Continued fraction

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${\displaystyle a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{a_{n}}}}}}}}}}$

A finite regular continued fraction, where ${\displaystyle n}$ is a non-negative integer, ${\displaystyle a_{0}}$ is an integer, and ${\displaystyle a_{i}}$ is a positive integer, for ${\displaystyle i=1,\ldots ,n}$.

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers ${\displaystyle a_{i}}$ are called the coefficients or terms of the continued fraction.

It is generally assumed that the numerator of all of the fractions is 1. If arbitrary values and/or functions are used in place of one or more of the numerators or the integers in the denominators, the resulting expression is a generalized continued fraction. When it is necessary to distinguish the first form from generalized continued fractions, the former may be called a simple or regular continued fraction, or said to be in canonical form.

Continued fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers. Every rational number ${\displaystyle p}$/${\displaystyle q}$ has two closely related expressions as a finite continued fraction, whose coefficients a_i can be determined by applying the Euclidean algorithm to ${\displaystyle (p,q)}$. The numerical value of an infinite continued fraction is irrational; it is defined from its infinite sequence of integers as the limit of a sequence of values for finite continued fractions. Each finite continued fraction of the sequence is obtained by using a finite prefix of the infinite continued fraction's defining sequence of integers. Moreover, every irrational number ${\displaystyle \alpha }$ is the value of a unique infinite regular continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the incommensurable values ${\displaystyle \alpha }$ and 1. This way of expressing real numbers (rational and irrational) is called their continued fraction representation.

The term continued fraction may also refer to representations of rational functions, arising in their analytic theory. For this use of the term, see Padé approximation and Chebyshev rational functions.

Motivation and notation

Consider, for example, the rational number 415/93, which is around 4.4624. As a first approximation, start with 4, which is the integer part; 415/93 = 4 + 43/93. The fractional part is the reciprocal of 93/43 which is about 2.1628. Use the integer part, 2, as an approximation for the reciprocal to obtain a second approximation of 4 + 1/2 = 4.5; 93/43 = 2 + 7/43. The remaining fractional part, 7/43, is the reciprocal of 43/7, and 43/7 is around 6.1429. Use 6 as an approximation for this to obtain 2 + 1/6 as an approximation for 93/43 and 4 + 1/2 + 1/6, about 4.4615, as the third approximation; 43/7 = 6 + 1/7. Finally, the fractional part, 1/7, is the reciprocal of 7, so its approximation in this scheme, 7, is exact (7/1 = 7 + 0/1) and produces the exact expression 4 + 1/2 + 1/6 + 1/7 for 415/93.

The expression 4 + 1/2 + 1/6 + 1/7 is called the continued fraction representation of 415/93. This can be represented by the abbreviated notation 415/93 = [4; 2, 6, 7]. (It is customary to replace only the first comma by a semicolon.) Some older textbooks use all commas in the (n + 1)-tuple, for example, [4, 2, 6, 7].

If the starting number is rational, then this process exactly parallels the Euclidean algorithm applied to the numerator and denominator of the number. In particular, it must terminate and produce a finite continued fraction representation of the number. The sequence of integers that occur in this representation is the sequence of successive quotients computed by the Euclidean algorithm. If the starting number is irrational, then the process continues indefinitely. This produces a sequence of approximations, all of which are rational numbers, and these converge to the starting number as a limit. This is the (infinite) continued fraction representation of the number. Examples of continued fraction representations of irrational numbers are:

• √19 = [4;2,1,3,1,2,8,2,1,3,1,2,8,...] (sequence A010124 in the OEIS). The pattern repeats indefinitely with a period of 6.
• e = [2;1,2,1,1,4,1,1,6,1,1,8,...] (sequence A003417 in the OEIS). The pattern repeats indefinitely with a period of 3 except that 2 is added to one of the terms in each cycle.
• π = [3;7,15,1,292,1,1,1,2,1,3,1,...] (sequence A001203 in the OEIS). No pattern has ever been found in this representation.
• Φ = [1;1,1,1,1,1,1,1,1,1,1,1,...] (sequence A000012 in the OEIS). The golden ratio, the irrational number that is the "most difficult" to approximate rationally. See: A property of the golden ratio φ.
• γ = [0;1,1,2,1,2,1,4,3,13,5,1,...] (sequence A002852 in the OEIS). The Euler–Mascheroni constant, which is expected but not known to be irrational, and whose continued fraction has no apparent pattern.

Continued fractions are, in some ways, more "mathematically natural" representations of a real number than other representations such as decimal representations, and they have several desirable properties:

• The continued fraction representation for a rational number is finite and only rational numbers have finite representations. In contrast, the decimal representation of a rational number may be finite, for example 137/1600 = 0.085625, or infinite with a repeating cycle, for example 4/27 = 0.148148148148...
• Every rational number has an essentially unique simple continued fraction representation. Each rational can be represented in exactly two ways, since [a₀;a₁,... a_n−1,a_n] = [a₀;a₁,... a_n−1,(a_n−1),1]. Usually the first, shorter one is chosen as the canonical representation.
• The simple continued fraction representation of an irrational number is unique. (However, additional representations are possible when using generalized continued fractions; see below.)
• The real numbers whose continued fraction eventually repeats are precisely the quadratic irrationals. For example, the repeating continued fraction [1;1,1,1,...] is the golden ratio, and the repeating continued fraction [1;2,2,2,...] is the square root of 2. In contrast, the decimal representations of quadratic irrationals are apparently random. The square roots of all (positive) integers that are not perfect squares are quadratic irrationals, and hence are unique periodic continued fractions.
• The successive approximations generated in finding the continued fraction representation of a number, that is, by truncating the continued fraction representation, are in a certain sense (described below) the "best possible".

Basic formula

A (generalized) continued fraction is an expression of the form

${\displaystyle a_{0}+{\cfrac {b_{1}}{a_{1}+{\cfrac {b_{2}}{a_{2}+{\cfrac {b_{3}}{a_{3}+{_{\ddots }}}}}}}}}$

where a_i and b_i can be any complex numbers.

When b_i = 1 for all i the expression is called a simple continued fraction. When the expression contains finitely many terms, it is called a finite continued fraction. When the expression contains infinitely many terms, it is called an infinite continued fraction. When the terms eventually repeat from some point onwards, the expression is called a periodic continued fraction.

Thus, all of the following illustrate valid finite simple continued fractions:

Examples of finite simple continued fractions

Formula	Numeric	Remarks
${\displaystyle \ a_{0}}$	${\displaystyle \ 2}$	All integers are a degenerate case
${\displaystyle \ a_{0}+{\cfrac {1}{a_{1}}}}$	${\displaystyle \ 2+{\cfrac {1}{3}}}$	Simplest possible fractional form
${\displaystyle \ a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}}}}}}$	${\displaystyle \ -3+{\cfrac {1}{2+{\cfrac {1}{18}}}}}$	First integer may be negative
${\displaystyle \ a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}}}}}}}}$	${\displaystyle \ {\cfrac {1}{15+{\cfrac {1}{1+{\cfrac {1}{102}}}}}}}$	First integer may be zero

For simple continued fractions of the form

${\displaystyle r=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{_{\ddots }}}}}}}}}$

the ${\displaystyle a_{n}}$ term can be calculated using the following recursive formula:

${\displaystyle a_{n}=\left\lfloor {\frac {N_{n}}{N_{n+1}}}\right\rfloor }$

where ${\displaystyle N_{n+1}=N_{n-1}{\bmod {N}}_{n}}$ and ${\displaystyle {\begin{cases}N_{0}=r\\N_{1}=1\end{cases}}}$

From which it can be understood that the ${\displaystyle a_{n}}$ sequence stops if ${\displaystyle N_{n+1}=0}$.

Calculating continued fraction representations

Consider a real number r. Let ${\displaystyle i=\lfloor r\rfloor }$ be the integer part of r and let ${\displaystyle f=r-i}$ be the fractional part of r. Then the continued fraction representation of r is ${\displaystyle [i;a_{1},a_{2},\ldots ]}$, where ${\displaystyle [a_{1};a_{2},\ldots ]}$ is the continued fraction representation of ${\displaystyle 1/f}$.

To calculate a continued fraction representation of a number r, write down the integer part (technically the floor) of r. Subtract this integer part from r. If the difference is 0, stop; otherwise find the reciprocal of the difference and repeat. The procedure will halt if and only if r is rational. This process can be efficiently implemented using the Euclidean algorithm when the number is rational. The table below shows an implementation of this procedure for the number 3.245, resulting in the continued fraction expansion [3; 4,12,4].

Find the simple continued fraction for ${\displaystyle 3.245={\frac {649}{200}}}$

Step	Real Number	Integer part	Fractional part	Simplified	Reciprocal of f
1	${\displaystyle r={\frac {649}{200}}}$	${\displaystyle i=3}$	${\displaystyle f={\frac {649}{200}}-3}$	${\displaystyle ={\frac {49}{200}}}$	${\displaystyle {\frac {1}{f}}={\frac {200}{49}}}$
2	${\displaystyle r={\frac {200}{49}}}$	${\displaystyle i=4}$	${\displaystyle f={\frac {200}{49}}-4}$	${\displaystyle ={\frac {4}{49}}}$	${\displaystyle {\frac {1}{f}}={\frac {49}{4}}}$
3	${\displaystyle r={\frac {49}{4}}}$	${\displaystyle i=12}$	${\displaystyle f={\frac {49}{4}}-12}$	${\displaystyle ={\frac {1}{4}}}$	${\displaystyle {\frac {1}{f}}={\frac {4}{1}}}$
4	${\displaystyle r=4}$	${\displaystyle i=4}$	${\displaystyle f=4-4}$	${\displaystyle =0}$	STOP
Continued fraction form for ${\displaystyle 3.245={\frac {649}{200}}=[3;4,12,4]}$ = 3 + 1/4 + 1/12 + 1/4

Notations

The integers ${\displaystyle a_{0}}$, ${\displaystyle a_{1}}$ etc., are called the coefficients or terms of the continued fraction. One can abbreviate the continued fraction

${\displaystyle x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}}}}}}}}$

in the notation of Carl Friedrich Gauss

${\displaystyle x=a_{0}+{\underset {i=1}{\overset {3}{\mathrm {K} }}}~{\frac {1}{a_{i}}}}$

or as

${\displaystyle x=[a_{0};a_{1},a_{2},a_{3}]}$,

or in the notation of Pringsheim as

${\displaystyle x=a_{0}+{\frac {1\mid }{\mid a_{1}}}+{\frac {1\mid }{\mid a_{2}}}+{\frac {1\mid }{\mid a_{3}}},}$

or in another related notation as

${\displaystyle x=a_{0}+{1 \over a_{1}+{}}{1 \over a_{2}+{}}{1 \over a_{3}{}}.}$

Sometimes angle brackets are used, like this:

${\displaystyle x=\left\langle a_{0};a_{1},a_{2},a_{3}\right\rangle .}$

The semicolon in the square and angle bracket notations is sometimes replaced by a comma.

One may also define infinite simple continued fractions as limits:

${\displaystyle [a_{0};a_{1},a_{2},a_{3},\,\ldots ]=\lim _{n\to \infty }[a_{0};a_{1},a_{2},\,\ldots ,a_{n}].}$

This limit exists for any choice of ${\displaystyle a_{0}}$ and positive integers ${\displaystyle a_{1},a_{2},\ldots }$.

Finite continued fractions

Every finite continued fraction represents a rational number, and every rational number can be represented in precisely two different ways as a finite continued fraction, with the conditions that the first coefficient is an integer and the other coefficients are positive integers. These two representations agree except in their final terms. In the longer representation the final term in the continued fraction is 1; the shorter representation drops the final 1, but increases the new final term by 1. The final element in the short representation is therefore always greater than 1, if present. In symbols:

[a₀; a₁, a₂, ..., a_{n − 1}, a_n, 1] = [a₀; a₁, a₂, ..., a_{n − 1}, a_n + 1].

[a₀; 1] = [a₀ + 1].

Reciprocals

The continued fraction representations of a positive rational number and its reciprocal are identical except for a shift one place left or right depending on whether the number is less than or greater than one respectively. In other words, the numbers represented by ${\displaystyle [a_{0};a_{1},a_{2},\ldots ,a_{n}]}$ and ${\displaystyle [0;a_{0},a_{1},\ldots ,a_{n}]}$ are reciprocals.

For instance if ${\displaystyle a}$ is an integer and ${\displaystyle x<1}$ then

${\displaystyle x=0+{\frac {1}{a+{\frac {1}{b}}}}}$ and ${\displaystyle {\frac {1}{x}}=a+{\frac {1}{b}}}$.

If ${\displaystyle x>1}$ then

${\displaystyle x=a+{\frac {1}{b}}}$ and ${\displaystyle {\frac {1}{x}}=0+{\frac {1}{a+{\frac {1}{b}}}}}$.

The last number that generates the remainder of the continued fraction is the same for both ${\displaystyle x}$ and its reciprocal.

For example,

${\displaystyle 2.25={\frac {9}{4}}=[2;4]}$ and ${\displaystyle {\frac {1}{2.25}}={\frac {4}{9}}=[0;2,4]}$.

Infinite continued fractions and convergents

Convergents approaching the golden ratio

Every infinite continued fraction is irrational, and every irrational number can be represented in precisely one way as an infinite continued fraction.

An infinite continued fraction representation for an irrational number is useful because its initial segments provide rational approximations to the number. These rational numbers are called the convergents of the continued fraction. The larger a term is in the continued fraction, the closer the corresponding convergent is to the irrational number being approximated. Numbers like π have occasional large terms in their continued fraction, which makes them easy to approximate with rational numbers. Other numbers like e have only small terms early in their continued fraction, which makes them more difficult to approximate rationally. The golden ratio Φ has terms equal to 1 everywhere—the smallest values possible—which makes Φ the most difficult number to approximate rationally. In this sense, therefore, it is the "most irrational" of all irrational numbers. Even-numbered convergents are smaller than the original number, while odd-numbered ones are larger.

For a continued fraction [a₀; a₁, a₂, ...], the first four convergents (numbered 0 through 3) are

a₀/1, a₁a₀ + 1/a₁, a₂(a₁a₀ + 1) + a₀/a₂a₁ + 1, a₃(a₂(a₁a₀ + 1) + a₀) + (a₁a₀ + 1)/ a₃(a₂a₁ + 1) + a₁.

The numerator of the third convergent is formed by multiplying the numerator of the second convergent by the third coefficient, and adding the numerator of the first convergent. The denominators are formed similarly. Therefore, each convergent can be expressed explicitly in terms of the continued fraction as the ratio of certain multivariate polynomials called continuants.

If successive convergents are found, with numerators h₁, h₂, ... and denominators k₁, k₂, ... then the relevant recursive relation is:

h_n = a_nh_{n − 1} + h_{n − 2},

k_n = a_nk_{n − 1} + k_{n − 2}.

The successive convergents are given by the formula

h_n/k_n = a_nh_{n − 1} + h_{n − 2}/a_nk_{n − 1} + k_{n − 2}.

Thus to incorporate a new term into a rational approximation, only the two previous convergents are necessary. The initial "convergents" (required for the first two terms) are ⁰⁄₁ and ¹⁄₀. For example, here are the convergents for [0;1,5,2,2].

n	−2	−1	0	1	2	3	4
an			0	1	5	2	2
hn	0	1	0	1	5	11	27
kn	1	0	1	1	6	13	32

When using the Babylonian method to generate successive approximations to the square root of an integer, if one starts with the lowest integer as first approximant, the rationals generated all appear in the list of convergents for the continued fraction. Specifically, the approximants will appear on the convergents list in positions 0, 1, 3, 7, 15, ... , 2^k−1, ... For example, the continued fraction expansion for √3 is [1;1,2,1,2,1,2,1,2,...]. Comparing the convergents with the approximants derived from the Babylonian method:

n	−2	−1	0	1	2	3	4	5	6	7
an			1	1	2	1	2	1	2	1
hn	0	1	1	2	5	7	19	26	71	97
kn	1	0	1	1	3	4	11	15	41	56

x₀ = 1 = 1/1

x₁ = 1/2(1 + 3/1) = 2/1 = 2

x₂ = 1/2(2 + 3/2) = 7/4

x₃ = 1/2(7/4 + 3/7/4) = 97/56

Properties

A Baire space is a topological space on infinite sequences of natural numbers. The infinite continued fraction provides a homeomorphism from the Baire space to the space of irrational real numbers (with the subspace topology inherited from the usual topology on the reals). The infinite continued fraction also provides a map between the quadratic irrationals and the dyadic rationals, and from other irrationals to the set of infinite strings of binary numbers (i.e. the Cantor set); this map is called the Minkowski question mark function. The mapping has interesting self-similar fractal properties; these are given by the modular group, which is the subgroup of Möbius transformations having integer values in the transform. Roughly speaking, continued fraction convergents can be taken to be Möbius transformations acting on the (hyperbolic) upper half-plane; this is what leads to the fractal self-symmetry.

The limit probability distribution of the coefficients in the continued fraction expansion of a random variable uniformly distributed in (0, 1) is the Gauss–Kuzmin distribution.

Some useful theorems

If ${\displaystyle a_{0}}$, ${\displaystyle a_{1}}$, ${\displaystyle a_{2}}$, ${\displaystyle \ldots }$ is an infinite sequence of positive integers, define the sequences ${\displaystyle h_{n}}$ and ${\displaystyle k_{n}}$ recursively:

${\displaystyle h_{n}=a_{n}h_{n-1}+h_{n-2}}$			${\displaystyle h_{-1}=1}$		${\displaystyle h_{-2}=0}$
${\displaystyle k_{n}=a_{n}k_{n-1}+k_{n-2}}$			${\displaystyle k_{-1}=0}$		${\displaystyle k_{-2}=1}$

Theorem 1. For any positive real number ${\displaystyle z}$

${\displaystyle \left[a_{0};a_{1},\,\dots ,a_{n-1},z\right]={\frac {zh_{n-1}+h_{n-2}}{zk_{n-1}+k_{n-2}}}.}$

Theorem 2. The convergents of [${\displaystyle a_{0}}$; ${\displaystyle a_{1}}$, ${\displaystyle a_{2}}$, ${\displaystyle \ldots }$] are given by

${\displaystyle \left[a_{0};a_{1},\,\dots ,a_{n}\right]={\frac {h_{n}}{k_{n}}}.}$

Theorem 3. If the ${\displaystyle n}$th convergent to a continued fraction is ${\displaystyle h_{n}}$/${\displaystyle k_{n}}$, then

${\displaystyle k_{n}h_{n-1}-k_{n-1}h_{n}=(-1)^{n}.}$

Corollary 1: Each convergent is in its lowest terms (for if ${\displaystyle h_{n}}$ and ${\displaystyle k_{n}}$ had a nontrivial common divisor it would divide ${\displaystyle k_{n}h_{n-1}-k_{n-1}h_{n}}$, which is impossible).

Corollary 2: The difference between successive convergents is a fraction whose numerator is unity:

${\displaystyle {\frac {h_{n}}{k_{n}}}-{\frac {h_{n-1}}{k_{n-1}}}={\frac {h_{n}k_{n-1}-k_{n}h_{n-1}}{k_{n}k_{n-1}}}={\frac {(-1)^{n+1}}{k_{n}k_{n-1}}}.}$

Corollary 3: The continued fraction is equivalent to a series of alternating terms:

${\displaystyle a_{0}+\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{k_{n}k_{n+1}}}.}$

Corollary 4: The matrix

${\displaystyle {\begin{bmatrix}h_{n}&h_{n-1}\\k_{n}&k_{n-1}\end{bmatrix}}}$

has determinant plus or minus one, and thus belongs to the group of ${\displaystyle 2\times 2}$ unimodular matrices ${\displaystyle \mathrm {GL} (2,\mathbb {Z} )}$.

Theorem 4. Each (${\displaystyle s}$th) convergent is nearer to a subsequent (${\displaystyle n}$th) convergent than any preceding (${\displaystyle r}$th) convergent is. In symbols, if the ${\displaystyle n}$th convergent is taken to be ${\displaystyle [a_{0};a_{1},\ldots ,a_{n}]=x_{n}}$, then

${\displaystyle \left|x_{r}-x_{n}\right|>\left|x_{s}-x_{n}\right|}$

for all ${\displaystyle r<s<n}$.

Corollary 1: The even convergents (before the ${\displaystyle n}$th) continually increase, but are always less than ${\displaystyle x_{n}}$.

Corollary 2: The odd convergents (before the ${\displaystyle n}$th) continually decrease, but are always greater than ${\displaystyle x_{n}}$.

Theorem 5.

${\displaystyle {\frac {1}{k_{n}(k_{n+1}+k_{n})}}<\left|x-{\frac {h_{n}}{k_{n}}}\right|<{\frac {1}{k_{n}k_{n+1}}}.}$

Corollary 1: A convergent is nearer to the limit of the continued fraction than any fraction whose denominator is less than that of the convergent.

Corollary 2: A convergent obtained by terminating the continued fraction just before a large term is a close approximation to the limit of the continued fraction.

Semiconvergents

If

${\displaystyle {\frac {h_{n-1}}{k_{n-1}}},{\frac {h_{n}}{k_{n}}}}$

are consecutive convergents, then any fractions of the form

${\displaystyle {\frac {h_{n-1}+mh_{n}}{k_{n-1}+mk_{n}}},}$

where ${\displaystyle m}$ is an integer such that ${\displaystyle 0\leq m\leq a_{n+1}}$, are called semiconvergents, secondary convergents, or intermediate fractions. The ${\displaystyle (m+1)}$-st semiconvergent equals the mediant of the ${\displaystyle m}$-th one and the convergent ${\displaystyle {\tfrac {h_{n}}{k_{n}}}}$. Sometimes the term is taken to mean that being a semiconvergent excludes the possibility of being a convergent (i.e., ${\displaystyle 0<m<a_{n+1}}$), rather than that a convergent is a kind of semiconvergent.

It follows that semiconvergents represent a monotonic sequence of fractions between the convergents ${\displaystyle {\tfrac {h_{n-1}}{k_{n-1}}}}$ (corresponding to ${\displaystyle m=0}$) and ${\displaystyle {\tfrac {h_{n+1}}{k_{n+1}}}}$ (corresponding to ${\displaystyle m=a_{n+1}}$). The consecutive semiconvergents ${\displaystyle {\tfrac {a}{b}}}$ and ${\displaystyle {\tfrac {c}{d}}}$ satisfy the property ${\displaystyle ad-bc=\pm 1}$.

If a rational approximation ${\displaystyle {\tfrac {p}{q}}}$ to a real number ${\displaystyle x}$ is such that the value ${\displaystyle \left|x-{\tfrac {p}{q}}\right|}$ is smaller than that of any approximation with a smaller denominator, then ${\displaystyle {\tfrac {p}{q}}}$ is a semiconvergent of the continued fraction expansion of ${\displaystyle x}$. The converse is not true, however.

Best rational approximations

See also: Diophantine approximation and Padé approximant

One can choose to define a best rational approximation to a real number x as a rational number n/d, d > 0, that is closer to x than any approximation with a smaller or equal denominator. The simple continued fraction for x can be used to generate all of the best rational approximations for x by applying these three rules:

1. Truncate the continued fraction, and reduce its last term by a chosen amount (possibly zero).
2. The reduced term cannot have less than half its original value.
3. If the final term is even, half its value is admissible only if the corresponding semiconvergent is better than the previous convergent. (See below.)

For example, 0.84375 has continued fraction [0;1,5,2,2]. Here are all of its best rational approximations.

Continued fraction	[0;1]	[0;1,3]	[0;1,4]	[0;1,5]	[0;1,5,2]	[0;1,5,2,1]	[0;1,5,2,2]
Rational approximation	1	3/4	4/5	5/6	11/13	16/19	27/32
Decimal equivalent	1	0.75	0.8	~0.83333	~0.84615	~0.84211	0.84375
Error	+18.519%	−11.111%	−5.1852%	−1.2346%	+0.28490%	−0.19493%	0%

Best rational approximants for π (green circle), e (blue diamond), ϕ (pink oblong), (√3)/2 (grey hexagon), 1/√2 (red octagon) and 1/√3 (orange triangle) calculated from their continued fraction expansions, plotted as slopes y/x with errors from their true values (black dashes)  

The strictly monotonic increase in the denominators as additional terms are included permits an algorithm to impose a limit, either on size of denominator or closeness of approximation.

The "half rule" mentioned above requires that when a_k is even, the halved term a_k/2 is admissible if and only if |x − [a₀ ; a₁, ..., a_{k − 1}]| > |x − [a₀ ; a₁, ..., a_{k − 1}, a_k/2]|  This is equivalent to: Shoemake (1995).

[a_k; a_{k − 1}, ..., a₁] > [a_k; a_{k + 1}, ...].

The convergents to x are "best approximations" in a much stronger sense than the one defined above. Namely, n/d is a convergent for x if and only if |dx − n| has the smallest value among the analogous expressions for all rational approximations m/c with c ≤ d; that is, we have |dx − n| < |cx − m| so long as c < d. (Note also that |d_kx − n_k| → 0 as k → ∞.)

Best rational within an interval

A rational that falls within the interval (x, y), for 0 < x < y, can be found with the continued fractions for x and y. When both x and y are irrational and

x = [a₀; a₁, a₂, ..., a_{k − 1}, a_k, a_{k + 1}, ...]

y = [a₀; a₁, a₂, ..., a_{k − 1}, b_k, b_{k + 1}, ...]

where x and y have identical continued fraction expansions up through a_k−1, a rational that falls within the interval (x, y) is given by the finite continued fraction,

z(x,y) = [a₀; a₁, a₂, ..., a_{k − 1}, min(a_k, b_k) + 1]

This rational will be best in the sense that no other rational in (x, y) will have a smaller numerator or a smaller denominator.

If x is rational, it will have two continued fraction representations that are finite, x₁ and x₂, and similarly a rational y will have two representations, y₁ and y₂. The coefficients beyond the last in any of these representations should be interpreted as +∞; and the best rational will be one of z(x₁, y₁), z(x₁, y₂), z(x₂, y₁), or z(x₂, y₂).

For example, the decimal representation 3.1416 could be rounded from any number in the interval [3.14155, 3.14165). The continued fraction representations of 3.14155 and 3.14165 are

3.14155 = [3; 7, 15, 2, 7, 1, 4, 1, 1] = [3; 7, 15, 2, 7, 1, 4, 2]

3.14165 = [3; 7, 16, 1, 3, 4, 2, 3, 1] = [3; 7, 16, 1, 3, 4, 2, 4]

and the best rational between these two is

[3; 7, 16] = 355/113 = 3.1415929....

Thus, 355/113 is the best rational number corresponding to the rounded decimal number 3.1416, in the sense that no other rational number that would be rounded to 3.1416 will have a smaller numerator or a smaller denominator.

Interval for a convergent

A rational number, which can be expressed as finite continued fraction in two ways,

z = [a₀; a₁, ..., a_{k − 1}, a_k, 1] = [a₀; a₁, ..., a_{k − 1}, a_k + 1] = p_k/q_k

will be one of the convergents for the continued fraction expansion of a number, if and only if the number is strictly between (see this proof)

x = [a₀; a₁, ..., a_{k − 1}, a_k, 2] = 2p_k - p_k-1/2q_k - q_k-1 and

y = [a₀; a₁, ..., a_{k − 1}, a_k + 2] = p_k + p_k-1/q_k + q_k-1

The numbers x and y are formed by incrementing the last coefficient in the two representations for z. It is the case that x < y when k is even, and x > y when k is odd.

For example, the number 355/113 has the continued fraction representations

355/113 = [3; 7, 15, 1] = [3; 7, 16]

and thus 355/113 is a convergent of any number strictly between

[3; 7, 15, 2]	=	688/219 ≈ 3.1415525
[3; 7, 17]	=	377/120 ≈ 3.1416667

Comparison

Consider x = [a₀; a₁, ...] and y = [b₀; b₁, ...]. If k is the smallest index for which a_k is unequal to b_k then x < y if (−1)^k(a_k − b_k) < 0 and y < x otherwise.

If there is no such k, but one expansion is shorter than the other, say x = [a₀; a₁, ..., a_n] and y = [b₀; b₁, ..., b_n, b_{n + 1}, ...] with a_i = b_i for 0 ≤ i ≤ n, then x < y if n is even and y < x if n is odd.

Continued fraction expansion of π and its convergents

To calculate the convergents of π we may set a₀ = ⌊π⌋ = 3, define u₁ = 1/π − 3 ≈ 7.0625 and a₁ = ⌊u₁⌋ = 7, u₂ = 1/u₁ − 7 ≈ 15.9966 and a₂ = ⌊u₂⌋ = 15, u₃ = 1/u₂ − 15 ≈ 1.0034. Continuing like this, one can determine the infinite continued fraction of π as

[3;7,15,1,292,1,1,...] (sequence A001203 in the OEIS).

The fourth convergent of π is [3;7,15,1] = 355/113 = 3.14159292035..., sometimes called Milü, which is fairly close to the true value of π.

Let us suppose that the quotients found are, as above, [3;7,15,1]. The following is a rule by which we can write down at once the convergent fractions which result from these quotients without developing the continued fraction.

The first quotient, supposed divided by unity, will give the first fraction, which will be too small, namely, 3/1. Then, multiplying the numerator and denominator of this fraction by the second quotient and adding unity to the numerator, we shall have the second fraction, 22/7, which will be too large. Multiplying in like manner the numerator and denominator of this fraction by the third quotient, and adding to the numerator the numerator of the preceding fraction, and to the denominator the denominator of the preceding fraction, we shall have the third fraction, which will be too small. Thus, the third quotient being 15, we have for our numerator (22 × 15 = 330) + 3 = 333, and for our denominator, (7 × 15 = 105) + 1 = 106. The third convergent, therefore, is 333/106. We proceed in the same manner for the fourth convergent. The fourth quotient being 1, we say 333 times 1 is 333, and this plus 22, the numerator of the fraction preceding, is 355; similarly, 106 times 1 is 106, and this plus 7 is 113. In this manner, by employing the four quotients [3;7,15,1], we obtain the four fractions:

3/1, 22/7, 333/106, 355/113, ....

The following Maple code will generate continued fraction expansions of pi

To sum up, the pattern is ${\displaystyle {\text{Numerator}}_{i}={\text{Numerator}}_{(i-1)}\cdot {\text{Quotient}}_{i}+{\text{Numerator}}_{(i-2)}}$ ${\displaystyle {\text{Denominator}}_{i}={\text{Denominator}}_{(i-1)}\cdot {\text{Quotient}}_{i}+{\text{Denominator}}_{(i-2)}}$

These convergents are alternately smaller and larger than the true value of π, and approach nearer and nearer to π. The difference between a given convergent and π is less than the reciprocal of the product of the denominators of that convergent and the next convergent. For example, the fraction 22/7 is greater than π, but 22/7 − π is less than 1/7 × 106 = 1/742 (in fact, 22/7 − π is just more than 1/791 = 1/7 × 113).

The demonstration of the foregoing properties is deduced from the fact that if we seek the difference between one of the convergent fractions and the next adjacent to it we shall obtain a fraction of which the numerator is always unity and the denominator the product of the two denominators. Thus the difference between 22/7 and 3/1 is 1/7, in excess; between 333/106 and 22/7, 1/742, in deficit; between 355/113 and 333/106, 1/11978, in excess; and so on. The result being, that by employing this series of differences we can express in another and very simple manner the fractions with which we are here concerned, by means of a second series of fractions of which the numerators are all unity and the denominators successively be the product of every two adjacent denominators. Instead of the fractions written above, we have thus the series:

3/1 + 1/1 × 7 − 1/7 × 106 + 1/106 × 113 − ...

The first term, as we see, is the first fraction; the first and second together give the second fraction, 22/7; the first, the second and the third give the third fraction 333/106, and so on with the rest; the result being that the series entire is equivalent to the original value.

Generalized continued fraction

Main article: Generalized continued fraction

A generalized continued fraction is an expression of the form

${\displaystyle x=b_{0}+{\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{b_{3}+{\cfrac {a_{4}}{b_{4}+\ddots \,}}}}}}}}}$

where the a_n (n > 0) are the partial numerators, the b_n are the partial denominators, and the leading term b₀ is called the integer part of the continued fraction.

To illustrate the use of generalized continued fractions, consider the following example. The sequence of partial denominators of the simple continued fraction of π does not show any obvious pattern:

${\displaystyle \pi =[3;7,15,1,292,1,1,1,2,1,3,1,\ldots ]}$

or

${\displaystyle \pi =3+{\cfrac {1}{7+{\cfrac {1}{15+{\cfrac {1}{1+{\cfrac {1}{292+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{3+{\cfrac {1}{1+\ddots }}}}}}}}}}}}}}}}}}}}}}}$

However, several generalized continued fractions for π have a perfectly regular structure, such as:

${\displaystyle \pi ={\cfrac {4}{1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+{\cfrac {7^{2}}{2+{\cfrac {9^{2}}{2+\ddots }}}}}}}}}}}}={\cfrac {4}{1+{\cfrac {1^{2}}{3+{\cfrac {2^{2}}{5+{\cfrac {3^{2}}{7+{\cfrac {4^{2}}{9+\ddots }}}}}}}}}}=3+{\cfrac {1^{2}}{6+{\cfrac {3^{2}}{6+{\cfrac {5^{2}}{6+{\cfrac {7^{2}}{6+{\cfrac {9^{2}}{6+\ddots }}}}}}}}}}}$

${\displaystyle \displaystyle \pi =2+{\cfrac {2}{1+{\cfrac {1}{1/2+{\cfrac {1}{1/3+{\cfrac {1}{1/4+\ddots }}}}}}}}=2+{\cfrac {2}{1+{\cfrac {1\cdot 2}{1+{\cfrac {2\cdot 3}{1+{\cfrac {3\cdot 4}{1+\ddots }}}}}}}}}$

${\displaystyle \displaystyle \pi =2+{\cfrac {4}{3+{\cfrac {1\cdot 3}{4+{\cfrac {3\cdot 5}{4+{\cfrac {5\cdot 7}{4+\ddots }}}}}}}}}$

The first two of these are special cases of the arctangent function with π = 4 arctan (1) and the fourth and fifth one can be derived using the Wallis product.

${\displaystyle \pi =3+{\cfrac {1^{3}}{6+{\cfrac {1^{3}+2^{3}}{6+{\cfrac {1^{3}+2^{3}+3^{3}+4^{3}}{6+{\cfrac {1^{3}+2^{3}+3^{3}+4^{3}+5^{3}+6^{3}}{6+{\cfrac {1^{3}+2^{3}+3^{3}+4^{3}+5^{3}+6^{3}+7^{3}+8^{3}}{6+\ddots }}}}}}}}}}}$

The continued fraction of ${\displaystyle \pi }$ above consisting of cubes uses the Nilakantha series and an exploit from Leonhard Euler.

Other continued fraction expansions

Periodic continued fractions

Main article: Periodic continued fraction

The numbers with periodic continued fraction expansion are precisely the irrational solutions of quadratic equations with rational coefficients; rational solutions have finite continued fraction expansions as previously stated. The simplest examples are the golden ratio φ = [1;1,1,1,1,1,...] and √2 = [1;2,2,2,2,...], while √14 = [3;1,2,1,6,1,2,1,6...] and √42 = [6;2,12,2,12,2,12...]. All irrational square roots of integers have a special form for the period; a symmetrical string, like the empty string (for √2) or 1,2,1 (for √14), followed by the double of the leading integer.

A property of the golden ratio φ

Because the continued fraction expansion for φ doesn't use any integers greater than 1, φ is one of the most "difficult" real numbers to approximate with rational numbers. Hurwitz's theorem states that any irrational number k can be approximated by infinitely many rational m/n with

${\displaystyle \left|k-{m \over n}\right|<{1 \over n^{2}{\sqrt {5}}}.}$

While virtually all real numbers k will eventually have infinitely many convergents m/n whose distance from k is significantly smaller than this limit, the convergents for φ (i.e., the numbers 5/3, 8/5, 13/8, 21/13, etc.) consistently "toe the boundary", keeping a distance of almost exactly ${\displaystyle {\scriptstyle {1 \over n^{2}{\sqrt {5}}}}}$ away from φ, thus never producing an approximation nearly as impressive as, for example, 355/113 for π. It can also be shown that every real number of the form a + bφ/c + dφ, where a, b, c, and d are integers such that a d − b c = ±1, shares this property with the golden ratio φ; and that all other real numbers can be more closely approximated.

Regular patterns in continued fractions

While there is no discernible pattern in the simple continued fraction expansion of π, there is one for e, the base of the natural logarithm:

${\displaystyle e=e^{1}=[2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,1,\dots ],}$

which is a special case of this general expression for positive integer n:

${\displaystyle e^{1/n}=[1;n-1,1,1,3n-1,1,1,5n-1,1,1,7n-1,1,1,\dots ]\,\!.}$

Another, more complex pattern appears in this continued fraction expansion for positive odd n:

${\displaystyle e^{2/n}=\left[1;{\frac {n-1}{2}},6n,{\frac {5n-1}{2}},1,1,{\frac {7n-1}{2}},18n,{\frac {11n-1}{2}},1,1,{\frac {13n-1}{2}},30n,{\frac {17n-1}{2}},1,1,\dots \right]\,\!,}$

with a special case for n = 1:

${\displaystyle e^{2}=[7;2,1,1,3,18,5,1,1,6,30,8,1,1,9,42,11,1,1,12,54,14,1,1\dots ,3k,12k+6,3k+2,1,1\dots ]\,\!.}$

Other continued fractions of this sort are

${\displaystyle \tanh(1/n)=[0;n,3n,5n,7n,9n,11n,13n,15n,17n,19n,\dots ]}$

where n is a positive integer; also, for integer n:

${\displaystyle \tan(1/n)=[0;n-1,1,3n-2,1,5n-2,1,7n-2,1,9n-2,1,\dots ]\,\!,}$

with a special case for n = 1:

${\displaystyle \tan(1)=[1;1,1,3,1,5,1,7,1,9,1,11,1,13,1,15,1,17,1,19,1,\dots ]\,\!.}$

If I_n(x) is the modified, or hyperbolic, Bessel function of the first kind, we may define a function on the rationals p/q by

${\displaystyle S(p/q)={\frac {I_{p/q}(2/q)}{I_{1+p/q}(2/q)}},}$

which is defined for all rational numbers, with p and q in lowest terms. Then for all nonnegative rationals, we have

${\displaystyle S(p/q)=[p+q;p+2q,p+3q,p+4q,\dots ],}$

with similar formulas for negative rationals; in particular we have

${\displaystyle S(0)=S(0/1)=[1;2,3,4,5,6,7,\dots ].}$

Many of the formulas can be proved using Gauss's continued fraction.

Typical continued fractions

Most irrational numbers do not have any periodic or regular behavior in their continued fraction expansion. Nevertheless, Khinchin proved that for almost all real numbers x, the a_i (for i = 1, 2, 3, ...) have an astonishing property: their geometric mean tends to a constant (known as Khinchin's constant, K ≈ 2.6854520010...) independent of the value of x. Paul Lévy showed that the nth root of the denominator of the nth convergent of the continued fraction expansion of almost all real numbers approaches an asymptotic limit, approximately 3.27582, which is known as Lévy's constant. Lochs' theorem states that nth convergent of the continued fraction expansion of almost all real numbers determines the number to an average accuracy of just over n decimal places.

Applications

Square roots

Generalized continued fractions are used in a method for computing square roots.

The identity

${\displaystyle {\sqrt {x}}=1+{\frac {x-1}{1+{\sqrt {x}}}}}$

(1)

leads via recursion to the generalized continued fraction for any square root:

${\displaystyle {\sqrt {x}}=1+{\cfrac {x-1}{2+{\cfrac {x-1}{2+{\cfrac {x-1}{2+{\ddots }}}}}}}}$

(2)

Pell's equation

Continued fractions play an essential role in the solution of Pell's equation. For example, for positive integers p and q, and non-square n, it is true that if p² − nq² = ±1, then p/q is a convergent of the regular continued fraction for √n. The converse holds if the period of the regular continued fraction for √n is 1, and in general the period describes which convergents give solutions to Pell's equation.

Dynamical systems

Continued fractions also play a role in the study of dynamical systems, where they tie together the Farey fractions which are seen in the Mandelbrot set with Minkowski's question mark function and the modular group Gamma.

The backwards shift operator for continued fractions is the map h(x) = 1/x − ⌊1/x⌋ called the Gauss map, which lops off digits of a continued fraction expansion: h([0; a₁, a₂, a₃, ...]) = [0; a₂, a₃, ...]. The transfer operator of this map is called the Gauss–Kuzmin–Wirsing operator. The distribution of the digits in continued fractions is given by the zero'th eigenvector of this operator, and is called the Gauss–Kuzmin distribution.

Eigenvalues and eigenvectors

The Lanczos algorithm uses a continued fraction expansion to iteratively approximate the eigenvalues and eigenvectors of a large sparse matrix.

Networking applications

Continued fractions have also been used in modelling optimization problems for wireless network virtualization to find a route between a source and a destination.

Examples of rational and irrational numbers

Number	r	0	1	2	3	4	5	6	7	8	9	10
123	ar	123
	ra	123
12.3	ar	12	3	3
	ra	12	37/3	123/10
1.23	ar	1	4	2	1	7
	ra	1	5/4	11/9	16/13	123/100
0.123	ar	0	8	7	1	2	5
	ra	0	1/8	7/57	8/65	23/187	123/1 000
Φ = ${\displaystyle {\frac {1+{\sqrt {5}}}{2}}}$	ar	1	1	1	1	1	1	1	1	1	1	1
	ra	1	2	3/2	5/3	8/5	13/8	21/13	34/21	55/34	89/55	144/89
-Φ = ${\displaystyle -{\frac {1+{\sqrt {5}}}{2}}}$	ar	-2	2	1	1	1	1	1	1	1	1	1
	ra	-2	-3/2	-5/3	-8/5	-13/8	-21/13	-34/21	-55/34	-89/55	-144/89	-233/144
${\displaystyle {\sqrt {2}}}$	ar	1	2	2	2	2	2	2	2	2	2	2
	ra	1	3/2	7/5	17/12	41/29	99/70	239/169	577/408	1 393/985	3 363/2 378	8 119/5 741
${\displaystyle {\frac {1}{\sqrt {2}}}}$	ar	0	1	2	2	2	2	2	2	2	2	2
	ra	0	1	2/3	5/7	12/17	29/41	70/99	169/239	408/577	985/1 393	2 378/3 363
${\displaystyle {\sqrt {3}}}$	ar	1	1	2	1	2	1	2	1	2	1	2
	ra	1	2	5/3	7/4	19/11	26/15	71/41	97/56	265/153	362/209	989/571
${\displaystyle {\frac {1}{\sqrt {3}}}}$	ar	0	1	1	2	1	2	1	2	1	2	1
	ra	0	1	1/2	3/5	4/7	11/19	15/26	41/71	56/97	153/265	209/362
${\displaystyle {\frac {\sqrt {3}}{2}}}$	ar	0	1	6	2	6	2	6	2	6	2	6
	ra	0	1	6/7	13/15	84/97	181/209	1 170/1 351	2 521/2 911	16 296/18 817	35 113/40 545	226 974/262 087
${\displaystyle {\sqrt[{3}]{2}}}$	ar	1	3	1	5	1	1	4	1	1	8	1
	ra	1	4/3	5/4	29/23	34/27	63/50	286/227	349/277	635/504	5 429/4 309	6 064/4 813
e	ar	2	1	2	1	1	4	1	1	6	1	1
	ra	2	3	8/3	11/4	19/7	87/32	106/39	193/71	1 264/465	1 457/536	2 721/1 001
π	ar	3	7	15	1	292	1	1	1	2	1	3
	ra	3	22/7	333/106	355/113	103 993/33 102	104 348/33 215	208 341/66 317	312 689/99 532	833 719/265 381	1 146 408/364 913	4 272 943/1 360 120
Number	r	0	1	2	3	4	5	6	7	8	9	10

ra: rational approximant obtained by expanding continued fraction up to a_r

History

• 300 BCE Euclid's Elements contains an algorithm for the greatest common divisor, whose modern version generates a continued fraction as the sequence of quotients of successive Euclidean divisions that occur in it.
• 499 The Aryabhatiya contains the solution of indeterminate equations using continued fractions
• 1572 Rafael Bombelli, L'Algebra Opera – method for the extraction of square roots which is related to continued fractions
• 1613 Pietro Cataldi, Trattato del modo brevissimo di trovar la radice quadra delli numeri – first notation for continued fractions

Cataldi represented a continued fraction as ${\displaystyle a_{0}}$ & ${\displaystyle {\frac {n_{1}}{d_{1}\cdot }}}$ & ${\displaystyle {\frac {n_{2}}{d_{2}\cdot }}}$ & ${\displaystyle {\frac {n_{3}}{d_{3}\cdot }}}$ with the dots indicating where the following fractions went.

• 1695 John Wallis, Opera Mathematica – introduction of the term "continued fraction"
• 1737 Leonhard Euler, De fractionibus continuis dissertatio – Provided the first then-comprehensive account of the properties of continued fractions, and included the first proof that the number e is irrational.
• 1748 Euler, Introductio in analysin infinitorum. Vol. I, Chapter 18 – proved the equivalence of a certain form of continued fraction and a generalized infinite series, proved that every rational number can be written as a finite continued fraction, and proved that the continued fraction of an irrational number is infinite.
• 1761 Johann Lambert – gave the first proof of the irrationality of π using a continued fraction for tan(x).
• 1768 Joseph-Louis Lagrange – provided the general solution to Pell's equation using continued fractions similar to Bombelli's
• 1770 Lagrange – proved that quadratic irrationals expand to periodic continued fractions.
• 1813 Carl Friedrich Gauss, Werke, Vol. 3, pp. 134–138 – derived a very general complex-valued continued fraction via a clever identity involving the hypergeometric function
• 1892 Henri Padé defined Padé approximant
• 1972 Bill Gosper – First exact algorithms for continued fraction arithmetic.

Mound Builders

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Mound_Builders 

Monks Mound, built c. 950–1100 CE and located at the Cahokia Mounds UNESCO World Heritage Site near Collinsville, Illinois, is the largest pre-Columbian earthwork in America north of Mesoamerica.

A number of pre-Columbian cultures are collectively termed "Mound Builders". The term does not refer to a specific people or archaeological culture, but refers to the characteristic mound earthworks erected for an extended period of more than 5,000 years. The "Mound Builder" cultures span the period of roughly 3500 BCE (the construction of Watson Brake) to the 16th century CE, including the Archaic period, Woodland period (Calusa culture, Adena and Hopewell cultures), and Mississippian period. Geographically, the cultures were present in the region of the Great Lakes, the Ohio River Valley, and the Mississippi River valley and its tributary waters.

The first mound building was an early marker of political and social complexity among the cultures in the Eastern United States. Watson Brake in Louisiana, constructed about 3500 BCE during the Middle Archaic period, is currently the oldest known and dated mound complex in North America. It is one of 11 mound complexes from this period found in the Lower Mississippi Valley. These cultures generally had developed hierarchical societies that had an elite. These commanded hundreds or even thousands of workers to dig up tons of earth with the hand tools available, move the soil long distances, and finally, workers to create the shape with layers of soils as directed by the builders.

From about 800 CE, the mound building cultures were dominated by the Mississippian culture, a large archaeological horizon, whose youngest descendants, the Plaquemine culture and the Fort Ancient culture, were still active at the time of European contact in the 16th century. One tribe of the Fort Ancient culture has been identified as the Mosopelea, presumably of southeast Ohio, who were speakers of an Ohio Valley Siouan language. The bearers of the Plaquemine culture were presumably speakers of the Natchez language isolate. The first description of these cultures is due to Spanish explorer Hernando de Soto, written between 1540 and 1542.

Mounds

A mound diagram of the platform mound showing the multiple layers of mound construction, mound structures such as temples or mortuaries, ramps with log stairs, and prior structures under later layers, multiple terraces, and intrusive burials

The namesake cultural trait of the Mound Builders was the building of mounds and other earthworks. These burial and ceremonial structures were typically flat-topped pyramids or platform mounds, flat-topped or rounded cones, elongated ridges, and sometimes a variety of other forms. They were generally built as part of complex villages. The early earthworks built in Louisiana around 3500 BCE are the only ones known to have been built by a hunter-gatherer culture, rather than a more settled culture based on agricultural surpluses.

The best-known flat-topped pyramidal structure is Monks Mound at Cahokia, near present-day Collinsville, Illinois. This mound appears to have been the main ceremonial and residential mound for the religious and political leaders; it is more than 100 feet (30 m) tall and is the largest pre-Columbian earthwork north of Mexico. This site had numerous mounds, some with conical or ridge tops, as well as a Woodhenge, and palisaded stockades protecting the large settlement and elite quarter. At its maximum about 1150 CE, Cahokia was an urban settlement with 20,000–30,000 people; this population was not exceeded by North American European settlements until after 1800.

A depiction of the Serpent Mound in southern Ohio, as published in the magazine The Century, April 1890

Some effigy mounds were constructed in the shapes or outlines of culturally significant animals. The most famous effigy mound, Serpent Mound in southern Ohio, ranges from 1 foot (0.30 m) to just over 3 feet (0.91 m) tall, 20 feet (6.1 m) wide, more than 1,330 feet (410 m) long, and shaped as an undulating serpent.

Early descriptions

Illustration of the Parkin site, thought to be the capital of the Province of Casqui visited by de Soto

Between 1540 and 1542, Hernando de Soto, the Spanish conquistador, traversed what became the southeastern United States. There he encountered many different mound-builder peoples who were perhaps descendants of the great Mississippian culture. De Soto observed people living in fortified towns with lofty mounds and plazas, and surmised that many of the mounds served as foundations for priestly temples. Near present-day Augusta, Georgia, de Soto encountered a group ruled by a queen, Cofitachequi. She told him that the mounds within her territory served as the burial places for nobles.

Engraving after Jacques le Moyne, showing the burial of a Timucua chief

The artist Jacques le Moyne, who had accompanied French settlers to northeastern Florida during the 1560s, likewise noted Native American groups using existing mounds and constructing others. He produced a series of watercolor paintings depicting scenes of native life. Although most of his paintings have been lost, some engravings were copied from the originals and published in 1591 by a Flemish company. Among these is a depiction of the burial of an aboriginal Floridian tribal chief, an occasion of great mourning and ceremony. The original caption reads:

Sometimes the deceased king of this province is buried with great solemnity, and his great cup from which he was accustomed to drink is placed on a tumulus with many arrows set about it.

— Jacques le Moyne, 1560s

Maturin Le Petit, a Jesuit priest, met the Natchez people, as did Le Page du Pratz (1758), a French explorer. Both observed them in the area that today is known as Mississippi. The Natchez were devout worshippers of the sun. Having a population of some 4,000, they occupied at least nine villages and were presided over by a paramount chief, known as the Great Sun, who wielded absolute power. Both observers noted the high temple mounds which the Natchez had built so that the Great Sun could commune with God, the sun. His large residence was built atop the highest mound, from "which, every morning, he greeted the rising sun, invoking thanks and blowing tobacco smoke to the four cardinal directions".

Archaeological surveys

A depiction of the Portsmouth Earthworks in Ancient Monuments of the Mississippi Valley

The most complete reference for these earthworks is Ancient Monuments of the Mississippi Valley, written by Ephraim G. Squier and Edwin H. Davis. It was published in 1848 by the Smithsonian Institution. Since many of the features which the authors documented have since been destroyed or diminished by farming and development, their surveys, sketches, and descriptions are still used by modern archaeologists. All of the sites which they identified as located in Kentucky came from the manuscripts of C. S. Rafinesque.

Chronology

Archaic era

Main article: Archaic period in the Americas

 

Illustration of Watson Brake, the oldest known mound complex in North America

 

Illustration of Poverty Point in West Carroll Parish, Louisiana

Radiocarbon dating has established the age of the earliest Archaic mound complex in southeastern Louisiana. One of the two Monte Sano Site mounds, excavated in 1967 before being destroyed for new construction at Baton Rouge, was dated at 6220 BP (plus or minus 140 years). Researchers at the time thought that such societies were not organizationally capable of this type of construction. It has since been dated as about 6500 BP, or 4500 BCE, although not all agree.

Watson Brake is located in the floodplain of the Ouachita River near Monroe in northern Louisiana. Securely dated to about 5,400 years ago (around 3500 BCE), in the Middle Archaic period, it consists of a formation of 11 mounds from 3 feet (0.91 m) to 25 feet (7.6 m) tall, connected by ridges to form an oval nearly 900 feet (270 m) across. In the Americas, building of complex earthwork mounds started at an early date, well before the pyramids of Egypt were constructed. Watson Brake was being constructed nearly 2,000 years before the better-known Poverty Point, and building continued for 500 years. Middle Archaic mound construction seems to have ceased about 2800 BCE, and scholars have not ascertained the reason, but it may have been because of changes in river patterns or other environmental factors.

With the 1990s dating of Watson Brake and similar complexes, scholars established that pre-agricultural, pre-ceramic American societies could organize to accomplish complex construction during extended periods of time, invalidating scholars' traditional ideas of Archaic society. Watson Brake was built by a hunter-gatherer society, the people of which occupied the area on only a seasonal basis, but where successive generations organized to build the complex mounds over a 500-year period. Their food consisted mostly of fish and deer, as well as available plants.

Poverty Point, built about 1500 BCE in what is now Louisiana, is a prominent example of Late Archaic mound-builder construction (around 2500 BCE – 1000 BCE). It is a striking complex of more than 1 square mile (2.6 km²), where six earthwork crescent ridges were built in concentric arrangement, interrupted by radial aisles. Three mounds are also part of the main complex, and evidence of residences extends for about 3 miles (4.8 km) along the bank of Bayou Macon. It is the major site among 100 associated with the Poverty Point culture and is one of the best-known early examples of earthwork monumental architecture. Unlike the localized societies during the Middle Archaic, this culture showed evidence of a wide trading network outside its area, which is one of its distinguishing characteristics.

Horr's Island, Florida, now a gated community next to Marco Island, when excavated by Michael Russo in 1980 found an Archaic Indian village site. Mound A was a burial mound that dated to 3400 BCE, making it the oldest known burial mound in North America.

Woodland period

Main article: Woodland period

 

Grave Creek Mound, Moundsville, West Virginia, Adena culture

The oldest mound associated with the Woodland period was the mortuary mound and pond complex at the Fort Center site in Glade County, Florida. 2012 excavations and dating by Thompson and Pluckhahn show that work began around 2600 BCE, seven centuries before the mound-builders in Ohio.

The Archaic period was followed by the Woodland period (circa 1000 BCE). Some well-understood examples are the Adena culture of Ohio, West Virginia, and parts of nearby states. The subsequent Hopewell culture built monuments from present-day Illinois to Ohio; it is renowned for its geometric earthworks. The Adena and Hopewell were not the only mound-building peoples during this time period. Contemporaneous mound-building cultures existed throughout what is now the Eastern United States, stretching as far south as Crystal River in western Florida. During this time, in parts of present-day Mississippi, Arkansas, and Louisiana, the Hopewellian Marksville culture degenerated and was succeeded by the Baytown culture. Reasons for degeneration include attacks from other tribes or the impact of severe climatic changes undermining agriculture.

Coles Creek culture

Illustration of Kings Crossing site in Warren County, Mississippi

 

Main article: Coles Creek culture

The Coles Creek culture is a Late Woodland culture (700–1200 CE) in the Lower Mississippi Valley in the Southern United States that marks a significant change of the cultural history of the area. Population and cultural and political complexity increased, especially by the end of the Coles Creek period. Although many of the classic traits of chiefdom societies were not yet made, by 1000 CE, the formation of simple elite polities had begun. Coles Creek sites are found in Arkansas, Louisiana, Oklahoma, Mississippi, and Texas. The Coles Creek culture is considered ancestral to the Plaquemine culture.

Mississippian cultures

Illustration of Cahokia with the large Monks Mound in the central precinct, encircled by a palisade, surrounded by four plazas, notably the Grand Plaza to the south

 

Main article: Mississippian culture

Around 900–1450 CE, the Mississippian culture developed and spread through the Eastern United States, primarily along the river valleys. The largest regional center where the Mississippian culture is first definitely developed is located in Illinois near the Mississippi, and is referred to presently as Cahokia. It had several regional variants including the Middle Mississippian culture of Cahokia, the South Appalachian Mississippian variant at Moundville and Etowah, the Plaquemine Mississippian variant in south Louisiana and Mississippi, and the Caddoan Mississippian culture of northwestern Louisiana, eastern Texas, and southwestern Arkansas. Like the mound builders of the Ohio, these people built gigantic mounds as burial and ceremonial places.

Fort Ancient culture

Artist's conception of the Fort Ancient culture SunWatch Indian Village

 

Main article: Fort Ancient culture

Fort Ancient is the name for a Native American culture that flourished from 1000 to 1650 CE among a people who predominantly inhabited land along the Ohio River in areas of modern-day southern Ohio, northern Kentucky, and western West Virginia.

Plaquemine culture

Illustration of the Holly Bluff site in Yazoo County, Mississippi

 

Main article: Plaquemine culture

A continuation of the Coles Creek culture in the lower Mississippi River Valley in western Mississippi and eastern Louisiana. Examples include the Medora site in West Baton Rouge Parish, Louisiana; and the Anna and Emerald Mound sites in Mississippi. Sites inhabited by Plaquemine peoples continued to be used as vacant ceremonial centers without large village areas much as their Coles Creek ancestors had done, although their layout began to show influences from Middle Mississippian peoples to the north. The Winterville and Holly Bluff (Lake George) sites in western Mississippi are good examples that exemplify this change of layout, but continuation of site usage. During the Terminal Coles Creek period (1150 to 1250 CE), contact increased with Mississippian cultures centered upriver near St. Louis, Missouri. This resulted in the adaption of new pottery techniques, as well as new ceremonial objects and possibly new social patterns during the Plaquemine period. As more Mississippian culture influences were absorbed, the Plaquemine area as a distinct culture began to shrink after CE 1350. Eventually, the last enclave of purely Plaquemine culture was the Natchez Bluffs area, while the Yazoo Basin and adjacent areas of Louisiana became a hybrid Plaquemine-Mississippian culture. This division was recorded by Europeans when they first arrived in the area. In the Natchez Bluffs area, the Taensa and Natchez people had held out against Mississippian influence and continued to use the same sites as their ancestors, and the Plaquemine culture is considered directly ancestral to these historic period groups encountered by Europeans. Groups who appear to have absorbed more Mississippian influence were identified as those tribes speaking the Tunican, Chitimachan, and Muskogean languages.

Disappearance

Following the description by Jacques le Moyne in 1560, the mound building cultures seem to have disappeared within the next century. However, there were also other European accounts, earlier than 1560, that gives a first-hand description of the enormous earth-built mounds being constructed by the Native Americans. One of them was Garcilaso de la Vega (c.1539–1616), a Spanish chronicler also known as "El Inca" because of his Incan mother, who was also the record-keeper of the infamous De Soto expedition that landed in present-day Florida on May 31, 1538. Garcilaso gave a first-hand description through his Historia de la Florida (published in 1605, Lisbon, as La Florida del Inca) describing how the Indians had built mounds and how the Native American Mound Cultures practiced their traditional way of life. De la Vega's accounts also include vital details about the Native American tribes' systems of government present in the South-East, tribal territories, and construction of mounds and temples. A few French expeditions in 1560s reported staying with Indian societies who had built mounds also.

Diseases

Later explorers to the same regions, only a few decades after mound-building settlements had been reported, found the regions largely depopulated with its residents vanished and the mounds untended. Conflicts with Europeans were dismissed by historians as the major cause of populations reduction, since only few clashes had occurred between the natives and the Europeans in the area during the same period. The most-widely accepted explanation behind the disappearances were the infectious diseases from the Old World, such as smallpox and influenza, which had decimated most of the Native Americans from the last mound-builder civilization. The Fort Ancient culture of the Ohio River valley is considered a "sister culture" of the Mississippian horizon, or one of the "Mississippianised" cultures adjacent to the main areal of the mound building cultures. This culture was also mostly extinct in the 17th century, but remnants may have survived into the first half of the 18th century. While this culture shows strong Mississippian influences, its bearers were most likely ethno-linguistically distinct from the Mississippians, possibly belonging to the Siouan phylum. The only tribal name associated with the Fort Ancient culture in the historical record are the Mosopelea, recorded by Jean-Baptiste-Louis Franquelin in 1684 as inhabiting eight villages north of the Ohio River. The Mosopelea are most likely identical to the Ofo (Oufé, Offogoula) recorded in the same area in the 18th century. The Ofo language was formerly classified as Muskogean but is now recognized as an eccentric member of the Western Siouan phylum. The late survival of the Fort Ancient culture is suggested by the remarkable amount of European made goods in the archaeological record which would have been acquired by trade even before direct European contact. These artefacts include brass and steel items, glassware, and melted down or broken goods reforged into new items. The Fort Ancient peoples are known to have been severely affected by disease in the 17th century (Beaver Wars period), and Carbon dating seems to indicate that they were wiped out by successive waves of disease.

Massacre and revolt

Because of the disappearance of the cultures by the end of the 17th century, the identification of the bearers of these cultures was an open question in 19th-century ethnography. Modern stratigraphic dating has established that the "Mound builders" have spanned the extended period of more than five millennia, so that any ethno-linguistic continuity is unlikely. The spread of the Mississippian culture from the late 1st millennium CE most likely involved cultural assimilation, in archaeological terminology called "Mississippianised" cultures.

19th-century ethnography assumed that the Mound-builders were an ancient prehistoric race with no direct connection to the Southeastern Woodland peoples of the historical period. A reference to this idea appears in the poem "The Prairies" (1832) by William Cullen Bryant.

The cultural stage of the Southeastern Woodland natives encountered in the 18th and 19th centuries by British colonists was deemed incompatible with the comparatively advanced stone, metal, and clay artifacts of the archaeological record. The age of the earthworks was also partly over-estimated. Caleb Atwater's misunderstanding of stratigraphy caused him to significantly overestimate the age of the earthworks. In his book, Antiquities Discovered in the Western States (1820), Atwater claimed that "Indian remains" were always found right beneath the surface of the earth, while artifacts associated with the Mound Builders were found fairly deep in the ground. Atwater argued that they must be from a different group of people. The discovery of metal artifacts further convinced people that the Mound Builders were not identical to the Southeast Woodland Native Americans of the 18th century.

It is now thought that the most likely bearers of the Plaquemine culture, a late variant of the Mississippian culture, were ancestral to the related Natchez and Taensa peoples. The Natchez language is a language isolate, supporting the scenario that after the collapse of the Mound builder cultures in the 17th century, there was an influx of unrelated peoples into the area. The Natchez are known to have historically occupied the Lower Mississippi Valley. They are first mentioned in French sources of around 1700, when they were centuered around the Grand Village close to present day Natchez, Mississippi. In 1729 the Natchez revolted, and massacred the French colony of Fort Rosalie, and the French retaliated by destroying all the Natchez villages. The remaining Natchez fled in scattered bands to live among the Chickasaw, Creek and Cherokee, whom they followed on the trail of tears when Indian removal policies of the mid 19th century forced them to relocate to Oklahoma. The Natchez language was extinct in the 20th century, with the death of the last known native speaker, Nancy Raven, in 1957.

Maps

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  Hopewell traditions

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  Adena culture

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  Troyville culture and Baytown culture

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  Coles Creek culture

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  Mississippian culture

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  Caddoan Mississippian culture

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  Fort Ancient culture

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  Plaquemine culture

Popular mythology

Alternative scenarios and hoaxes

See also: Archaeology and the Book of Mormon and Zelph

Based on the idea that the origins of the mound builders lay with a mysterious ancient people, there were various other suggestions belonging to the more general genre of Pre-Columbian trans-oceanic contact theories, specifically involving Vikings, Atlantis, and the Ten Lost Tribes of Israel, summarised by Feder (2006) under the heading of "The Myth of a Vanished Race".

Benjamin Smith Barton in his Observations on Some Parts of Natural History (1787) proposed the theory that the Mound Builders were associated with "Danes", i.e. with the Norse colonization of North America. In 1797, Barton reconsidered his position and correctly identified the mounds as part of indigenous prehistory.

Notable for the association with the Ten Lost Tribes is the Book of Mormon (1830). In this narrative, the Jaredites (3000–2000 BCE) and an Israelite group in 590 BCE (termed Nephites, Lamanites, and Mulekites). While the Nephites, Lamanites, and Mulekites were all of Jewish origin coming from Israel around 590 BCE, the Jaradites were a non-Abrahamic people separate in all aspects, except in a belief in Jehovah, from the Nephites. The Book of Mormon depicts these settlers building magnificent cities, which were destroyed by warfare about CE 385. The Book of Mormon can be placed in the tradition of the "Mound-Builder literature" of the period. Dahl (1961) states that it is "the most famous and certainly the most influential of all Mound-Builder literature". Later authors placing the Book of Mormon in this context include Silverberg (1969), Brodie (1971), Kennedy (1994) and Garlinghouse (2001).

Some nineteenth-century archaeological finds (e.g., earth and timber fortifications and towns, the use of a plaster-like cement, ancient roads, metal points and implements, copper breastplates, head-plates, textiles, pearls, native North American inscriptions, North American elephant remains etc.) were well-publicized at the time of the publication of the Book of Mormon and there is incorporation of some of these ideas into the narrative. References are made in the Book of Mormon to then-current understanding of pre-Columbian civilizations, including the formative Mesoamerican civilizations such as the (Pre-Classic) Olmec, Maya, and Zapotec.

Lafcadio Hearn in 1876 suggested that the mounds were built by people from the lost continent of Atlantis. The Reverend Landon West in 1901 claimed that Serpent Mound in Ohio was built by God, or by man inspired by him. He believed that God built the mound and placed it as a symbol of the story of the Garden of Eden.

More recently, Black nationalist websites claiming association with the Moorish Science Temple of America, have taken up the Atlantean ("Mu") association of the Mound Builders. Similarly, the "Washitaw Nation", a group associated with the Moorish Science Temple of America established in the 1990s, has been associated with mound-building in Black nationalist online articles of the early 2000s.

Kinderhook plates

On April 23, 1843, nine men unearthed human bones and six small, bell-shaped plates in Kinderhook, Illinois. Both sides of the plates apparently contained some sort of ancient writings. The plates, later known as the Kinderhook plates, were presented to Joseph Smith, who was an American religious leader and founder of Mormonism. He was reported to have said, "I have translated a portion of them, and find they contain the history of the person with whom they were found. He was a descendant of Ham, through the loins of Pharaoh, king of Egypt, and that he received his kingdom from the Ruler of heaven and earth."

In later letters, two eyewitnesses among the nine people who were present at the excavation of the plates, W. P. Harris and Wilbur Fugate, confessed that the whole thing was a hoax. With a group of other conspirators, they had manufactured the plates to give them the appearance of antiquity, buried them in a mound, and later pretended to excavate them, all for the purpose of trapping Joseph Smith into pretending to translate. However, it was not until 1980, when the only remaining plate was forensically examined, that the plates were conclusively determined to be, in fact, a nineteenth-century production.Up to 1980 most Latter-day Saints rejected the confessions and believed the plates were authentic. Not only did skeptics accept the confessors’ statements, but some continue to this day to argue that Joseph Smith pretended to translate a portion of the faked plates, claiming that he could not have been a true prophet.

The purpose for faking the Kinderhook plates is mainly to prove that the book of Mormon is true according to the witness Fulgate, "We learn there was a Mormon present when the plates were found, who it is said, leaped for joy at the discovery, and remarked that it would go to prove the authenticity of the Book of Mormon. The Mormons wanted to take the plates to Joe Smith, but we refused to let them go. Some time afterward a man assuming the name of Savage, of Quincy, borrowed the plates of Wiley to show to his literary friends there, and took them to Joe Smith."

Newark Holy Stone

Keystone

On June 29, 1860, David Wyrick, the surveyor of Licking County near Newark, discovered the so called Keystone in a shallow excavation at the monumental Newark Earthworks, which is an extraordinary set of ancient geometric enclosures created by Indigenous people. There he dug up a four-sided, plumb-bob-shaped stone with Hebrew letters engraved on each of its faces. The local Episcopal minister John W. McCarty translated the four inscriptions as "Law of the Lord," “Word of the Lord," “Holy of Holies," and "King of the Earth." Charles Whittlesey, who was one of the foremost archaeologists at that time, pronounced the stone to be authentic. The Newark Holy Stones, if genuine, would provide support for monogenesis, since they would establish that American Indians could be encompassed within Biblical history.

Decalogue Stone

After his first expedition, Wyrick uncovered a small stone box that was found to contain an intricately carved slab of black limestone covered with archaic-looking Hebrew letters along with a representation of a man in flowing robes. When translated, once again by McCarty, the inscription was found to include the entire Ten Commandments, and the robed figure was identified as Moses. Naturally enough, it became known as the Decalogue Stone.

Rather than being found beneath only a foot or two of soil, the Decalogue Stone was claimed to have been buried beneath a forty-foot-tall stone mound. Instead of modern Hebrew typography, the characters on the stone were blocky and appeared to be an ancient form of the Hebrew alphabet. Finally, the stone bore no resemblance to any modern Masonic artifact. In 1870, Whittlesey declared finally that the Holy Stones and other similar artifacts were "Archaeological Frauds."

Giants

In 19th-century America, a number of popular mythologies surrounding the origin of the mounds were in circulation, typically involving the mounds being built by a race of giants. A New York Times article from 1897 described a mound in Wisconsin in which a giant human skeleton measuring over 9 feet (2.7 m) in length was found. From 1886, another New York Times article described water receding from a mound in Cartersville, Georgia, which uncovered acres of skulls and bones, some of which were said to be gigantic. Two thigh bones were measured with the height of their owners estimated at 14 feet (4.3 m). President Lincoln made reference to the giants whose bones fill the mounds of America.

But still there is more. It calls up the indefinite past. When Columbus first sought this continent – when Christ suffered on the cross – when Moses led Israel through the Red-Sea – nay, even, when Adam first came from the hand of his Maker – then as now, Niagara was roaring here. The eyes of that species of extinct giants, whose bones fill the mounds of America, have gazed on Niagara, as ours do now. Co[n]temporary with the whole race of men, and older than the first man, Niagara is strong, and fresh to-day as ten thousand years ago. The Mammoth and Mastodon – now so long dead, that fragments of their monstrous bones, alone testify, that they ever lived, have gazed on Niagara. In that long – long time, never still for a single moment. Never dried, never froze, never slept, never rested.

The antiquarian author William Pidgeon in 1858 created fraudulent surveys of mound groups that did not exist.

Beginning in the 1880s, the supposed origin of the earthworks with a race of giants was increasingly recognized as spurious. Pidgeon's fraudulent claims about the archaeological record was shown to be a hoax by Theodore Lewis in 1886. A major factor contributing to public acceptance of the earthworks as a regular part of North American prehistory was the 1894 report by Cyrus Thomas of the Bureau of American Ethnology. Earlier authors making a similar case include Thomas Jefferson, who excavated a mound and from the artifacts and burial practices, noted similarities between mound-builder funeral practices and those of Native Americans in his time.

Walam Olum

The Walam Olum hoax had considerable influence on perceptions of the Mound Builders. In 1836, Constantine Samuel Rafinesque published his translation of a text he claimed had been written in pictographs on wooden tablets. This text explained that the Lenape Indians originated in Asia, told of their passage over the Bering Strait, and narrated their subsequent migration across the North American continent. This "Walam Olum" tells of battles with native peoples already in America before the Lenape arrived. People hearing of the account believed that the "original people" were the Mound Builders, and that the Lenape overthrew them and destroyed their culture. David Oestreicher later asserted that Rafinesque's account was a hoax. He argued that the Walam Olum glyphs derived from Chinese, Egyptian, and Mayan alphabets. Meanwhile, the belief that the Native Americans destroyed the mound-builder culture had gained widespread acceptance.