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Agatharchides

Agatharchides or Agatharchus (Ἀγαθαρχίδης or Ἀγάθαρχος, Agatharchos) of Cnidus was a Greek historian and geographer (flourished 2nd century BC).

Life
Agatharchides is believed to have been born at Cnidus, hence his appellation. As Stanley M. Burstein notes, the "evidence for Agatharchides' life is meagre." Photius describes him as a threptos, a kind of assistant of servile origin, to Cinnaeus and states that he was later a secretary to Heraclides Lembus. Cinnaeus served as a counselor to Ptolemy VI; Heraclides is best known for negotiating the treaty that ended Antiochus IV's invasion of Egypt in 169 BC.

Agatharchides furnishes few clues about his own life. At the conclusion of his On the Erythraean Sea, he apologizes for being unable to complete his work "since our age is unable to similarly bear the toil" and "as a result of the disturbances in Egypt" he could no longer access the official records (a fragment cited by Photius in his Bibliotheca Cod. 250.110, 460b). There are two possible occasions when this could have happened: the first was in 145 BC, when Ptolemy VIII purged Alexandria of the intellectuals who supported his rivals for the throne; and in 132 BC after Ptolemy, who had been driven from his kingdom by a rebellion in Alexandria, returned and exacted reprisals on that city. While most scholars have favored the later date, Burstein argues for the earlier one.

Extracts from the first book of his Erythraean Sea, written in the first person and advocating a military campaign into the lands south of Egypt, led early scholars to deduce that Agatharchides was an important political figure of his time, and served as a guardian to one of the sons of Ptolemy VIII. Dodwell endeavored to show that it the younger son, Alexander, and objects to Soter, that he reigned conjointly with his mother. This, however, was the case with Alexander likewise. Wesseling and Henry Fynes Clinton think the elder brother to be the one meant, for Soter was more likely to have been a minor on his accession in 117 BC than Alexander in 107 BC, ten years after their father's death; the second edition of the Oxford Classical Dictionary article on Agatharchides agrees that the son was Soter. Moreover, Dodwell's date would leave too short an interval between the publication of Agatharchides's work on the Erythraean Sea (about 113 BC), and the work of Artemidorus. However at least as early as 1810, when B. G. Niebuhr pointed out that these excerpts were from a speech, and not part of the narrative of his book, this theory has been recognized as conflicting with other known historical facts.

Writings
Agatharchides was not well known in ancient times. Of his two major works, Affairs in Asia (Ta kata ten Asian) in ten books, and Affairs in Europe (Ta kata ten Europen) in forty-nine books, only a few fragments survive, too few to provide us with any sense of the contents of either work. However, for his On the Erythraean Sea (Peri tes Erythras Thalasses or De Mari Erythraeo) in five books, almost the entire fifth book, a geographical treatise on the Horn of Africa and the lands around the Red Sea, has survived almost intact. According to Burstein, "the comparative soberness of Agatharchides' treatment compared to previous accounts and the wealth of information contained in it led to a quick recognition . . . [that it was] a valuable summary of the results of Ptolemaic exploration."

In the first book of On the Erythraean Sea was a discussion respecting the origin of the name. In the fifth Agatharchides described the mode of life amongst the Sabaeans in Arabia, and the Ichthyophagi, or fish-eaters, the way in which elephants were caught by the elephant-eaters, and the mode of working the gold mines in the mountains of Egypt, near the Red Sea. His account of the Ichthyophagi and of the mode of working the gold mines, has been copied by Diodorus (iii.12-18). Amongst other extraordinary animals he mentions the camelopard, which was found in the country of the Troglodytae, and the rhinoceros.

Material from this book is quoted directly or indirectly by Diodorus Siculus, Strabo, Pliny the Elder, Aelian (Claudius Aelianus), Josephus and other authors. Although Agatharchides' work was superseded by more detailed accounts in the 2nd century AD, Photius found a copy of Erythraean Sea in the 9th century, from which he preserved extensive extracts in his Bibliotheca. Photius states that Agatharchides wrote in the Attic dialect, with a style that was dignified and perspicuous, and abounded in sententious passages—inspiring a favorable opinion from Photius. In the composition of his speeches Agatharchides was an imitator of Thucydides, whom he equalled in dignity and excelled in clearness. He was acquainted with the language of the Aethiopians (de Rubr. M. p. 46), and appears to have been the first who discovered the true cause of the yearly inundations of the Nile. (Diod. i. 41.)

An Agatharchides, of Samos, is mentioned by Plutarch, as the author of a work on Persia, and one περὶ λίθων. J.A. Fabricius, however, conjectures that the true reading is Agathyrsides, not Agatharchides. (Dodwell in Hudson's Geogr. Script. Gr. Minores; Clinton, Fasti Hell. iii, p. 535.)

Namesake
The crater Agatharchides on the Moon is named in his honour.

title: Agatharchides


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Wayford Bridge

Wayford Bridge is a village on the River Ant on the A149 road, near Stalham in Norfolk, England within The Broads National Park.

Due to height restrictions under the road bridge it is the most northerly point of navigation on the Norfolk Broads for boats over 7 ft in height above the water line. Broads boats can continue to Dilham, though the channel is narrow and has limited places in which to turn around. Wayford Bridge is the southern end of the North Walsham & Dilham Canal.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Wayford Bridge


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Extended Euclidean algorithm

In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers a and b, also the coefficients of Bézout's identity, which are integers x and y such that
 * $$ax + by = \gcd(a, b).$$

This is a certifying algorithm, because the gcd is the only number that can simultaneously satisfy this equation and divide the inputs. It allows one to compute also, with almost no extra cost, the quotients of a and b by their greatest common divisor.

also refers to a very similar algorithm for computing the polynomial greatest common divisor and the coefficients of Bézout's identity of two univariate polynomials.

The extended Euclidean algorithm is particularly useful when a and b are coprime. With that provision, x is the modular multiplicative inverse of a modulo b, and y is the modular multiplicative inverse of b modulo a. Similarly, the polynomial extended Euclidean algorithm allows one to compute the multiplicative inverse in algebraic field extensions and, in particular in finite fields of non prime order. It follows that both extended Euclidean algorithms are widely used in cryptography. In particular, the computation of the modular multiplicative inverse is an essential step in the derivation of key-pairs in the RSA public-key encryption method.

Description
The standard Euclidean algorithm proceeds by a succession of Euclidean divisions whose quotients are not used. Only the remainders are kept. For the extended algorithm, the successive quotients are used. More precisely, the standard Euclidean algorithm with a and b as input, consists of computing a sequence $$q_1,\ldots, q_k$$ of quotients and a sequence $$r_0,\ldots, r_{k+1}$$ of remainders such that

\begin{align} r_0 & =a \\ r_1 & =b \\ & \,\,\,\vdots \\ r_{i+1} & =r_{i-1}-q_i r_i \quad \text {and} \quad 0\le r_{i+1} < |r_i| \quad\text{(this defines }q_i)\\ & \,\,\, \vdots \end{align} $$ It is the main property of Euclidean division that the inequalities on the right define uniquely $$q_i$$ and $$r_{i+1}$$ from $$r_{i-1}$$ and $$r_{i}.$$

The computation stops when one reaches a remainder $$r_{k+1}$$ which is zero; the greatest common divisor is then the last non zero remainder $$r_{k}.$$

The extended Euclidean algorithm proceeds similarly, but adds two other sequences, as follows

\begin{align} r_0 & =a & r_1 & =b \\ s_0 & =1 & s_1 & =0 \\ t_0 & =0 & t_1 & =1 \\ & \,\,\,\vdots & & \,\,\,\vdots \\ r_{i+1} & =r_{i-1}-q_i r_i & \text {and } 0 & \le r_{i+1} < |r_i| & \text{(this defines } q_i \text{)}\\ s_{i+1} & =s_{i-1}-q_i s_i \\ t_{i+1} & =t_{i-1}-q_i t_i \\ & \,\,\, \vdots \end{align} $$

The computation also stops when $$r_{k+1}=0$$ and gives
 * $$r_k$$ is the greatest common divisor of the input $$a=r_0$$ and $$b=r_1.$$
 * The Bézout coefficients are $$s_k$$ and $$t_k,$$ that is $$\gcd(a,b)=r_k=as_k+bt_k$$
 * The quotients of a and b by their greatest common divisor are given by $$s_{k+1}=\pm\frac{b}{\gcd(a,b)}$$ and $$t_{k+1}=\pm\frac{a}{\gcd(a,b)}$$

Moreover, if a and b are both positive and $$\gcd(a,b)\neq\min(a,b)$$, then
 * $$|s_i| \leq \left\lfloor\frac{b}{2\gcd(a,b)}\right\rfloor\quad \text{and} \quad |t_i| \leq \left\lfloor\frac{a}{2\gcd(a,b)}\right\rfloor$$

for $$0\leq i \leq k,$$ where $$\lfloor x\rfloor$$ denotes the integral part of $x$, that is the greatest integer not greater than $x$.

This implies that the pair of Bézout's coefficients provided by the extended Euclidean algorithm is the minimal pair of Bézout coefficients, as being the unique pair satisfying both above inequalities.

Also it means that the algorithm can be done without integer overflow by a computer program using integers of a fixed size that is larger than that of a and b.

Example
The following table shows how the extended Euclidean algorithm proceeds with input and. The greatest common divisor is the last non zero entry, in the column "remainder". The computation stops at row 6, because the remainder in it is. Bézout coefficients appear in the last two entries of the second-to-last row. In fact, it is easy to verify that × + ×  =. Finally the last two entries  and  of the last row are, up to the sign, the quotients of the input  and  by the greatest common divisor.

Proof
As $$ 0\le r_{i+1}<|r_i|, $$ the sequence of the $$ r_i $$ is a decreasing sequence of nonnegative integers (from i = 2 on). Thus it must stop with some $$ r_{k+1}=0.$$ This proves that the algorithm stops eventually.

As $$ r_{i+1}= r_{i-1} - r_i q_i,$$ the greatest common divisor is the same for $$(r_{i-1}, r_i)$$ and $$(r_{i}, r_{i+1}).$$ This shows that the greatest common divisor of the input $$a=r_0, b=r_1 $$ is the same as that of $$ r_k, r_{k+1}=0.$$ This proves that $$ r_k $$ is the greatest common divisor of a and b. (Until this point, the proof is the same as that of the classical Euclidean algorithm.)

As $$ a=r_0$$ and $$ b=r_1,$$ we have $$as_i+bt_i=r_i$$ for i = 0 and 1. The relation follows by induction for all $$i>1$$: $$r_{i+1} = r_{i-1} - r_i q_i = (as_{i-1}+bt_{i-1}) - (as_i+bt_i)q_i = (as_{i-1}-as_iq_i) + (bt_{i-1}-bt_iq_i) = as_{i+1}+bt_{i+1}.$$

Thus $$s_k$$ and $$t_k$$ are Bézout coefficients.

Consider the matrix $$A_i=\begin{pmatrix} s_{i-1} & s_i\\ t_{i-1} & t_i \end{pmatrix}.$$

The recurrence relation may be rewritten in matrix form $$A_{i+1} = A_i \cdot \begin{pmatrix} 0 & 1\\ 1 & -q_i \end{pmatrix}.$$

The matrix $$A_1$$ is the identity matrix and its determinant is one. The determinant of the rightmost matrix in the preceding formula is −1. It follows that the determinant of $$A_i$$ is $$(-1)^{i-1}.$$ In particular, for $$i=k+1,$$ we have $$s_k t_{k+1} - t_k s_{k+1} = (-1)^k.$$ Viewing this as a Bézout's identity, this shows that $$s_{k+1}$$ and $$t_{k+1}$$ are coprime. The relation $$as_{k+1}+bt_{k+1}=0$$ that has been proved above and Euclid's lemma show that $$s_{k+1}$$ divides $b$, that is that $$b=ds_{k+1}$$ for some integer $d$. Dividing by $$s_{k+1}$$ the relation $$as_{k+1}+bt_{k+1}=0$$ gives $$a=-dt_{k+1}.$$ So, $$s_{k+1}$$ and $$-t_{k+1}$$ are coprime integers that are the quotients of $a$ and $b$ by a common factor, which is thus their greatest common divisor or its opposite.

To prove the last assertion, assume that a and b are both positive and $$\gcd(a,b)\neq\min(a,b)$$. Then, $$a \neq b $$, and if $$a < b$$, it can be seen that the s and t sequences for (a,b) under the EEA are, up to initial 0s and 1s, the t and s sequences for (b,a). The definitions then show that the (a,b) case reduces to the (b,a) case. So assume that $$a > b$$ without loss of generality.

It can be seen that $$s_2$$ is 1 and $$s_3$$ (which exists by $$\gcd(a,b)\neq\min(a,b)$$) is a negative integer. Thereafter, the $$s_i$$ alternate in sign and strictly increase in magnitude, which follows inductively from the definitions and the fact that $$q_i\geq 1$$ for $$1 \leq i \leq k$$, the case $$i = 1$$ holds because $$a > b$$. The same is true for the $$t_i$$ after the first few terms, for the same reason. Furthermore, it is easy to see that $$q_k \geq 2$$ (when a and b are both positive and $$\gcd(a,b)\neq\min(a,b)$$). Thus, noticing that $$|s_{k+1}| = |s_{k-1}| + q_k |s_k|$$, we obtain $$|s_{k+1}|=\left |\frac{b}{\gcd(a,b)} \right | \geq 2|s_k| \qquad \text{and} \qquad |t_{k+1}|= \left |\frac{a}{\gcd(a,b)} \right | \geq 2|t_k|.$$

This, accompanied by the fact that $$s_k,t_k$$ are larger than or equal to in absolute value than any previous $$s_i$$ or $$t_i$$ respectively completed the proof.

Polynomial extended Euclidean algorithm
For univariate polynomials with coefficients in a field, everything works similarly, Euclidean division, Bézout's identity and extended Euclidean algorithm. The first difference is that, in the Euclidean division and the algorithm, the inequality $$0\le r_{i+1}<|r_i|$$ has to be replaced by an inequality on the degrees $$\deg r_{i+1}<\deg r_i.$$ Otherwise, everything which precedes in this article remains the same, simply by replacing integers by polynomials.

A second difference lies in the bound on the size of the Bézout coefficients provided by the extended Euclidean algorithm, which is more accurate in the polynomial case, leading to the following theorem.

If a and b are two nonzero polynomials, then the extended Euclidean algorithm produces the unique pair of polynomials (s, t) such that
 * $$as+bt=\gcd(a,b)$$

and
 * $$\deg s < \deg b - \deg (\gcd(a,b)), \quad \deg t < \deg a - \deg (\gcd(a,b)).$$

A third difference is that, in the polynomial case, the greatest common divisor is defined only up to the multiplication by a non zero constant. There are several ways to define unambiguously a greatest common divisor.

In mathematics, it is common to require that the greatest common divisor be a monic polynomial. To get this, it suffices to divide every element of the output by the leading coefficient of $$r_{k}.$$ This allows that, if a and b are coprime, one gets 1 in the right-hand side of Bézout's inequality. Otherwise, one may get any non-zero constant. In computer algebra, the polynomials commonly have integer coefficients, and this way of normalizing the greatest common divisor introduces too many fractions to be convenient.

The second way to normalize the greatest common divisor in the case of polynomials with integer coefficients is to divide every output by the content of $$r_{k},$$ to get a primitive greatest common divisor. If the input polynomials are coprime, this normalisation also provides a greatest common divisor equal to 1. The drawback of this approach is that a lot of fractions should be computed and simplified during the computation.

A third approach consists in extending the algorithm of subresultant pseudo-remainder sequences in a way that is similar to the extension of the Euclidean algorithm to the extended Euclidean algorithm. This allows that, when starting with polynomials with integer coefficients, all polynomials that are computed have integer coefficients. Moreover, every computed remainder $$r_i$$ is a subresultant polynomial. In particular, if the input polynomials are coprime, then the Bézout's identity becomes
 * $$as+bt=\operatorname{Res}(a,b),$$

where $$\operatorname{Res}(a,b)$$ denotes the resultant of a and b. In this form of Bézout's identity, there is no denominator in the formula. If one divides everything by the resultant one gets the classical Bézout's identity, with an explicit common denominator for the rational numbers that appear in it.

Pseudocode
To implement the algorithm that is described above, one should first remark that only the two last values of the indexed variables are needed at each step. Thus, for saving memory, each indexed variable must be replaced by just two variables.

For simplicity, the following algorithm (and the other algorithms in this article) uses parallel assignments. In a programming language which does not have this feature, the parallel assignments need to be simulated with an auxiliary variable. For example, the first one, (old_r, r) := (r, old_r - quotient * r) is equivalent to prov := r; r := old_r - quotient × prov; old_r := prov; and similarly for the other parallel assignments. This leads to the following code:

function extended_gcd(a, b)    (old_r, r) := (a, b)     (old_s, s) := (1, 0) (old_t, t) := (0, 1) while r ≠ 0 do quotient := old_r div r        (old_r, r) := (r, old_r − quotient × r)         (old_s, s) := (s, old_s − quotient × s)         (old_t, t) := (t, old_t − quotient × t)     output "Bézout coefficients:", (old_s, old_t) output "greatest common divisor:", old_r output "quotients by the gcd:", (t, s)

The quotients of a and b by their greatest common divisor, which is output, may have an incorrect sign. This is easy to correct at the end of the computation but has not been done here for simplifying the code. Similarly, if either a or b is zero and the other is negative, the greatest common divisor that is output is negative, and all the signs of the output must be changed.

Finally, notice that in Bézout's identity, $$ax + by = \gcd(a, b)$$, one can solve for $$y$$ given $$a, b, x, \gcd(a, b)$$. Thus, an optimization to the above algorithm is to compute only the $$s_k$$ sequence (which yields the Bézout coefficient $$x$$), and then compute $$y$$ at the end:

function extended_gcd(a, b)    s := 0;    old_s := 1 r := b;   old_r := a     while r ≠ 0 do quotient := old_r div r        (old_r, r) := (r, old_r − quotient × r)         (old_s, s) := (s, old_s − quotient × s)     if b ≠ 0 then bezout_t := (old_r − old_s × a) div b    else bezout_t := 0 output "Bézout coefficients:", (old_s, bezout_t) output "greatest common divisor:", old_r

However, in many cases this is not really an optimization: whereas the former algorithm is not susceptible to overflow when used with machine integers (that is, integers with a fixed upper bound of digits), the multiplication of old_s * a in computation of bezout_t can overflow, limiting this optimization to inputs which can be represented in less than half the maximal size. When using integers of unbounded size, the time needed for multiplication and division grows quadratically with the size of the integers. This implies that the "optimisation" replaces a sequence of multiplications/divisions of small integers by a single multiplication/division, which requires more computing time than the operations that it replaces, taken together.

Simplification of fractions
A fraction $a⁄b$ is in canonical simplified form if $a$ and $b$ are coprime and $b$ is positive. This canonical simplified form can be obtained by replacing the three output lines of the preceding pseudo code by if $s = 0$ then output "Division by zero" if $s < 0$ then $s := −s$; $t := −t$   (for avoiding negative denominators) if $s = 1$ then output $-t$       (for avoiding denominators equal to 1) output $-t⁄s$

The proof of this algorithm relies on the fact that $s$ and $t$ are two coprime integers such that $as + bt = 0$, and thus $$\frac{a}{b} = -\frac{t}{s}$$. To get the canonical simplified form, it suffices to move the minus sign for having a positive denominator.

If $b$ divides $a$ evenly, the algorithm executes only one iteration, and we have $s = 1$ at the end of the algorithm. It is the only case where the output is an integer.

Computing multiplicative inverses in modular structures
The extended Euclidean algorithm is the essential tool for computing multiplicative inverses in modular structures, typically the modular integers and the algebraic field extensions. A notable instance of the latter case are the finite fields of non-prime order.

Modular integers
If $n$ is a positive integer, the ring $Z/nZ$ may be identified with the set ${0, 1, ..., n-1}$ of the remainders of Euclidean division by $n$, the addition and the multiplication consisting in taking the remainder by $n$ of the result of the addition and the multiplication of integers. An element $a$ of $Z/nZ$ has a multiplicative inverse (that is, it is a unit) if it is coprime to $n$. In particular, if $n$ is prime, $a$ has a multiplicative inverse if it is not zero (modulo $n$). Thus $Z/nZ$ is a field if and only if $n$ is prime.

Bézout's identity asserts that $a$ and $n$ are coprime if and only if there exist integers $s$ and $t$ such that
 * $$ns+at=1$$

Reducing this identity modulo $n$ gives
 * $$at \equiv 1 \mod n.$$

Thus $t$, or, more exactly, the remainder of the division of $t$ by $n$, is the multiplicative inverse of $a$ modulo $n$.

To adapt the extended Euclidean algorithm to this problem, one should remark that the Bézout coefficient of $n$ is not needed, and thus does not need to be computed. Also, for getting a result which is positive and lower than n, one may use the fact that the integer $t$ provided by the algorithm satisfies $|t| < n$. That is, if $t < 0$, one must add $n$ to it at the end. This results in the pseudocode, in which the input n is an integer larger than 1. function inverse(a, n)    t := 0;     newt := 1 r := n;    newr := a     while newr ≠ 0 do quotient := r div newr (t, newt) := (newt, t − quotient × newt) (r, newr) := (newr, r − quotient × newr) if r > 1 then return "a is not invertible" if t < 0 then t := t + n    return t

Simple algebraic field extensions
The extended Euclidean algorithm is also the main tool for computing multiplicative inverses in simple algebraic field extensions. An important case, widely used in cryptography and coding theory, is that of finite fields of non-prime order. In fact, if $p$ is a prime number, and $q = p^{d}$, the field of order $q$ is a simple algebraic extension of the prime field of $p$ elements, generated by a root of an irreducible polynomial of degree $d$.

A simple algebraic extension $L$ of a field $K$, generated by the root of an irreducible polynomial $p$ of degree $d$ may be identified to the quotient ring $$K[X]/\langle p\rangle,$$, and its elements are in bijective correspondence with the polynomials of degree less than $d$. The addition in $L$ is the addition of polynomials. The multiplication in $L$ is the remainder of the Euclidean division by $p$ of the product of polynomials. Thus, to complete the arithmetic in $L$, it remains only to define how to compute multiplicative inverses. This is done by the extended Euclidean algorithm.

The algorithm is very similar to that provided above for computing the modular multiplicative inverse. There are two main differences: firstly the last but one line is not needed, because the Bézout coefficient that is provided always has a degree less than $d$. Secondly, the greatest common divisor which is provided, when the input polynomials are coprime, may be any non zero elements of $K$; this Bézout coefficient (a polynomial generally of positive degree) has thus to be multiplied by the inverse of this element of $K$. In the pseudocode which follows, $p$ is a polynomial of degree greater than one, and $a$ is a polynomial.

function inverse(a, p)    t := 0;     newt := 1 r := p;    newr := a     while newr ≠ 0 do quotient := r div newr (r, newr) := (newr, r − quotient × newr) (t, newt) := (newt, t − quotient × newt) if degree(r) > 0 then return "Either p is not irreducible or a is a multiple of p"    return (1/r) × t

Example
For example, if the polynomial used to define the finite field GF(28) is $p = x^{8} + x^{4} + x^{3} + x + 1$, and $a = x^{6} + x^{4} + x + 1$ is the element whose inverse is desired, then performing the algorithm results in the computation described in the following table. Let us recall that in fields of order 2n, one has −z = z and z + z = 0 for every element z in the field). Since 1 is the only nonzero element of GF(2), the adjustment in the last line of the pseudocode is not needed.

Thus, the inverse is $p = x^{8} + x^{4} + x^{3} + x + 1$, as can be confirmed by multiplying the two elements together, and taking the remainder by $p$ of the result.

The case of more than two numbers
One can handle the case of more than two numbers iteratively. First we show that $$\gcd(a,b,c) = \gcd(\gcd(a,b),c)$$. To prove this let $$d=\gcd(a,b,c)$$. By definition of gcd $$d$$ is a divisor of $$a$$ and $$b$$. Thus $$\gcd(a,b)=k d$$ for some $$k$$. Similarly $$d$$ is a divisor of $$c$$ so $$c=jd$$ for some $$j$$. Let $$u=\gcd(k,j)$$. By our construction of $$u$$, $$ud | a,b,c$$ but since $$d$$ is the greatest divisor $$u$$ is a unit. And since $$ud=\gcd(\gcd(a,b),c)$$ the result is proven.

So if $$na + mb = \gcd(a,b)$$ then there are $$x$$ and $$y$$ such that $$x\gcd(a,b) + yc = \gcd(a,b,c)$$ so the final equation will be

 * $$x(na + mb) + yc = (xn)a + (xm)b + yc = \gcd(a,b,c).\,$$

So then to apply to n numbers we use induction

 * $$\gcd(a_1,a_2,\dots,a_n) =\gcd(a_1,\, \gcd(a_2,\, \gcd(a_3,\dots, \gcd(a_{n-1}\,,a_n))),\dots),$$

with the equations following directly.

title: Extended Euclidean algorithm


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Pollard

Pollard may refer to:

Places in the United States

 * Pollard, Alabama, a town
 * Pollard, Arkansas, a city
 * Pollard, Kansas, an unincorporated community

People

 * Pollard (surname), a list of people with the surname
 * Pollard Hopewell (between 1786 and 1789 – 1813), midshipman in the United States Navy
 * Charles Pollard Olivier (1884–1975), American astronomer
 * James Pollard Espy (1785–1860), American meteorologist
 * Ngoia Pollard Napaltjarri (c. 1948–2022), Australian indigenous (Warlpiri people) artist
 * Thomas Pollard Sampson (1875–1961), Australian architect

Flora and fauna

 * Pollard, a tree affected by pollarding (cropping of the upper branches)
 * Pollard, a deer which has cast its antlers
 * Pollard or polled livestock, hornless livestock of normally-horned species
 * Pollard, the European chub (Squalius cephalus), a freshwater fish

Mathematics

 * Several algorithms created by British mathematician John Pollard:
 * Pollard's kangaroo algorithm
 * Pollard's p − 1 algorithm
 * Pollard's rho algorithm

Other uses

 * Pollard (coin), a medieval coin made in Europe in imitation of the English penny, outlawed under Edward I
 * Pollard (novel), a novel by Laura Beatty
 * Pollard, a mixture of fine bran and a small amount of flour
 * Pollard railway station, former station in Victoria, Australia
 * Pollard script, a writing system devised in 1905 for the A-Hmao language

title: Pollard


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Coltishall

Coltishall is a village on the River Bure, west of Wroxham, in the English county of Norfolk. The village is located within the Norfolk Broads.

Etymology
Coltishall's name is of Old English origin and first attested in the Domesday Book of 1086 in the forms Cokereshala and Coketeshala. From 1200 onwards, it is attested in the contracted form Couteshal(e), in which form it has more or less remained to the present day (the l in the modern spelling is due to hypercorrection). The second part of the name is thought to derive from the Old English word halh ('nook') but the origin of the first part is uncertain; one guess is that it was an otherwise unattested personal name Coccede or Cohhede, and thus meant 'Coccede's nook'. But The Cambridge Dictionary of English Place-Names concludes that "the recorded forms are too few and contradictory for satisfactory explanation".

History
In the Domesday Book, Coltishall is recorded as a settlement of 16 households in the hundred of South Erpingham. The village was divided between the estates of William de Warenne and Roger de Poitou.

In 1231, Coltishall was made a 'free-town' by King Henry III. Furthermore, from the mid-Eighteenth Century, Coltishall was a centre for the malting industry with many wherries being built in the village.

In 1939, RAF Coltishall was opened as a base for the Hawker Hurricanes of No. 242 Squadron RAF, with the famous fighter ace Douglas Bader being based in Coltishall during the Second World War. RAF Coltishall continue to be used by the Royal Air Force until its closure in November 2006 following a Ministry of Defence review. The site is today operated as HM Prison Bure, which houses Male Category C prisoners.

Horstead watermill, close to the village, was one of the most photographed mills in Britain until it burned down in 1963.

Geography
According to the 2011 Census, Coltishall has a population of 1,503 residents living in 692 households.

Coltishall falls within the constituency of Broadland and is represented at Parliament by Jerome Mayhew MP of the Conservative Party.

St. John the Baptist's Church
Coltishall's parish church is of Norman origin and is dedicated to John the Baptist. The church features good examples of James Powell and Sons stained glass windows and evidence of the wealth of the malting families who have lived in Coltishall throughout history.

Notable residents

 * John Alen- Archbishop of Dublin, 1529–1534
 * Sir John Alan- Lord Chancellor of Ireland

Climate
Climate in this area has mild differences between highs and lows, and there is adequate rainfall year-round. The Köppen climate classification subtype for this climate is "Cfb" (Marine West Coast Climate/Oceanic climate).

title: Coltishall


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1964 New York World's Fair

The 1964–1965 New York World's Fair was an international exposition at Flushing Meadows–Corona Park in Queens, New York City, United States. The fair included exhibitions, activities, performances, films, art, and food presented by 80 nations, 24 U.S. states, and nearly 350 American companies. The 646 acre fairground consisted of five sections: the Federal and State, International, Transportation, Lake Amusement, and Industrial areas. The fair was themed to "peace through understanding" and was centered on the Unisphere, a stainless-steel model of the Earth. Initially, the fair had 139 pavilions, in addition to 34 concessions and shows.

The site had previously hosted the 1939 New York World's Fair. Several businessmen devised plans for a 1964 fair in the 1950s, and the New York World's Fair 1964 Corporation (WFC) was formed in 1959. Although U.S. president Dwight D. Eisenhower approved the fair, the Bureau International des Expositions never recognized it. Construction began in late 1960, and over a hundred exhibitors signed up for the fair over the next three years. The fair ran for two six-month seasons: April 22 to October 18, 1964, and April 21 to October 17, 1965. Despite initial projections of 70 million visitors, the fair had just over 51.6 million guests. After the fair, some pavilions were preserved or relocated, but most structures were demolished.

The fair showcased mid-20th-century American culture and technology. The different sections were designed in various architectural styles, though anyone could host an exhibit if they could afford to rent the land and pay for a pavilion. The fairground had several amusement and transport rides, as well as various plazas and fountains. The fair had 198 restaurants at its peak, and dishes served at these restaurants, such as Belgian waffles, were popularized through the fair. There were more than 30 entertainment events at the fair, in addition to 40 theaters and various music performances. Exhibitors also displayed sculptures, visual art, and artifacts, along with consumer products such as electronics and cars. The contemporary press criticized the event as a financial failure. Nonetheless, the fair helped influence 21st-century technologies and popularized consumer products such as Belgian waffles and the Ford Mustang.

Development
The site of the 1964 World's Fair, Flushing Meadows–Corona Park in Queens was originally a natural wetland straddling the Flushing River. The site was the Corona Ash Dumps in the early 20th century before it was selected for the 1939–1940 World's Fair. The 1939 fair was themed to "the world of tomorrow" and was highly unprofitable, recouping only 32% of its original cost. After the 1939 fair, the site was used as a park, although the site fell into disrepair due to a lack of funds. The development of the 1964 fair coincided with social upheavals in the early 1960s, including the civil rights movement, Vietnam War protests, and the aftermath of U.S. president John F. Kennedy's assassination.

World's Fair Corporation
The idea for the 1964 fair was conceived by a group of businessmen. Among them was Robert Kopple, a lawyer who first discussed the idea at a family dinner in 1958 before suggesting it at a meeting of the Mutual Admiration Society the following year. The year 1964 was nominally selected to coincide with the 300th anniversary of the British conquest of the Dutch colony of New Netherland. Kopple and two friends, Charles Preusse and Thomas J. Deegan, met with 35 potential financiers at the 21 Club restaurant. New York City mayor Robert F. Wagner Jr. and parks commissioner Robert Moses formally endorsed the proposal in August 1959, and seventy-five businessmen formed the New York World's Fair 1964 Corporation (WFC) that month. Moses, who saw a 1964 fair as a means to develop the Flushing Meadows site, offered to let the WFC use Flushing Meadows for a nominal fee. The fairground would include not only the 1939 World's Fair site but also a part of the nearby Kissena Corridor Park.

The bid still needed approval from the United States Congress and the Bureau International des Expositions (BIE), the French organization that was in charge of approving world's fairs. With Los Angeles and Washington, D.C., submitting competing bids, U.S. president Dwight D. Eisenhower appointed a committee to review the 1964 world's fair bids in October 1959, and Eisenhower approved the New York bid later that month. Seventy-five nations had informally indicated an intention to attend the fair by late 1959, and the WFC began looking for a president and three additional managers in early 1960. Moses was tentatively selected as the WFC's president that March, despite Kopple's objections that Moses was too old. In turn, Moses would not take the job unless Kopple resigned, as the two men had disagreed bitterly over the canceled Mid-Manhattan Elevated Expressway. After Kopple quit the WFC, Moses formally became the WFC's president that May. Moses wanted the fair to run for two years, and consultants for the WFC predicted that the fair would have 70 million visitors during that time.

Moses traveled to Paris to ask for the BIE's recognition of the fair. Although the BIE had allowed the WFC to begin planning the fair in November 1959, BIE officials decided not to give formal recognition to the fair. Under BIE rules, world's fairs could run for only one 6-month period, though the WFC had tried to request an exemption. The New York fair would also charge rent to foreign governments, contravening another BIE rule that prevented rent from being charged to exhibitors. In addition, the BIE allowed only one exposition per country every ten years. These rules were not immutable; for example, the BIE had recognized the 1939 fair, even though the previous exposition had run for two seasons. However, Moses refused to negotiate with BIE officials and treated them derisively, belittling the BIE as a "bunch of clowns in Paris". As such, the BIE instead decided to approve the 1962 Seattle World's Fair, and the BIE directed its members to not host official exhibits at the fair.

Financing and initial exhibitors
The WFC planned to issue $500 million in bonds, which were later decreased to $150 million. Moses proclaimed that the 1964 fair would be a "billion-dollar" event, though this included expenses for related projects such as roads and the nearby Shea Stadium. The WFC leased about 646 acre from the city government in May 1960. Moses hired former lieutenant governor Charles Poletti and military engineer William Everett Potter to organize the exhibits. A design committee had proposed one massive, doughnut-shaped pavilion. Moses rejected the plan, and the design committee had been forced out by the end of the year. Moses did not devise any master plan for the fair; he wanted to save the WFC money by having exhibitors erect most of their own pavilions, The city government implemented a building code and health code, which Potter enforced. Nearly all buildings were to be temporary structures.

The 1964 fair was to be themed to "peace through understanding". WFC member Jerome Weinstein had devised the motto, which was inspired by an ideal that Kopple had wanted for his daughters. Moses wrote that the fair was meant "to assist in educating the peoples of the world as to the interdependence of nations and the need for universal lasting peace". Exhibits were to be divided into five areas, including a transportation area operated by the Port of New York Authority. The original plans called for an amusement park area, which was canceled after the WFC could not find an operator. By August 1960, the first ten exhibitors had applied for space at the fair, and architectural blueprints had been submitted for the fair's first pavilion. The WFC began sending delegations abroad to invite foreign governments to the fair.

The group began issuing $67.5 million in promissory notes in late 1960 to fund construction; the WFC later reduced the amount to $64 million (consisting of $40 million in notes plus $24 million from the city). At the time, the WFC's finance chairman predicted that the fair would earn over $200 million. By the end of 1960, seven countries had agreed to sponsor exhibits. and one-third of the industrial pavilion sites had been leased. Moses announced in early 1961 that the Unisphere would be built as the fair's symbol, and the WFC also hired the Pinkerton agency to provide security and first-aid services. A report published that January indicated that the fair itself would cost $768 million, although much of the cost would be paid by individual exhibitors.

Construction
Exhibitors designed their own pavilions, and the construction contractors hired members of local labor unions to build the structures. Wagner predicted that 10,000 people would be employed during construction. The WFC hosted "preview days" where selected guests could view the construction. The Travelers Companies built information centers across the U.S. to promote the fair, and local chapters of the Elks, Kiwanis, and Rotary clubs promoted the fair nationwide. The WFC issued collectible bronze and silver medallions, manufactured by the Medallic Art Company. Commemorative postage stamps were issued to celebrate the fair, both in the U.S. and in other nations. Some New York license plates also had slogans advertising the fair. Several hotels were built nearby to accommodate fair visitors, and public transit and roads to the fair were also upgraded. The WFC opened an information office to answer visitors' questions. Private businesses promoted their products for the fair, and discounted tickets were also sold in advance of the opening.

1961 and 1962
William Whipple Jr., the fair's chief engineer, said in September 1960 that exhibitors would be able to begin erecting pavilions by 1962. Construction of the first building, an administration structure, began in August 1960 and was finished in January 1961. In early 1961, almost all of Flushing Meadows–Corona Park was closed to accommodate the fair's construction, and the Long Island Rail Road (LIRR)'s World's Fair station opened. Workers also moved trees and diverted parts of the Flushing River into tunnels. By April 1961, thirty-four countries had accepted invitations to the fair, and the city agreed to spend $24 million improving the park. Moses obtained additional funds secretively from the city government; according to Moses's biographer Robert Caro, the city government may have spent as much as $60 million on the fair. The WFC announced in May that it would proceed with the planned amusement area around Meadow Lake, hiring the billionaire H. L. Hunt to operate the rides. By mid-1961, the WFC had privately raised $25 million and was predicting a $53 million profit. The groundbreaking ceremony for the first 1964 Fair pavilion took place that June. The WFC struggled to sell the remaining bonds and had sold around $30 million in promissory notes (three-fourths of the total) by the end of 1961. During the fair's construction, civil-rights activists expressed concerns that that the WFC's leadership included very few African Americans. Even after Moses met with activists, he still did not appoint African Americans to leadership positions, which attracted controversy amid the ongoing civil rights movement. The WFC eventually hired an African American executive to the fair's international division in 1962. Later that year, New York governor Nelson Rockefeller formed a committee to investigate persistent complaints about discrimination within the WFC.

Meanwhile, by the beginning of 1962, more than 60 nations, the governments of 30 U.S. states, and 50 companies had agreed to exhibit at the fair. The WFC also created a scale model of the fairground. The LIRR constructed a siding from the Port Washington Branch, allowing trains to deliver material. At a luncheon in March 1962, Moses lamented that construction had fallen behind schedule. The WFC had allocated $6 million to advertise the fair by mid-1962, and Deegan predicted that the fair's participants would spend another $75 million of their own money on promotion. The WFC also tried to attract Latin American countries to the fair. By late 1962, exhibits for the fair were being finalized, and many pavilions were under construction. Either 68 or 71 nations had announced plans for exhibits at the fair by then, though only 35 countries had formally leased space. Additionally, 125 businesses had expressed interest, and the WFC had finished installing utilities on the fairground. At the end of 1962, a small number of state and international pavilions were being built, while work in the industrial and transportation areas was further along. Groundbreaking ceremonies were hosted for many of the international pavilions.

1963 and 1964
The World's Fair Housing Bureau was formed in early 1963 to coordinate the development of hotel rooms for the fair. Despite commitments from various state and national governments, only some of these governments were actively constructing pavilions. The WFC also wanted to hire 40 concessionaires and sell 70 intellectual property licenses, which the corporation hoped would raise $130 million. On April 22, 1963, exactly a year before the fair's opening, U.S. President John F. Kennedy activated a countdown clock to the fair's opening. At the time, only 48 of the 200 proposed buildings had even started construction, even though all major structures had to be under construction by the following month. The press building opened that May, and an insurance syndicate was formed that June to protect the exhibits. By mid-1963, civil-rights groups were protesting the lack of racial diversity in the fair's development and filed a lawsuit to halt construction. That July, Moses denied rumors that construction had fallen behind schedule.

Materials from overseas began arriving in August 1963, though work on 50 structures still had not started by the next month. Moses became increasingly hostile toward journalists who doubted that the fair would be completed on time. There were also disagreements over whether students should receive discounted tickets; Moses opposed the plan, though the city government ultimately forced him to sell discounted tickets. The first pavilion, the Port Authority Heliport, opened in October 1963. The same month, Hunt resigned as the amusement area's operator, following disagreements over ticket prices and rides. Work on many pavilions was behind schedule due to funding issues, labor shortages, and bad weather. There were other issues, such as labor strikes, exhibitor withdrawals, and continuing racial tensions. Despite these difficulties, many pavilions were nearly completed by late 1963, and the WFC had sold 3.8 million advance tickets by year's end. To draw attention to the fair, the WFC displayed models of exhibits at the Time-Life Building in Manhattan. Exhibits were installed through late 1963 and early 1964, and the WFC also borrowed $3 million to fund the fair's completion.

In January 1964, the Chicago Tribune reported that the site was filled with raw material, incomplete building frames, and unpaved roads. That month, WFC officials said that work on 26 buildings was behind schedule, and they sought to demolish a pavilion that would not be ready for the fair's opening. They also signed a document outlining how profits from the fair were to be used. By that February, most of the major pavilions and attractions were complete, but Whipple estimated that up to 10 pavilions would not be finished before the fair's opening. The same month, the WFC stopped selling advance tickets, having sold 28 million tickets. There were still 4,800 workers on site in late March, when the state government began hiring people for the fair. Before the fair had even opened, the WFC had spent $30 million.

Opening

The night before the fair opened, the TV series The Bell Telephone Hour broadcast an opening celebration. When the World's Fair officially opened at 9:00 a.m. on April 22, 1964, the first visitor was a college student from New Jersey. The opening was celebrated with speeches by Robert Moses, Nelson Rockefeller, and U.S. president Lyndon B. Johnson. The same day, Johnson dedicated the US Pavilion, while Rockefeller and Moses dedicated the New York State Pavilion. During the ceremonies, hundreds of civil rights activists organized a sit-in and were arrested. The civil-rights group Congress of Racial Equality (CORE) proposed a "stall-in" to block roads leading to the fair, but few activists participated. The opening ceremony ultimately attracted 90,000 attendees, less than half the predicted number, in part due to inclement weather. The WFC banned picketing on the grounds, prompting lawsuits from civil-rights groups; a federal judge later ruled that protesters could give out handbills to passersby.

The New York Times wrote that 15 pavilions and three amusement attractions were not finished by opening day. One pavilion, the Belgian Village, was not completed until the end of the 1964 season, though it did operate for part of that year. Some pavilions could not open on schedule because the artifacts in the pavilions had been damaged, while others were incomplete. Exhibitors also accused workers of delaying some pavilions' construction to collect overtime pay. Three incomplete pavilions were abandoned completely, and work on other pavilions continued for several months after the opening.

Fairground
The fairground was divided into five regions. Exhibits for individual U.S. states and the U.S. federal government were concentrated in the Federal & State Area at the center of the fairground near the Unisphere. The international exhibits were concentrated in the International Area—a group of pavilions surrounding the Unisphere. Industry pavilions were concentrated around the Industrial Area on the eastern end near the Van Wyck Expressway. The Transportation Area was on the western side of the fairground. South of the Long Island Expressway, connected with the rest of the fair only via one overpass, was the Lake Amusement Area  (known as the Lake Area during 1965). Eight gates provided access to the fairground.

The 1964 World's Fair had 139 pavilions on opening day, in addition to 34 concessions and shows. Of the pavilions and shows, either 121 or 124 were free, and the rest required an additional payment. Scattered across the fairground were 5,300 trees, 3,500 benches, 1,400 telephones, and 60 mailboxes. There were also bank branches, picnic areas, and restrooms. Accessible bathrooms, wheelchair rental stands, and Braille guidebooks were provided for disabled visitors. There were also several hotels nearby, albeit few campgrounds. A spokesman for the 1964 fair said the exposition was supposed to be "cultural and sophisticated", and Deegan claimed that the exposition would be the "greatest single event in history".

Pavilions
Each section was designed in various architectural styles,  and many of the pavilions were designed in a Space Age style. Some pavilions used experimental designs; for example, the Bell System pavilion was supported by massive cantilevers, while the IBM pavilion was designed in the shape of a giant egg. Most structures were designed so they could be demolished easily after the fair and rebuilt elsewhere. Anyone could rent exhibition space if they could afford the land's rent and the pavilion's construction cost. Thus, the space was dominated by large corporations. Private companies spent a combined $300 million on their pavilions. The Big Three car manufacturers—Chrysler, Ford, and General Motors—alone spent a combined $110 million on attractions such as artificial islands and the Magic Skyway and Futurama car rides.

Twenty-three state pavilions were built. The fair included exhibits from 24 states, including Alaska, Florida, Hawaii, Illinois, Louisiana, Maryland, Minnesota, Missouri, Montana, New Jersey, New York, Oklahoma, Tennessee, Texas, West Virginia, Wisconsin, and the six states in New England. In addition, New York City had its own pavilion, as did the neighborhood of Hollywood, Los Angeles. Nineteen of the state pavilions were in the federal and state section, and three of the other four state pavilions were clustered around Meadow Lake at the southern end of the fair. None of the state governments had to pay rent for the land. State governments still had to fund the buildings themselves, and 20 states and Washington, D.C., did not pay for exhibits at the fair.

There were 45 pavilions in the International Area, most of which featured foreign countries' exhibits. Individual exhibits were presented by 66 nations, including the United States (whose pavilion was in the Federal and State Area). If nations that were represented only by one city or region are included, the fair featured attractions from 80 countries. Foreign nations rented the land from the WFC, and they also paid for staff lodging, food, and other expenses. Many nations from Asia, Africa, and Central and South America, though relatively few from Europe, exhibited at the fair. Other countries such as the United Kingdom, France, and Italy boycotted the fair because the BIE had not approved it. Because of BIE members' lack of participation, only six major countries—Egypt, India, Indonesia, Japan, Mexico, and Pakistan—had official exhibits at the fair. Some BIE members hosted unofficial exhibits or were represented by private companies' exhibits, and several countries were represented solely by an industry exhibit. Still other countries were represented by regional pavilions, such as those for the Caribbean and Africa. Many of the international pavilions sold merchandise.

The Industrial Area had 43 pavilions, representing nearly 350 American companies. Most of the companies were consolidated within four exhibit buildings, though about three dozen companies had their own pavilions. Corporations rented land from the WFC, while religious organizations were given the land for free. Large firms such as Bell Telephone Company, DuPont, IBM, Kodak, RCA, The Travelers Companies, and US Royal Tires, participated. The 1964 fair included only a few companies in the food, chemical, tobacco, cosmetic, or pharmaceutical industries. Transportation companies, including the Big Three, displayed products in the Transportation Area section of the fairground. Several of the industry pavilions offered free merchandise or other sponsorships, which often attracted customers. Moses also provided about 7 acre for religious groups and invited every major sect of Christianity to the fair. Eight religious pavilions were built, each of which was staffed by volunteers. Some exhibits were never built (such as the Soviet and Israel pavilions) but were displayed on official maps, causing confusion among visitors.

Amusement
In addition to pavilions, the Lake Area included several rides and attractions during 1964. There was a circus operated by John Ringling North, which performed in a 5,000-seat tent. Nearby was a wax museum. The amusement area also included attractions such as a children's play area, puppet show, and porpoise show. A lake cruise traveled off the shore of Meadow Lake, and an early log flume ride was also installed at the fair. A replica of the ship Santa María was also displayed in the lake. There was a 10,000-seat outdoor theater in the amusement area as well. The Flushing Meadows Carousel was formed through the combination of two Coney Island carousels.

The Florida pavilion took over much of the Lake Area in 1965, and two amusement areas called Carnival and Continental Park were added at that time. Outside the Lake Area were the Fiesta Playground and the Sculpture Continuum Playground.

Transportation
Within the Lake Area, the American Machine and Foundry Company (AMF) constructed a monorail with two 4000 ft loops. The line was a suspended monorail with seven 80-passenger trains, each two cars long. Another transport attraction at the fair was the Swiss Sky Ride, a ski lift or aerial gondola that was sponsored by the Swiss government. During the 1964 season, visitors could rent one of 147 Greyhound Escorters, driven by chauffeurs. In addition, 61 Glide-a-Ride trolleys served the fairground during both seasons.

The fairground was accessed by numerous highways on Long Island, which had been upgraded to provide access to the fair. An expanded World's Fair Marina provided access via Flushing Bay. There was a short-lived ferry service to Manhattan, as well as other ferry routes to various ports in New York and New Jersey. A helicopter shuttle ran to the Pan Am Building and Lower Manhattan heliports. Local buses, airport shuttle buses, the subway, and the LIRR all stopped near the fairground. The R33S and R36WF subway cars were constructed for the 7 route, which ran to the Willets Point station near the fair. Although a dedicated subway line had served the 1939 fair, no such route was built for 1964. A luxury bus service carried "distinguished guests" to and from the fair. There were 20,000 parking spaces, and shuttle buses transported people from the parking lots to the main gate.

Other features
The fairground had nine fountains and eleven reflecting pools. Seven of the fountains had custom designs, although none of these custom fountains remain. Placed at the center of the fair was the Unisphere, which was constructed by American Bridge Company. Weighing 700,000 lb, the globe was created to symbolize "man's achievements on a shrinking globe in an expanding universe". At the far east end of Flushing Meadows–Corona Park, the Fountain of the Planets (Pool of Industry) could spray water up to 625 ft high, and it hosted nightly fireworks displays and music performances. The Unisphere and Fountain of the Planets are connected via the Fountain of the Fairs, which included a five-section reflecting pool and two rectangular pools.

The fairground had 28 mi, 39 mi, or 40 mi of paths, and there were also numerous plazas. Throughout the fairground were information booths operated by Greyhound Bus. Near the northern end of the fairground was a customs building, where customs officials examined items bound for the fair's international pavilions and concessionaires. There was also a press building next to the Grand Central Parkway, with a reporters' bullpen, offices for major news agencies, a press conference room, and offices. On the fairground was the 22-room Atomedic Hospital, which was staffed 24 hours a day, in addition to five first-aid stations. About 300 closed-circuit televisions were installed across the fairground, and a film studio for independent filmmakers was also built. Pinkerton matrons operated a lost-child bureau with activities and games for lost children.

Culture
Both foreign cultures and American technologies were featured at the fair. While WFC rules technically prevented the fair's officials from influencing the design or contents of any exhibits, the WFC still retained a significant impact on the contents of exhibits. For example, developing nations were encouraged to showcase their art and culture, rather than technology, and WFC officials pressured Islamic nations to emphasize their religion.

Cuisine
The restaurant had a large number of eateries. When the fair opened, it had 110 to 114 eateries, including 61 within pavilions. The World's Fair had 31 standalone restaurants operated by the Brass Rail firm (including 6 specialty restaurants and 25 fine-dining restaurants). Restaurant Associates was supposed to operate several restaurants at the World's Fair, but its contract was canceled because of a dispute over signage, and Brass Rail instead received the contract. Ten of Brass Rail's restaurants were designed by Victor Lundy and had canopies shaped like bunches of white balloons. For the 1965 season, the fair was expanded to include 198 restaurants.

Cuisine sold at the fair included Belgian waffles, 7 Up drinks, dumplings, pizza, tacos, kimchi, Turkish coffee, tandoori chicken, and hummus. Many of these became popular in New York City and in the U.S. after the fair closed. The Thailand pavilion included North America's first Thai restaurant, while the Malaysia pavilion served Tiger Beer and satay. During the 1964 season, many meals cost 99 cents because any food below $1 could not be taxed. Brewers spent millions of dollars convincing exhibitors to sell their beers.

Performances
There were more than 30 entertainment events at the fair. Moses disdained carnival-style attractions, saying that there would be "no whiskered women, tattooed giants, nudes on ice. [...] The appeal of a world's fair should not be entirely below the Adam's apple." As the WFC's president, Moses also reserved the right to ban any project from the fair. Shows that appealed to prurient interests, like seminude dancing, were thus ruled out. An exception was an adult-only musical puppet show, Les Poupées de Paris. The lack of adult shows may have contributed to the amusement area's unpopularity in 1964. The ban on adult shows was relaxed in 1965, and nine discotheques opened at the fair during that season. A striptease show in the Louisiana pavilion lasted two performances in May 1965 before it was canceled.

Musical and theatrical performances took place at several pavilions, and there were fireworks and water shows at the Pool of Industry. Among the theatrical shows were the revues To Broadway With Love, Wonder World (which lasted two months), and Summer Time Revue. DuPont presented a musical revue in its own pavilion, The Wonderful World of Chemistry. A controversial minstrel show in the Louisiana pavilion was canceled after two days. International pavilions, such as the African, Indonesia, and Spain pavilions, also hosted dance and other live shows.

Sporting events, such as wrestling, boxing, gymnastics, fencing, judo, and weightlifting, took place at the fair. Some of these events were presented as part of the 1964 Summer Olympics tryouts in New York City. The Dick Button's Ice-Travaganza ice-skating show was hosted at the New York City Pavilion. The Auto Thrill Show was hosted in the Transportation Area. A parade traveled across the fairground every day, and there were fireworks and water shows at the Pool of Industry. Some shows, like To Broadway With Love and the Ice-Travaganza, closed within a few months of the fair's opening. The evangelist Billy Graham gave sermons every day at the Billy Graham Pavilion.

Music and film
The fairground itself did not emphasize music, although a "World's Fair Festival" did take place at Lincoln Center in Manhattan. Popular and classical music was broadcast from 800 lampposts on the fairground. The fair also had an official band: Cities Service's World's Band of America, a 50-piece ensemble headed by conductor Paul Lavalle. Other ensembles performed throughout the fair, including Guy Lombardo, the United States Marine Band, and the United States Navy Steel Band. During 1964, the amusement area hosted rock-and-roll concerts that were popular among local youth; Moses, who abhorred the genre, canceled these concerts when he learned about them.

The fairground contained about 40 theaters, most of which were housed within pavilions. Fifty films were produced for these pavilions. There were several religious films including Parable at the Protestant and Orthodox pavilion; Man in the 5th Dimension at the Billy Graham pavilion; and Man's Search for Happiness at the LDS pavilion. The Johnson's Wax pavilion screened a film titled To Be Alive!, which later won an Academy Award, The WFC also produced a film promoting the fair in seven languages in 1964.

Art and artifacts
Originally, the WFC made no effort to coordinate art exhibitions at the 1964 World's Fair, and Moses did not wish to subsidize art exhibits on the fairground. Moses did not want to pay for an art pavilion himself. After commentators spoke about the lack of art at the fair, Moses changed his mind, allowing states to display art in their own pavilions. For example, ten pop artists designed art for the New York State Pavilion's Theaterama. The Fine Arts pavilion displayed 250 contemporary artists' work.

Foreign nations displayed art and artifacts at the fair as well. Spain displayed works from artists such as Francisco Goya, El Greco, Joan Miró, Pablo Picasso, and Diego Velázquez. The United Arab Republic displayed artifacts from several historical eras; the Sudan pavilion displayed a 1,300-year-old Madonna fresco; the Jordan pavilion showcased the Dead Sea Scrolls; and the Republic of China pavilion displayed Chinese jade. The Mexico pavilion displayed art during the 1965 season, including Mesoamerican pieces alongside works by José Clemente Orozco, Diego Rivera, David Alfaro Siqueiros, and Rufino Tamayo. Though art was also displayed in various gift shops throughout the fair, WFC officials generally only publicized artwork that was exhibited in pavilions.

There were 95 sculptures at the fair, including five permanent sculptures. Four of the sculptures remain in Flushing Meadows–Corona Park: Forms in Transit by Theodore Roszak, Freedom of the Human Spirit by Marshall Fredericks, Free Form by José de Rivera, and Rocket Thrower by Donald De Lue. Another sculpture, Armillary Sphere by Paul Manship, was vandalized, and the remaining pieces stolen in 1980. The non-permanent sculptures included Pietà by Michelangelo, at the Vatican pavilion, which was one of the fair's most popular exhibits.

Consumer products
The 1964 World's Fair introduced and showcased many consumer products, in what one magazine called "the ultimate marketing bonanza of [its] time". For example, color television was popularized at the fair, and the Ford Mustang was launched just before its appearance at the fair. Also showcased at the fair were technologies such as Picturephones and IBM computers, as well as electronic devices that could display personalized data to visitors. Some pavilions incorporated personal computers into their exhibits, and many visitors saw touchtone phones for the first time while at the fair. Other innovations never became popular, such as thermonuclear fusion power plants, undersea hotels, underground houses, jet packs, and Corfam synthetic leather.

Operation
The WFC originally predicted a daily attendance of 225,000. Deegan predicted at least 6.7 million foreign visitors, out of an estimated total of 70 million. Visitors 13 and older were originally charged the adult admission price of $2.00, while children 2–12 years old were charged $1.00. The WFC sold discounted tickets in packs of 20; some major companies like AT&T bought hundreds of thousands of tickets for employees. Students paid 25 cents if they visited with their teachers, and the WFC sold certificates that allowed a class of 25 students to enter the fair for $6.25. Moses predicted that the fair would sell $120 million worth of tickets, with a net profit of $40 million. Initially, city officials predicted that people would spend $5 billion in the city due to the fair, an estimate that was later reduced to $2.5 billion.

Though the fair employed up to 20,000 people, the WFC directly employed only about 180 to 200 people. Conversely, there were 3,000 Pinkerton employees on the grounds, including firefighters, police officers, medics, matrons, and ticket sellers. Nine garbage trucks, nine EMS vehicles, 25 police cars, and three fire engines traveled the fairgrounds, and nine city health inspectors examined all restaurants on the fairground. United Press International was the fair's official photographer, while United World Films had exclusive rights to produce and publish films about the fair. Allied Maintenance was the only maintenance firm allowed to work at the fair, and it charged exorbitant fees, earning $10 million during 1964 alone. Allied also handled deliveries during the 1964 season and was replaced by the Rentar Corporation the following year. Other companies, such as Hertz and Cities Service, sponsored free services or events. The WFC selected symbols of a boy and a girl as the fair's mascots.

Exhibitors were required to operate from 10 a.m. to 10 p.m. daily, although the fairground opened at 9 a.m. Exhibits were prepared and cleaned throughout the night, though the Vatican pavilion was the only attraction with a live-in caretaker. Many exhibitors hired a racially diverse staff as well. In addition, the WFC required each exhibitor to purchase insurance from the firm of Campo & Roberts, which earned $3 million from insurance commissions.

1964 season
Famous and wealthy visitors, including government officials and heads of state, visited the fairground in the weeks after it opened. The fair needed 220,000 daily visitors just to break even on its operating expenses, which totaled $300,000 per day. In its first week, the fair recorded nearly a million visitors, and during the first month, the fair saw 150,000 daily visitors (60 percent of initial projections). Several problems arose throughout the fair. Disputes occurred over labor unions, maintenance fees, and a mural in the Jordan pavilion. Thefts and breakdowns also occurred regularly. Exhibitors complained about high rental rates and insufficient maintenance of the fairground. The Lake Amusement Area was especially unprofitable; it had few attractions and could not be easily accessed. Many of the most popular exhibits charged an additional fee, and visitors often did not bring enough money for food or for higher-priced exhibits.

The WFC unsuccessfully attempted to entice visitors by offering discounts to taxi drivers and improving fairground lighting, and the WFC was planning promotional campaigns by the end of June. The J. Walter Thompson Company advertised the fair in New York City–area media. By mid-1964, some exhibitors had gone out of business, including the two largest shows in the Lake Amusement Area. Employees, especially the 4,000 college students who worked the fair, faced occupational burnout. Despite the troubles, the WFC was able to buy back one-quarter of its promissory notes in mid-1964, and exhibitors publicly downplayed their grievances with the WFC to attract visitors. The industrial and international pavilions were profitable compared with the amusement area, and Moses said he was unbothered by the Amusement Area's financial troubles.

The fair struggled to attract even 200,000 daily visitors, even during July and August, when students were on summer break. The fair had recorded 13.4 million visitors by the season's midpoint in July 1964, and it received 5.8 million visitors that August, the highest of any month during the 1964 season. Nearly half of visitors came from the New York City area, and would-be visitors expressed fears about crime and unrest. Attendance declined significantly in September when children returned to school. In response, Moses complained that journalists were tarnishing the fair's reputation and accused them of suppressing attendance. An exhibitors' committee made several recommendations for increasing attendance, but Moses rejected nearly all the suggestions.

The fair closed for the season on October 18, 1964. There had been 33 million visitors, including 27 million who paid admission. The New York Times cited several reasons for the reduced attendance figures, including crime fears, long queues, and high prices. Additionally, the WFC had paid several contractors far more than it should have, and the fair's operating expenditures during 1964 amounted to $33.3 million, twice the original budget. Moses had projected a $53 million surplus, but the surplus stood at only $12.6 million at season's end, barely enough to pay back the city government. The WFC reduced its estimated total profit for both seasons to $30 million. Despite the financial problems, many industrial pavilions had long queues and tens of thousands of daily visitors, and the General Motors and Vatican pavilions each saw more than 10 million visitors during 1964. The New York Times reported that many international exhibitors were pleased with the fair but wanted someone else to operate it.

Off-season
Between the 1964 and 1965 seasons, the WFC hired 400 security guards to oversee the fairground, though exhibitors were obligated to maintain and guard their own pavilions. The WFC planned to spend $1.3 million on renovations, and 3,000 workers began winterizing the fairground in November 1964. The WFC planned to create a promotional film and advertisements for the fair, and it kept some of the paths and fountains illuminated. Deegan said several existing pavilions would be renovated, and a dozen new restaurants would be added. Moses also traveled around the world to convince foreign exhibitors to display additional artifacts, such as a Gutenberg Bible and Spanish artwork, during 1965.

WFC officials claimed that attendance would rise and anticipated 37.5 million visitors during the 1965 season. The prediction was unrealistic: previous world's fairs typically had fewer visitors during their second season, and no new pavilions were being planned. Furthermore, in its balance sheet, the WFC counted profits from advance ticket sales as part of its income for 1964, which meant that revenue would be much lower than expected during 1965. Nonetheless, unless the fair had at least 37.5 million visitors in 1965, it would not be profitable. WFC officials, fearing reprisal from Moses, waited weeks to tell him about the fair's financial troubles. Ultimately, Moses told Wagner in November 1964 that the WFC might not be able to repay the city's $24 million loan. The WFC's financial advisors raised suspicions of financial management the next month.

Several of the WFC's financial advisors quit in January 1965, following bitter disputes, and the WFC requested $3.5 million to reopen the fair. City controller Abraham Beame began auditing the WFC, and the WFC fired Deegan's public-relations firm (which had been receiving $300,000 annually for four years) following criticism over the firm's compensation. The WFC's internal audit had found a $17.5 million deficit, but Beame's audit was delayed for several months due to lawsuits. By February 1965, at least fourteen exhibitors from the 1964 season had declared bankruptcy. Franklin National Bank offered to lend the WFC $3.5 million, but the WFC indicated that it needed only $1 million. Though city officials wanted to oust Moses as the WFC's president, he ultimately retained his position. When Moses said he would spend $6.4 million to renovate Flushing Meadows–Corona Park before repaying debts, the WFC's finance chairman resigned. Two Marine Midland Bank branches provided a $1 million loan to the WFC that March, which the WFC repaid two months later.

Meanwhile, during the off-season, several exhibitors renovated and modified their pavilions, spending over $7 million in total. At least fifty exhibits were upgraded, and five major attractions were added, along with free entertainments and science demonstrations. New artwork and films were added to several pavilions. The struggling Lake Amusement Area became the Lake Area. The WFC asked the New York City Transit Authority to increase subway service to the fair, and 26 exhibitors collaborated on a promotional campaign. Fifty-three exhibitors proposed that the first week of the 1965 season be called Fair Festival Week, to which Wagner agreed. The WFC produced a promotional film, To the Fair, and individual exhibitors also produced their own films. To reduce its debts, the WFC decreased its budget for the 1965 season and fired some employees.

1965 season
More than 150,000 people attended the reopening of the fair on April 21, 1965. Ethiopian long-distance runners Abebe Bikila and Mamo Wolde participated in a ceremonial half marathon, running from Central Park in Manhattan to Singer Bowl at the fairground. Unlike the 1964 opening ceremony, no protests occurred at the fair's reopening, and almost all exhibits were completed on time. For the 1965 season, adult admission fees were raised to $2.50. During the first 20 days of the 1965 season, attendance declined 22 percent compared with the same time period in 1964, putting many exhibitors at risk of bankruptcy. In addition, fewer visitors were paying at the gates, as more than half of visitors carried advance tickets. Exhibitors requested that admission fees be reduced and that a reduced-price evening admission ticket be sold. Moses refused both proposals, and several exhibitors threatened to close their pavilions before reneging.

At the beginning of the 1965 season, there were issues such as race-related protests, Vietnam War protests, a controversy over a racially insensitive song in one pavilion, and disputes between Jewish and Arab exhibitors. Vandalism also increased due to the reduced police presence, and a fairgoer was murdered that May. WFC officials also tried to invalidate their January 1964 agreement for disbursing the fair's profits, and exhibitors continued to lose money due to lower-than-expected attendance. Fewer visitors came during the evening, but the WFC again rejected a proposal for discounted evening admission in July 1965. Despite increased attendance in mid-1965, the fair continued to record decreased revenue compared with 1964. Many exhibitors recorded substantial losses from the costs of their pavilions. By August 1965, the WFC was preparing to clear the fairground after the fair, though 13 exhibitors had declared bankruptcy and could not afford to demolish or move their pavilions.

Beame's interim report, published at the end of August, found that the WFC had squandered money by not awarding contracts through competitive bidding and by spending nearly everything it had on expenses incurred before and during 1964. Despite Moses's denials of wrongdoing, Queens district attorney Frank D. O'Connor opened a criminal inquiry into the WFC shortly afterward. Moses also installed highway signs promoting the fair and refused to remove them, even after city traffic commissioner Henry A. Barnes called the signs a safety hazard. By mid-September, estimates of the fair's total attendance had been reduced from 70 to 50 million. By the end of September, the fair had recorded 17 million visitors during the 1965 season, less than half the number of visitors needed to break even. At this point, the WFC had barely enough money to pay its weekly expenses.

Toward the end of the 1965 season, there was a sustained increase in attendance, and the fair recorded more than 250,000 daily visitors for three weeks straight. There were so many visitors that exhibitors worried that people would be dissuaded by the overcrowding. Architect and writer Robert A. M. Stern attributed the attendance increases to a prevailing feeling that the 1964 fair would be one of the last lavish world's fairs. Pope Paul VI visited the fair on October 4, 1965, during the first-ever papal visit to the United States. The fair ended on October 17, 1965, and recorded its highest-ever daily attendance, 446,953, on its final day. The fair's final day was chaotic, with reports of vandalism and theft. In total, the fair had recorded 51,607,448 admissions, seven million more than the 1939 fair and ten million more than Expo 58. The GM and Vatican pavilions had been the most popular. Nonetheless, the fair had lost an additional $1 million in 1965 and had a deficit of up to $40 million at the time of closure. The New York Times partly attributed the fair's underperformance to Moses's stubborn attitude and refusal to take advice.

Site clearing
Moses predicted that the WFC would have to spend $11.6 million to clear the site. He recommended that most pavilions be demolished after the fair ended. By mid-1965, the WFC proposed preserving 19 structures, while the remaining pavilions were offered to anyone who could afford to relocate it. Several exhibitors chose to sell off their buildings due to the high cost of demolition, including U.S. Steel and Thailand. Other exhibitors sold off the contents of their pavilions, and people offered to salvage specific parts of pavilions. Any other buildings had to be demolished within 90 days of the fair's closure. Each exhibitor was supposed to have placed money in escrow, or posted a bond, to cover the demolition costs, but most exhibitors had not done so. As such, the WFC feared that exhibitors would merely abandon their pavilions.

Demolition began the day after the fair closed; the rubble from most pavilions was dumped into Flushing Bay. By November, the site was filled with "rubbish and rats". Scientists temporarily preserved three of the fair's buildings for structural tests. The deadline for demolishing the other buildings was extended to December 1966, and about two dozen structures remained by mid-1966. Moses suggested that his Triborough Bridge and Tunnel Authority provide funding to convert the fairground into a park, and the city government took over Flushing Meadows–Corona Park from the WFC in June 1967. Few improvements were made to the park for several years, while many of the remaining structures were vandalized. According to a 1986 report, the city government had to spend $107 million ($ million in ) to turn the fairground into a park. In the 1980s, there was a failed proposal for an 1989 World's Fair on the site to mark the 1964 fair's 25th anniversary. The paths remain almost unchanged through the 21st century.

Remaining structures

Some of the structures from the World's Fair remain in Flushing Meadows–Corona Park, including the fair's symbol, the Unisphere. Near the Unisphere is the Column of Jerash from Jordan's pavilion, a stone bench marking the site of the Vatican pavilion, and a plaque on the site of the Garden of Meditation. The New York City Pavilion hosts the Queens Museum, and the New York Hall of Science was also preserved as a museum. The western side of the fairground includes the Port Authority pavilion, which became the Terrace on the Park banquet hall; the Winston Churchill Tribute, which became an aviary for the Queens Zoo; and the Flushing Meadows Carousel. The New York State Pavilion is largely unused, but its former Theaterama is used by Queens Theatre in the Park. The World's Fair Marina still operates along Flushing Bay. Other buildings remained for several years before being demolished, including the Travel and Transportation Pavilion, the Federal Pavilion, the Aquacade amphitheater, and the Singer Bowl.

Other structures were relocated at the end of the fair. Among these were the Austria, Christian Science, Denmark, Indonesia, Japan, Malaysia, Mormon, Parker Pen, Spain, Switzerland, Thailand, and Wisconsin pavilions, in addition to the Uniroyal Giant Tire and Golden Rondelle Theater. In some cases, only part of a pavilion was preserved due to the high cost of preserving the full pavilion. The LIRR, Mormon, Socony Mobil, and West Berlin pavilions, as well as the monorail, Poupees des Paris, and wax museum, were preserved within New York. Other objects were sent further afield, including parts of the Hollywood and Ireland pavilions, the Coca-Cola pavilion's carillon, the It's a Small World ride, Progressland carousel, and Swiss Sky Ride. Some pavilions also became traveling exhibitions, such as Clairol's "color carousel" and Sinclair Oil's dinosaur exhibits.

Profitability and effect on other world's fairs
Profits from the fair would have been used to improve Flushing Meadows–Corona Park, and many of its restaurants broke even. On the fair's closing day, the WFC had $11.58 million in cash. However, Moses anticipated that the WFC would not be able to repay the remaining $22.4 million in promissory notes, and the WFC was expecting to default on 60 percent of the bonds it had issued. Beame determined in December 1965 that the fair had actually lost $20.1 million in 1964 due to improper management. The WFC later agreed to pay noteholders another $4 million in mid-1966, even as it struggled to fulfill its obligations to restore the fairground.

During the late 1960s, the WFC was separately investigated on charges of financial mismanagement regarding the Belgian Village pavilion, and the state government also opened a racketeering investigation into the fair's construction. The city government received $1.5 million in profits from the fair in 1972, following several years of legal disputes. Ultimately, the fair recorded a $21.1 million net loss, and bondholders received about one-third of their original investment back. In total, the WFC and other public agencies had spent $83.832 million on permanent improvements to the site during the 1964 fair.

In part because of the 1964 fair's unprofitability, many industrial exhibitors were reluctant to sponsor major exhibits at the next world's fair, Expo 67 in Montreal, and two other American cities withdrew proposals for world's fairs in the 1960s and 1970s. Citing the 1964 fair, Expo 67's organizers invested heavily in amusement attractions, sought and received BIE approval, and constructed their pavilions ahead of schedule. Expo 67 officials also sought positive press coverage for their fair, a significant departure from Moses's negative reaction to every perceived criticism.

Contemporary
Before the fair opened, The Washington Post called the fair a "mixed boon" to New York City because BIE members had boycotted the fair. In late 1963, just before the fair opened, an Associated Press reporter wrote that the 1964 fair was "a big city cousin to the familiar county fair" but that it was also becoming one of the United States' most comprehensive exhibits of industries and businesses. A British newspaper called the fair "a great big grown-up Disneyland".

After the fair's opening, Life and Ebony magazines called it one of mankind's largest expositions, and Newsweek wrote that the attractions and pavilions were "hard to resist". Several writers criticized the large number of industrial exhibits at the fair, and observers also complained about the wastefulness of the pavilions' temporary nature. During the second season, Time magazine wrote that the fair was unsuccessful because of the long queues, meager exhibits, high prices, and overwhelmingly large number of attractions for visitors. When the fair closed, a Wall Street Journal reporter wrote that the fair had failed because it "lacked coordination, a common purpose that could be transmitted to the community". Russell Lynes said the concept of the fair was flawed because instant communication between various countries was already possible.

There was also commentary on the conflicting architectural styles, which were controversial even before the official opening. John Canaday of The New York Times wrote in 1961 that he would be surprised if the fair were not "a mess and disaster architecturally". After the fair opened, Ada Louise Huxtable of The New York Times wrote that the fair was architecturally "grotesque", while other critics lambasted the structures as befitting Coney Island or a street fair. The critic Vincent Scully Jr. derided the fair in a Life magazine article, "If This Is Architecture, God Help Us". Conversely, Time magazine wrote in June 1964 that the fair had "grace and substance" despite the presence of some "tacky" attractions, a sentiment repeated in The Atlanta Journal-Constitution. The American Institute of Architects gave awards to several pavilions for "excellence in design". A Newsday reporter described the fair as "both garish and subtle, tawdry and tasteful, ephemeral and lasting". After the fair closed, architectural critic Wolf Von Eckardt lambasted the fair as "a frightening image of ourselves" because of its "chaotic" architecture.

Retrospective
In 1967, New York Times reporter Robert Alden wrote that the 1964 fair benefited from "participation of private industry on a massive scale" but that Expo 67 had more participating countries. Another Times writer said in 1989 that "the 1964 fair was not as self-conscious a portrayal of the future so much as a display of contemporary American achievements". The same year, a Newsday reporter wrote that the 1964 fair had occurred at a time when audiences were no longer awed by cultural and technological innovations. Robert A. M. Stern wrote in 1995 that the fair had been so attractive in part because "it was out of place amid the realities of life in the nuclear age".

David W. Dunlap wrote in 2001 that the 1964 fair was still ingrained in the public imagination, even though it had been "a tailfin-tacky celebration of jet-age technological hubris" and an "unhappy final chapter" to Moses's career as New York City's main urban planner. According to the author Lawrence R. Samuel, the fair's motto of "peace through understanding" was overshadowed by the fair's focus on profits, and a Bloomberg reporter similarly wrote in 2013 that the fair had been dominated by its corporate exhibitors. Joseph Tirella wrote in 2013 that, even though "peace through understanding continues to elude us", the United States had become more ethnically diverse due to the Immigration and Nationality Act of 1965, which was enacted just before the fair ended.

The New York Daily News wrote in 2012 that the remaining structures from the fair "have provided [Flushing Meadows–Corona Park] with some of its most striking structures". For the fair's 50th anniversary, Smithsonian magazine wrote that the fair's "limitless faith in material and social progress" had been counteracted by social upheaval in the U.S. that took place during the mid-1960s. The New York Times wrote in 2018 that the fair was not only a showcase for futuristic technology, but also a place where "foreigners could broadcast their best wares and fairgoers could catch a glimpse into their far-off cultures."

Economic and regional influence
The fair was credited with increasing tourism in New York City even before the formal opening. The city's hotels were often overcrowded during the fair's 1964 season, and garages were also overcrowded as well. Broadway theaters recorded increased ticket sales, as did other visitor attractions such as the Empire State Building and Radio City Music Hall. Though major department stores and restaurants saw increased business, other merchants reported that the fair had not had a measurable impact on their business. To avoid the crowds, many residents also left the city during the fair. Attendance at local amusement parks declined during the fair, and some parks, such as Freedomland U.S.A., never recovered.

At the end of the first season, the city's Convention and Visitors Bureau estimated that the city had earned $400 million due to the fair. The Christian Science Monitor wrote in 1965 that the fair had only partially benefited the area's economy. After the fair ended, The New York Times estimated that the fair had increased local restaurant profits by 8%, while stores in Midtown Manhattan saw their profits increase by approximately 4%. The highways leading to the fairground remained in use after the fair's closure, and residential development in Queens increased following the fair.

Cultural influence and media
At the fair's 50th anniversary, a writer for amNewYork Metro wrote that the fair had helped influence 21st-century technologies and that it had highlighted Flushing Meadows–Corona Park as an attraction in itself. The fair helped popularize several consumer products such as Belgian waffles and the Ford Mustang. The 1964 World's Fair included several exhibits and technologies that were later included in Disney parks, including Audio-Animatronics. Disney's other exhibits at the World's Fair—It's a Small World, the Progressland carousel, and the Magic Skyway—were also incorporated into Disney parks.

The fair has been the subject of documentary films such as The 1964 World's Fair (1996), After the Fair (2014), and Peace Through Understanding: The 1964–65 New York World's Fair. The fair and its structures have also been depicted in works of popular media; for example, the New York State Pavilion and Unisphere were shown in the films Men in Black and Iron Man 2. Historians have created websites and written books about the fair. The fair has also been the subject of several exhibitions, including at the Flushing Council on Culture and the Arts, the Queens Museum, Flushing Town Hall, and the Long Island Museum of American Art, History, and Carriages. The fair's 50th anniversary in 2014 was celebrated with six months of parties, exhibits, and other events across Queens.

Hobbyists have collected memorabilia from the fair, and several collectors have created the World's Fair Collectors Society. Collectors have preserved objects such as bracelets, medallions, ponchos, purses, and pocketknives. The Smithsonian Institution and Queens Museum also own objects from the World's Fair, and there have been efforts over the years to develop a world's fair museum.

title: 1964 New York World's Fair