again, by geometry, if the base of a right-angled triangle = 1, and its altitude = 1/2, the hypothenuse is 1/2 √5. Subtracting the altitude from the hypothenuse, we have remaining 1/2 (√5 - 1), and this subtracted from the base leaves 1/2 (3 - √5 ).

This process is the method in geometry of dividing a line in extreme and mean ratio.

If two quantities are in extreme and mean ratio, their sum is to the greater as the greater is to the smaller, or as the smaller is to their difference; hence the surds A list of five terms labeled 1° through 5°, each expressed as a fraction with a numerator involving a linear combination of square roots of 5 and integers. The terms are: 1° = 1/2 (√5 − 1), 2° = 1/2 (3 − √5), 3° = 1/2 (2√5 − 4), 4° = 1/2 (7 − 3√5), and 5° = 1/2 (5√5 − 11), followed by an indication of continuation ("&c.").

each equal to the difference of the two preceding, are in geometrical progression with the ratio of the extreme and mean proportion. The first term 1/2 (√5 - 1) is equal to the common ratio, which we will indicate by k; hence these surds are the successive powers of the first; that is

The equation defines a sequence of powers of k: the first term is k to the power of 1, the second term is k squared, the third term is k cubed, and the nth term is k raised to the power of n.

The reciprocal of each of these surds is obtained by changing the sign - to +, since the difference of the squares of the two terms within the parenthesis of each surd is equal to 4; moreover each surd may be represented by a continued fraction of the form A nested continued fraction expression with three layers: the outermost fraction has numerator 1 and denominator a plus or minus 1 divided by a plus or minus 1 divided by a plus or minus b plus or minus c.

in which a is the rational term of the surd, and the signs are + for the odd surds, and - for the even surds; thus,

A continued fraction expression showing the equalities 1°=1/(1 + 1/1), 2°=1/(3 - 1/(3 - 1/3 - &c.)), 3°=1/(4 + 1/(4 + 1/(4 + &c.))), and 4°=1/(7 - 1/(7 - 1/(7 - &c.)). Each degree is represented as a nested fraction with continued terms indicated by "&c." The fractions form sequences with alternating addition and subtraction in their denominators.

The odd surds 1°, 3°, 5°, &c., may also be expressed by continued fractions of the form A continued fraction expression starting with 1 divided by (a + 1) minus a fraction with numerator 1 and denominator 1 plus a fraction continuing the pattern. The pattern involves alternating additions and subtractions of 1 and fractions, all involving the variable "a" and the number 1, extending indefinitely.

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And the even surds 2°, 4°, 6°, &c., by continued fractions of the form

Continued fraction expressions are shown. The top portion presents a continued fraction starting with 1/(a-1) and proceeding with nested fractions involving terms of the form (a-2) and repeated additions of 1 in the denominators. Below, three cases are labeled 1º, 2º, and 3º, each equated to 1 and displayed as different continued fractions involving sequential integers with both additions and subtractions of 1 in the denominators, each terminating with "1 + 1/(1 + &c.)" or "1 + 1/(4 - &c.)" to indicate continuation.
The figure presents three continued fraction representations for the integers 4°, 5°, and 6°. Each expression consists of nested fractions with numerators involving integers and complex numbers (i), and denominators containing integers, complex numbers, and continued fractions indicated by "&c." The first expression (4° = 1) starts with the fraction 1 divided by (6 + 1/(1 + 1/5 + i/(1 + 1/5 + &c.)). The second (5° = 1) begins with 1/(12 - 1/(1 + 1/(1 - i/(1 + 1/11 - &c.)))), and the third (6° = 1) starts with 1/(17 + 1/(1 + 1/(1 + i/(1 + 1/16 + &c.)))). Each continued fraction expands by alternating sums and subtractions involving i and integers, illustrating complex continued fractions converging to the respective integers.

§2. Geometrical Properties of the Phyllotactic Surds.

1. If leaves be supposed to follow each other in succession around their axis, with the constant interval k between them, we have the typical arrangement from which other arrangements may be produced by variations following a very simple law. Let the radii of the following figures represent the directions of the leaves in their typical arrangement. The numbers at the ends of the radii indicate the order of the leaves. Series of five diagrams labeled Fig. 1 through Fig. 5 showing lines radiating from a central point with numbered endpoints. Fig. 1 shows two solid lines meeting at a right angle, with endpoints labeled 1, 2, and 9. Fig. 2 adds a third dotted line to the right of Fig. 1, endpoints labeled 1, 2, 3, and 9. Fig. 3 shows four radiating lines, two solid and two dotted, with endpoints numbered 1 through 6. Fig. 4 has eight radiating lines, alternating solid and dotted, with endpoints numbered 1 through 8. Fig. 5 has thirteen lines radiating from the center, a mix of solid and dotted lines, with endpoints numbered 1 through 13. The diagrams show incremental addition of lines and labels in a radial pattern.

The leaves 1 and 2 of Fig. 1 divide the circumference into the spaces 1° and 2°, or into the fractions k and k2, and therefore in extreme and mean ratio. The leaf 3 of Fig. 2 divides the space 1° into the spaces 2° and 3°; the leaves 4 and 5 of Fig. 3 divide each of the spaces 2° into the spaces 3° and 4°; the leaves 6, 7, and 8, of Fig. 4, divide each of the spaces 3° into the spaces 4° and 5°; and the leaves 9, 10, 11, 12, and 13, of Fig. 5, divide each of the spaces 4° into the spaces 5° and 6°; — all in the same ratio, namely, that expressed by k. In general any leaf whatever, falling between two older leaves, divides the included space in extreme and mean ratio. This arrangement effects the most rapid and thorough distribution of the leaves around their axis.

If we consider each leaf as produced by that older leaf which stands nearest to it, we find that 1 produces 2, then 3, and then 4, while 2 produces 5; then 1 produces 6, 2 7, 3 8; then 1 9, 2 10, 3 11, 4 12, 5 13; then 1 14, 2 15, &c. Leaf 1 in producing 2, 3, 4, 6, 9, 14, &c., introduces successively, and alternately on its left and right, the intervals 1°, 2°, 3°, 4°, 5°, 6°, &c., which are the successive powers k, k2, k3, k4, k5, k6, &c. of the first and largest interval.

The suppression of any interval kn brings the leaf which introduces this interval over leaf 1, and produces one of the systems of arrangement expressed by the fractions

1/2, 1/3, 2/5, &c.

The interval 3° of Fig. 2 may be suppressed by moving back 2 one half, and 3 two halves of the interval 3°. This will give the alternate system expressed by the fraction The image shows the equation 1 = k² + 1/k², where k is a variable. The equation expresses that the sum of the square of k and the square of the reciprocal of k equals one.

The suppression of interval 4° by moving forward the leaves 2, 3, and 4 respectively 1/3, 2/3, and 3/3 of the interval 4°, gives the system expressed by the fraction

Equation showing the derivative with respect to s of the function y, expressed as dy/ds = k^2 - f k^4.

In the same manner, the suppression of the intervals 5°, 6°, &c. gives the systems

Two algebraic expressions show the sum and difference of two third powers. The first equation is "x/2 = k^3 + t k^6." The second equation is "y/2 = k^3 - t k^6, &c." The notation "&c." indicates continuation or other related expressions.

If we subtract each member of these equations from unity, we obtain the following:

A mathematical expression labeled (2) showing equalities involving powers of \( k \) and powers of \(-k\). The equalities are: \[ \tfrac{1}{2} = k - \tfrac{1}{2} k^2 = k + \tfrac{1}{2} (-k)^2, \quad \tfrac{3}{2} = k + \tfrac{1}{2} k^4 = k + \tfrac{1}{2} (-k)^4, \quad \tfrac{5}{2} = k - \tfrac{1}{2} k^6 = k + \tfrac{1}{2} (-k)^6, \quad \tfrac{7}{2} = k + \tfrac{1}{2} k^8 = k + \tfrac{1}{2} (-k)^8, \text{ etc.} \]

Certain anomalous forms of arrangement occur in nature, which cannot be expressed by any of the approximations of k. It has been shown, however, that all forms may be expressed by approximations of the fraction

A mathematical expression representing the fraction with numerator 1 and denominator m plus k, written as \( \frac{1}{m+k} \).

in which m is an integer.

The distribution produced by this more general typical arrangement is, in effect, the same as that which we have

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discussed; that is, the intervals 1°, 2°, 3°, &c. of this arrangement are in geometrical progression with the ratio of the extreme and mean proportion for

Mathematical expressions defining a sequence \(n^\circ\) in terms of variables \(m\) and \(k\). The first term is \( \frac{1}{m+k} \), the second term is \( 1 - \frac{m}{m+k} = \frac{k}{m+k} \), the third term is \( \frac{1}{m+k} - \frac{k}{m+k} = \frac{k^2}{m+k} \), the fourth term is \( \frac{k}{m+k} - \frac{k^2}{m+k} = \frac{k^3}{m+k} \), and the general term is \( \frac{k^{n-1}}{m+k} \).

2. If the ratio of the mean motions of two planets be indicated by r, and if x represent the fraction of the circumference between two successive heliocentric conjunctions, then

The figure presents a sequence of algebraic manipulations involving variables r, x, and k. It starts with the definition r = x / (1 + x). Next, it solves this equation for x, yielding x = r / (1 - r). Finally, assuming r = k², it substitutes and simplifies to show that x = k, through the steps x = k² / (1 - k²) = (k² / k) = k.
and if r be any one of the approximations of k2, 1/2, 1/3, 2/5, &c., then x is equal to the corresponding approximation of k; 1, 1/2, 2/5, &c.- Thus, if r = 2/5, x = 2/5 ÷ 3/5 = 2/3.

Since, as Professor Peirce has shown, the mean motions of the planets form a progression in which the ratios are nearly expressed by a series of phyllotactic fractions, it follows that the intervals between the successive conjunctions of neighboring planets are also expressed by these fractions, and that their points of conjunction are thus distributed around the sun as leaves are around their axes, in the several systems. If the ratio of the mean motions of two planets were exactly k\ their points of conjunction would be distributed in the most rapid and thorough manner.

The idea of thorough distribution (in the leaves of plants with reference to the formation of wood, and in the conjunctions of planets with reference to their mutual perturbations) seems to be a central thought or typical principle of these natural arrangements. Variations from the typical arrangement produce in plants specific forms of symmetry, while in the planets they fulfil conditions as yet imperfectly understood.

The progression of the mean motions ought not to be geometrical, lest through the common ratio there occur constant repetitions of the same configurations in the whole system of planets, which would cause their action through one another to be always of the same character.

The ratios of this progression, therefore, while each approaches the typical ratio k2, ought to differ as much as possible among themselves, and most for the larger planets; as in fact they do.

§3. The Phyllotactic Function.

Required the form of the function which by the substitution for the variable of the common series of numbers

1, 2, 3, 4, 5, 6, &c.

gives the phyllotactic series of numbers

1, 1, 2, 3, 5, 8, &c.

Each of the phyllotactic numbers is equal to the sum of the two preceding; hence the required function is subject to the condition

Mathematical text discussing the function \( f(x) \) and its finite difference \(\Delta f\left(x+\frac{1}{2}\right) = f(x-1)\). It examines exponentials \(a^x, a^{x+1}, a^{x+2}, \dots\) arranged with their differences and establishes a condition \(a^2 - a = 1\). The roots \(\frac{1}{2}(1 + \sqrt{5})\) and \(\frac{1}{2}(1 - \sqrt{5})\) are identified as \(k\) and \(-k\), respectively. The general form of \(f\) is given as \(A k^x + B (-k)^x\), with conditions on \(f(0)\) and \(f(1)\) to determine \(A\) and \(B\). The text derives the expression for phyllotactic fractions and provides values for integer arguments \(x = 1, 2, 3, 4, \dots\), resulting in sequences such as \(0, 1, \frac{1}{2}, \frac{2}{3}, \dots\) and \(1, 3, 4, 7, \dots\). The discussion links these expressions to the rational terms of surds and includes formulas involving powers of \(k\) and their combinations.