The closed-form problem arises when new ways are introduced for specifying mathematical objects, such as limits, series, and integrals: given an object specified with such tools, a natural problem is to find, if possible, a closed-form expression of this object; that is, an expression of this object in terms of previous ways of specifying it.
More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only nth-roots and field operations In fact, field theory
allows showing that if a solution of a polynomial equation has a closed
form involving exponentials, logarithms or trigonometric functions,
then it has also a closed form that does not involve these functions.
There are expressions in radicals for all solutions of cubic equations (degree 3) and quartic equations (degree 4). The size of these expressions increases significantly with the degree, limiting their usefulness.
In higher degrees, the Abel–Ruffini theorem
states that there are equations whose solutions cannot be expressed in
radicals, and, thus, have no closed forms. A simple example is the
equation Galois theory provides an algorithmic method for deciding whether a particular polynomial equation can be solved in radicals.
The fundamental problem of symbolic integration is thus, given an
elementary function specified by a closed-form expression, to decide
whether its antiderivative is an elementary function, and, if it is, to
find a closed-form expression for this antiderivative.
For rational functions; that is, for fractions of two polynomial functions;
antiderivatives are not always rational fractions, but are always
elementary functions that may involve logarithms and polynomial roots.
This is usually proved with partial fraction decomposition. The need for logarithms and polynomial roots is illustrated by the formula
Changing the basic functions to include additional functions can change the set of equations with closed-form solutions. Many cumulative distribution functions cannot be expressed in closed form, unless one considers special functions such as the error function or gamma function to be basic. It is possible to solve the quintic equation if general hypergeometric functions
are included, although the solution is far too complicated
algebraically to be useful. For many practical computer applications,
it is entirely reasonable to assume that the gamma function and other
special functions are basic since numerical implementations are widely
available.
Analytic expression
This is a term that is sometimes understood as a synonym for closed-form (see "Wolfram Mathworld".) but this usage is contested (see "Math Stackexchange".).
It is unclear the extent to which this term is genuinely in use as
opposed to the result of uncited earlier versions of this page.
Similarly, an equation or system of equations is said to have a closed-form solutionif and only if at least one solution can be expressed as a closed-form expression; and it is said to have an analytic solution
if and only if at least one solution can be expressed as an analytic
expression. There is a subtle distinction between a "closed-form function" and a "closed-form number" in the discussion of a "closed-form solution", discussed in (Chow 1999) and below. A closed-form or analytic solution is sometimes referred to as an explicit solution.
The expression:
is not in closed form because the summation entails an infinite number of elementary operations. However, by summing a geometric series this expression can be expressed in the closed form:
The integral of a closed-form expression may or may not itself be
expressible as a closed-form expression. This study is referred to as differential Galois theory, by analogy with algebraic Galois theory.
The basic theorem of differential Galois theory is due to Joseph Liouville in the 1830s and 1840s and hence referred to as Liouville's theorem.
A standard example of an elementary function whose antiderivative does not have a closed-form expression is: whose one antiderivative is (up to a multiplicative constant) the error function:
Mathematical modelling and computer simulation
Equations or systems too complex for closed-form or analytic solutions can often be analysed by mathematical modelling and computer simulation (for an example in physics, see).
Three subfields of the complex numbersC
have been suggested as encoding the notion of a "closed-form number";
in increasing order of generality, these are the Liouvillian numbers
(not to be confused with Liouville numbers in the sense of rational approximation), EL numbers and elementary numbers. The Liouvillian numbers, denoted L, form the smallest algebraically closed subfield of C closed under exponentiation and logarithm (formally, intersection of all such subfields)—that is, numbers which involve explicit exponentiation and logarithms, but allow explicit and implicit polynomials (roots of polynomials); this is defined in (Ritt 1948, p. 60). L was originally referred to as elementary numbers,
but this term is now used more broadly to refer to numbers defined
explicitly or implicitly in terms of algebraic operations, exponentials,
and logarithms. A narrower definition proposed in (Chow 1999, pp. 441–442), denoted E, and referred to as EL numbers, is the smallest subfield of C closed under exponentiation and logarithm—this need not be algebraically closed, and corresponds to explicit
algebraic, exponential, and logarithmic operations. "EL" stands both
for "exponential–logarithmic" and as an abbreviation for "elementary".
Whether a number is a closed-form number is related to whether a number is transcendental. Formally, Liouvillian numbers and elementary numbers contain the algebraic numbers,
and they include some but not all transcendental numbers. In contrast,
EL numbers do not contain all algebraic numbers, but do include some
transcendental numbers. Closed-form numbers can be studied via transcendental number theory, in which a major result is the Gelfond–Schneider theorem, and a major open question is Schanuel's conjecture.
Numerical computations
For
purposes of numeric computations, being in closed form is not in
general necessary, as many limits and integrals can be efficiently
computed. Some equations have no closed form solution, such as those
that represent the Three-body problem or the Hodgkin–Huxley model. Therefore, the future states of these systems must be computed numerically.
Conversion from numerical forms
There is software that attempts to find closed-form expressions for numerical values, including RIES, identify in Maple and SymPy, Plouffe's Inverter, and the Inverse Symbolic Calculator.
A flying saucer, or flying disc, is a purported type of disc-shaped unidentified flying object (UFO). The term was coined in 1947 by the United States (US) news media for the objects pilot Kenneth Arnold claimed flew alongside his airplane above Washington State. Newspapers reported Arnold's story with speed estimates implausible for aircraft of the period. The story preceded a wave of hundreds of sightings across the United States, including the Roswell incident and the Flight 105 UFO sighting. A National Guard pilot died in pursuit of a flying saucer in 1948, and civilian research groups and conspiracy theories developed around
the topic. The concept quickly spread to other countries. Early reports
speculated about secret military technology, but flying saucers became
synonymous with aliens by 1950. The more general military terms
unidentified flying object (UFO) and unidentified anomalous phenomena
(UAP) have gradually replaced the term over time.
Aerial photo taken over Lake Cote, Costa Rica, by Sergio Loaiza (1971)
In science fiction, UFO sightings, UFO conspiracy theories, and broader popular culture, saucers are typically piloted by nonhuman beings. Most reported sightings
describe saucers in the distance and do not mention a crew.
Descriptions of the craft vary considerably. Early reports emphasized
speed, but the descriptions shifted over the decades to the objects
mostly hovering. They are generally said to be round, sometimes with a
protrusion on top, but details of the shape vary between reports.
Witnesses describe flying saucers as silent or deafening, with lights of
every color, and flying alone or in formation. Size estimates range
from small enough to fit in a living room to over 2,000 feet (610 m) in diameter. Sightings are most frequent at night. Astronomer Donald Howard Menzel concluded that the reports were too varied to all be describing the same type of objects. Experts have identified most reported saucers as known phenomena, including astronomical objects such as Venus, airborne objects such as balloons, and optical phenomena such as sun dogs.
1950s pop culture embraced flying saucers. The discs appeared in
film, television, literature, music, toys, and advertising. Their
reports influenced religious movements and were the subject of military
investigations. The shape became visual shorthand for alien invaders.
During the 1960s, saucers waned in popularity as UFOs were reported and
depicted in other shapes. Discs ceased to be viewed as the standard
shape for alien spacecraft but are still often depicted, sometimes for
their retro value to evoke the early Cold War era.
Reports of fantastical aircraft predate the first flying saucers. In antiquity, mysterious lights in the sky were interpreted as spiritual phenomena. In the 1800s, many newspapers reported massive airships with glowing
lights and humming engines. These are often seen as precursors to flying
saucer and UFO sightings. On January 25, 1878, the Denison Daily News
printed an article in which John Martin, a local farmer, reported an
object resembling a balloon flying "at wonderful speed". The newspaper
said it appeared to be about the size of a saucer from his perspective,
one of the first uses of the word "saucer" in association with a UFO. An outbreak of a number of sightings of mystery airships occurred in America in 1896 and 1897. During World War II, Allied pilots reported balls of light following their planes. They named the lights foo fighters and believed they were advanced Axis aircraft.
Many aspects of the typical flying saucer first appeared in science fiction. French sociologist Bertrand Méheust noted, for example, Jean de La Hire's 1908 novel La Roue fulgurante (The Lightning Wheel). In the novel, a flying disc-shaped machine abducts the protagonists via a beam of light.[11]: 206–8 Science fiction magazine Amazing Stories began publishing "The Shaver Mystery" in 1945. Written by Richard Sharpe Shaver and edited by Raymond A. Palmer,
they were science fiction tales about technologically advanced
"detrimental robots" that abducted humans, but the stories were
presented as a true account of Shaver's life. Until the magazine ceased printing The Shaver Mystery, Amazing Stories' letter column was regularly full of readers sharing their own purportedly true sightings of the robots.
Before the flying saucer was coined as a term, fantasy artwork in pulp magazines depicted flying discs. Skeptical physicist Milton Rothman
noted the appearance of so-called flying saucers in the fantasy artwork
of 1930s pulp science fiction magazines, by artists such as Frank R. Paul. One of Paul's earliest depictions of a flying saucer appeared on the cover of the November 1929 issue of Hugo Gernsback's pulp science fiction magazine Science Wonder Stories. Science fiction illustrator Frank Wu wrote:
The point is that the idea of space
vehicles shaped like flying saucers was imprinted in the national
psyche for many years prior to 1947, when the Roswell incident took
place. It didn't take much stretching for the first observers of UFOs to
assume that the unknown objects hovering in the sky had the same disk
shape as the science fictional vehicles.
Origins
The modern flying saucer concept, including the association with aliens, can be traced to the 1947 Kenneth Arnold UFO sighting.[16][6] On June 24, 1947, businessman and amateur pilot Kenneth Arnold landed at the Yakima, Washington airstrip. He told staff and friends that he'd seen nine unusual airborne objects. Arnold estimated their speed at 1,700 miles per hour, beyond the capabilities of known aircraft. Newspapers soon contacted Arnold for interviews. The East Oregonian reported his supposed aircraft as "saucer-like". In a June 26 radio interview, Arnold described them as "something like a
pie plate that was cut in half with a sort of a convex triangle in the
rear". Headline writers coined the terms "flying saucer" and "flying disk" (or "disc") for the story.Arnold later told CBS News
that the early coverage "did not quote me properly [...] when I
described how they flew, I said that they flew like they take a saucer
and throw it across the water. Most of the newspapers misunderstood and
misquoted that, too. They said that I said that they were saucer-like; I
said that they flew in a saucer-like fashion." The circular shape of typical flying saucers may be due to reporters mistaking Arnold's "saucer-like" description of motion.
The public was divided on the potential origin of the saucers.
Arnold told military intelligence officers he suspected the discs were
experimental aircraft, and early newspapers reported Arnold saying, "I
don't know what they were—unless they were guided missiles." News media speculated on a Soviet origin, and many war veterans connected them to the foo fighters seen during World War II.
A Gallup Poll found that 90% of Americans were aware of the saucer
stories, and 16 percent believed they were secret military weapons, most
likely American. The most common explanation given was some type of
illusion or mirage. Less than one percent believed they were alien craft. One report from Seattle, Washington, described a hammer and sickle painted onto a flying disc.
The stories spread to other countries, where they were influenced by
local political and social concerns. In Europe, which was still
recovering from the Second World War, saucers were often reported with
rocket-like features. German newspapers reported flying saucers that
exploded or had tails of fire. The names for the discs were largely derived from the English "flying saucer" including the French soucoupe volante, Spanish platillo volante, Portuguese disco voador, Swedish flygande tefat, German fliegende Untertasse, and Italian disco volante.
The 1947 sightings peaked in the days after the Fourth of July and declined rapidly through mid-July. Multiple organizations offered $1,000 rewards for hard proof. In the widely reported July 7, 1947, Twin Falls saucer hoax, four teenagers in Idaho fabricated a crashed disc from jukebox parts. On July 8, the Army Air Force base at Roswell, New Mexico, issued a press release saying that they had recovered a "flying disc" from a nearby ranch; the so-called Roswell UFO incident made front-page news.International media covered the military's announcement of a crashed
disc, but within 24 hours were reporting the military's retraction and
explanation that the material was balloon debris. By July 11, the most widely reported story was a North Hollywood
resident's claim that a 30-inch galvanized iron disc containing glass
radio tubes had crashed in his garden. Newspapers quoted Fire Battalion
Chief Wallace Newcombe's assessment, "It doesn't look to me like it
could fly."
By the 1950s, the term "flying saucer" was widely associated with extraterrestrial life. After commercial pilots Clarence Chiles and John Whitted reported
a glowing cylindrical object flying past their plane in 1948, the US
Air Force began to seriously investigate the possibility of an alien
origin, but also concluded that reported discs "seem inconsistent with
the requirements for space travel."
In a 1950 interview on flying saucers, Kenneth Arnold said, "if it's
not made by our science or our Army Air Forces, I am inclined to believe
it's of an extra-terrestrial origin". This extraterrestrial hypothesis was accompanied by other unusual theories. Meade Layne speculated that they came from an alternate dimension. Many people claimed to be the inventors of the discs but could offer no evidence. From 1947 to 1970, there was a broad range of overlapping and
contradictory explanations for the saucers' origin and purpose, even
among proponents.
Beliefs about flying saucers were influenced by pulp science fiction. Amazing Stories
editor Ray Palmer transitioned from publishing the purportedly true
Shaver Mystery, to publishing and organizing UFO investigations. In 1946, Palmer published Fred Crisman's letters about his encounters with underground beings. The following year, Crisman sent Palmer pale metallic fragments along
with a report from his employee, Harold Dahl, about a malfunctioning
flying saucer. Palmer recruited Kenneth Arnold to investigate Crisman and Dahl's Maury Island incident. The fragments turned out to be slag from a local smelter, but the men in black that Crisman and Dahl claimed were following them would become a common element in later UFO literature. Gray Barker popularized the idea of "men in black" who intimidate or silence UFO witnesses in his book They Knew Too Much About Flying Saucers. Palmer launched the magazine Fate in 1948, claiming to offer "the truth about flying saucers". It was the first of many non-fiction paranormal magazines, a genre that flourished in the 1950s.
A flying saucer movement developed during the 1950s.
It was influenced by scientific research, occult practices, pop
culture, existing religions, and earlier myths. In reports and in
popular media such as the 1951 film The Day the Earth Stood Still, saucers and their pilots were characterized as messengers. The first wave of so-called contactees, George Van Tassel, George Adamski, Truman Bethurum, Orfeo Angelucci, and George Hunt Williamson, all claimed to have ridden aboard the saucers and brought back messages for humanity. New religions and institutions arose around the contactees. Van Tassel built the Integratron, a domed structure near Landers, California, intended to facilitate further contact with aliens, physical rejuvenation, and a kind of spiritual time travel. According to George King, he founded the Aetherius Society—a new religious movement influenced by theosophy—at the direct instruction of an extraterrestrial. Some existing religions began to incorporate flying saucers. The Nation of Islam taught that the end of the world would be brought about by the "Mother Wheel" or "Mother Plane", a flying saucer half a mile wide. During the same time that Margaret Murray's "Old Religion" or witch-cult hypothesis
was being discredited in academic circles, its core idea—a lost
civilization remembered in myth—was being embraced in pulp fiction,
occult groups, and the growing UFO movement. Several authors speculated that ancient astronauts piloting UFOs were the cause of myths and religions. Schoolteacher Robert Dione wrote God Drives a Flying Saucer to reframe biblical miracles and the Miracle of the Sun as the work of humanoid aliens piloting flying saucers. Later, Erich von Däniken released Chariots of the Gods?, a work of pseudoscience that attributed ancient artifacts and monuments to its purported ancient astronauts.
1952 spike in UFO reports
Ufology developed as a parallel social movement. Well-known Variety columnist Frank Scully published Behind the Flying Saucers in 1950. The book presents the Aztec, New Mexico, crashed saucer hoax as the true account of an alien craft that "gently pancaked to earth like Sonja Henie
imitating a dying swan" and was recovered by the United States
government. The hoaxers were convicted of fraud for selling useless dowsing
equipment to the oil industry based on a claimed alien origin, but the
book described one of the men as a doctor with "more degrees than a
thermometer".Donald Keyhoe took a "nuts and bolts" approach to the idea of the government covering up alien life in his 1950 book The Flying Saucers Are Real. When the popular and respected Life magazine ran "Have We Visitors From Space?" in 1952, taking seriously ideas of alien visitors, a wave of sightings followed. The 1952 sightings spurred Leonard H. Stringfield
to form an early UFO investigation group called the Civilian
Investigating Group for Aerial Phenomena and to publish research on
UFOs.Albert K. Bender started his own International Flying Saucer Bureau in Bridgeport, Connecticut, in 1952. Influenced by these works, James W. Moseley began to tour the country interviewing witnesses and distributing a newsletter for the growing saucer subculture.
Within a decade of the first saucer sightings, reports spread to
other countries, leading to the emergence of local groups and
ufologists. Antonio Ribera started Centro de Estudios Interplanetarios in Spain, and Edgar Jarrold founded the Australia Flying Saucer Bureau. In France, UFO groups overlapped with occult groups and the anti-nuclear movement.
Reports have been more often made in the countries where UFO groups are
in operation, such as the United States, France, Spain, the United
Kingdom, Brazil, Chile, and Argentina. By the end of the decade, The Case for the UFO author Morris K. Jessup
reflected on his field: "This embryonic science is as full of cults,
feuds, and dogmas as a dog is of fleas. There are probably more opinions
about the nature and purpose of UFO's as there are Ufologers."
Scientist
Walther Riedel said Adamski faked this 1952 UFO photo (top) using GE
light bulbs for landing struts. Adamski is believed to have also used a
1930s gas lantern (bottom).
UFO photography emerged as a subgenre of documentary photography, showing often blurry or abstract discs framed by otherwise everyday settings. Notable examples include the 1950 McMinnville photographs, the Passaic UFO photographs, and the photographs of contactee George Adamski. Some of the alleged flying saucer photographs of the era were hoaxes, created using everyday objects such as hubcaps. German rocket scientist Walther Johannes Riedel analyzed George Adamski's UFO photos and found them to be faked. The UFO's "landing struts" were General Electric light bulbs with GE logos visible on them. UFO researcher Joel Carpenter identified the body of Adamski's "flying saucer" as the lampshade from a 1930s pressure lantern.
Flying saucers are now considered retro and emblematic of the 1950s and of science fiction B movies. The term "flying saucer" was gradually supplanted by "UFO" and later "UAP". Discs ceased to be the standard shape in UFO reports, and a broader variety of objects were reported. Recent reports more often describe spherical and triangular UFOs.
Description
Identification
A sun dog caused by ice crystals, visible to the left of the sun
Experts have identified the majority of flying saucer and broader UFO reports with known phenomena. British government investigations in the 1950s found that the vast majority of reports were misidentifications or hoaxes.
Common explanations for saucer sightings include the planet Venus,
weather phenomena such as ice crystals, balloons, and airborne trash. The US Navy and General Mills launched thousands of top-secret Skyhook spy balloons
by the mid-1950s. Because they floated at high altitude, it was
difficult to judge the speed of the massive balloons, and they were
widely reported as flying saucers. Kentucky Air National Guard pilot Thomas Mantell died while pursuing an unknown round object "of tremendous size", later identified as a Skyhook balloon. News media reported Mantell as having crashed "chasing [a] flying
saucer", and some lost Skyhook balloons were tracked down using news
reports of UFO sightings.
In the mid-1950s, psychologists began to study why people believed in flying saucers despite the lack of evidence. French psychiatrist Georges Heuyer viewed the phenomenon as a kind of global folie à deux, or shared delusion, triggered by fear of a possible nuclear holocaust. In the 1970s, French UFO researcher Michel Monnerie
compared reports that were later identified with those that remained
unexplained. Monnerie found no difference in the frequency of paranormal
phenomena reported alongside the sightings identified later as mundane
known objects. These findings led him to develop the thesis that the
saucer-specific experiences were a "psychosocial" process of myth-making
triggered by but not caused by aerial phenomena. This psychosocial UFO hypothesis became a popular explanation in France.
Reported sightings
Sketches of reported flying saucers (from the UK National Archives)
Eyewitness descriptions differ in reported appearance, movement, and purpose. In a 1963 overview of flying saucers, astronomer Donald Howard Menzel found some broad traits across sightings but noted that "no two reports describe exactly the same kind of UFO."
Menzel found saucers were usually reported as round but included
objects shaped like dining saucers, teardrops, cigars, kidney beans, the
planet Saturn, and yarn spindles.
Saucers often were reported with a dome or knob-shaped protrusion on
the top side. Size estimates ranged from 20 feet to over 2,000 feet
(610 m) in diameter. Menzel found saucers reported in nearly every
color, often glowing or flashing. The sightings had little consistency
in reported movement. Witnesses described hearing sounds ranging from a
thunderclap to total silence. Sightings typically took place at night,
around sunset or sunrise. Almost all witnesses described distant saucers
in flight. Menzel concluded, "No single phenomenon could possibly display such infinite variety."
If a witness describes a saucer's crew, they usually regard them as extraterrestrial. Grey aliens gradually became the most reported type of pilot, but a vast range of beings have been reported. The diversity was greater in the 1950s and early 1960s, when witnesses
reported the aliens variously as hairy, hairless, monstrous, gorgeous,
gigantic, dwarfish, robotic, insectoid, avian, Nordic, or grey-skinned. Historian Greg Eghigian argues that this gradual standardization
indicates a cultural process to create a broadly recognizable design.
Witnesses consistently describe and depict flying saucers as
ahead of contemporary technology. When comparing the 1947 saucer reports
to the mystery airships of the 1800s, sociologist Robert Bartholomew found that the claimed observations "reflected popular social and cultural expectations of each period". The mystery airship sightings of the 1800s included details such as metal hulls, propellers, searchlights, and large wings. The 1947 sightings—occurring months before Chuck Yeager broke the sound barrier—emphasized the "incredible speed" of flying saucers. While most 1947 reports focused on speed, this fell to 41 percent in 1971 and 22 percent in 1986. In the 1950s, hovering flying saucers were associated with contactees and hoaxes. By 1986, almost half of reported UFOs were said to hover slowly or remain motionless.
Fictional portrayals
In
popular media, flying saucers underwent a change in motion similar to
the shift in eyewitness reports. Early portrayals emphasized high speed
maneuvers, but later media gradually shifted to slowly hovering discs. Early films such as The Flying Saucer (1950) and film serials such as Bruce Gentry – Daredevil of the Skies (1949), show saucers streaking past at high speeds. The Day the Earth Stood Still (1951) mentions high speeds tracked by radar but also includes a slow landing scene. The 1960s television series The Invaders prominently features a slow landing scene in every episode. Many later iconic flying saucer films, including Close Encounters of the Third Kind (1977) and Fire in the Sky (1993), depict hovering and slow movements.
Since the late 1940s, flying discs have increasingly become
associated with a cultural conception of aliens that reflects the social
and political anxieties of the 20th century. Fictional flying saucers
represent concerns about atomic warfare, the Cold War, loss of bodily integrity, xenophobia, government secrecy, and the question of whether humanity is alone in the universe. Reports from witnesses influenced popular media, which led to greater interest in flying saucers. For the film Earth vs. the Flying Saucers, producer Charles H. Schneer adapted Donald Keyhoe's UFO books for the screenplay, while special effects artist Ray Harryhausen consulted with contactee George Adamski about the saucer design. No correlation has been found between the release of major UFO films and spikes in sightings. A disc, often domed or emitting a beam of light, has become visual shorthand for aliens. In 2017, the flying saucer emoji was added to Unicode.
Although the symbol now signifies alien life, similar motifs had
unrelated religious and astronomical meanings in the past. Some
ufologists have attempted to re-interpret premodern art to support pseudohistorical
claims of ancient alien interactions with humanity. Ufologists claim
that early portrayals of flying discs can establish a historical basis
for their existence as physical craft or some other type of external
phenomena. However, experts have consistently explained purported
portrayals of ancient UFOs as artifacts of the cultures producing them.
For example, Italian Renaissance painter Carlo Crivelli put a
disc-shaped element in his 1486 altarpiece The Annunciation, with Saint Emidius
that art historian Massimo Polidoro described as "a vortex of angels in
the clouds". The artists and audiences of the time understood it as an
artistic device representing the influence of the Christian God, not
extraterrestrials. The device is seen more clearly in many contemporary
works, notably Luca Signorelli's 1491 Annunciation.
Literature
1946
1957
Flying disc-shaped craft depicted on pulp magazines from 1946 and 1957
Several precursors to modern flying saucers appeared in science fiction literature, including The Shaver Mystery. Richard Sharpe Shaver's stories about a secret technologically advanced civilization of "detrimental robots" inside the earth were published as a true account of his life. Backlash from the science fiction community carried over to UFO literature. Saucers did appear in conventional science fiction, but a genre emerged that treated fantastical stories as either true or plausibly true. The debut issue of Mystic
magazine asked readers, "When you read this story, you will tell
yourself that it is fiction; the editors assure you that it is. But what
if—it isn't?" The Fortec Conspiracy, a science fiction novel, both drew from and fed into crashed saucer rumors. Major newspapers rarely did reviews for saucer books but printed their
sensationalist advertisements claiming to prove that flying saucers had
landed or were being covered up. Cultural studies scholar Jonathan Gray
describes this type of widely-viewed alarmist ad as a paratext (related
to the central text but not a part of it), which can reach a much
broader audience than the text itself.
Advertisement formatted similar to a newspaper article
Advertisements leveraged cultural interest in flying saucers from the
earliest reports. Magazines were promoted as offering skeptical,
debunking explanations for the phenomenon. From 1947 into the 1970s,
marketing leveraged the discs' potential as advanced technology. By the
1980s, saucers in advertisement were used to evoke awe towards their
potential pilots more than futurism.
Aliens and flying discs were common in 1950s science fiction comics that flourished after the Golden Age of Comic Books. Launched in the 1960s, the comic book anthologyUFO Flying Saucers featured illustrations of supposedly real sightings. The opening to its first issue declared, "Our scientists have seen them! Our airmen have fought them!" As the 1950s progressed, former pulp readers turned their attention to the growing medium of television.
Film and television
Film poster for a drive-in theater showing Forbidden Planet
Many early portrayals of flying saucers linked them to the Cold War. The 1949 film serial Bruce Gentry – Daredevil of the Skies featured a man-made flying saucer, and the 1950 film The Flying Saucer focused on Cold War espionage. Saucer films in the 1950s featured alien pilots, but many continued to center on Cold War fears. The Thing from Another World
(1951) was a loose adaptation of John W. Campbell's "Who Goes There?",
updated to include aliens and relocated to Alaska, where Americans
feared a Russian attack. Later that year, The Day the Earth Stood Still had its human-looking alien Klaatu give audiences explicit warnings about a possible nuclear holocaust.The Day the Earth Stood Still and The Thing from Another World
were financial successes that established the market for an "alien
visitor" subgenre of science fiction that merged flying saucers into
existing space opera tropes. Slowly hovering discs, such as the one from the landing scene in The Day the Earth Stood Still, appeared throughout science fiction, including It Came from Outer Space (1953), Earth vs. the Flying Saucers (1956), and the television series The Invaders. While contactees described aliens as benevolent messengers, Hollywood films often depicted them as monstrous antagonists.
Other countries adapted the largely American phenomenon at
different times, adding elements of the local culture. Early British
films were low-budget productions such as Devil Girl from Mars (1954) and Stranger from Venus (1954). Japanese filmmakers incorporated flying discs and alien invaders into the tokusatsu tradition in mid-50s films such as Fearful Attack of the Flying Saucers and Warning from Space. Indian cinema began to incorporate alien invaders in the 1960s, starting with the Tamil-language Kalai Arasi. An adaptation of Bankubabur Bandhu by Satyajit Ray was never completed but may have influenced other works of science fiction. In Spain, alien-themed television shows became popular in the 1960s.
Flying saucers quickly spread to other genres. In Metro-Goldwyn-Mayer's big-budget Forbidden Planet, a futuristic 1956 adaptation of William Shakespeare's play The Tempest, humans travel through space in the United Planets Cruiser C-57D, a ship resembling a flying saucer. The Twilight Zoneepisodes "The Monsters Are Due on Maple Street", "Third from the Sun", "Death Ship", "To Serve Man", "The Invaders", and "On Thursday We Leave for Home", all make use of the iconic saucer from Forbidden Planet.
The C-57D was followed by other disc-shaped spacecraft in broader science fiction, such as the Jupiter 2 from the television series Lost in Space (1965–1968). Saucers appeared in the television series Babylon 5 (1994–1998) as starships used by a race called the Vree. Aliens in the film Independence Day (1996) attacked humanity in giant city-sized saucer-shaped spaceships.
Flying saucers were supplanted by other concepts and fell out of favor with Hollywood filmmakers. After 1956, American saucer films were mainly B movies. Plan 9 from Outer Space is infamous for its "pie-pan" saucers dangled from visible piano wire. Television shows and British films continued to depict flying discs and alien invaders into the 1960s. Various saucer designs have appeared in Doctor Who, such as those used by the Daleks in Daleks' Invasion Earth 2150 A.D. or the Cybermen in "The Tenth Planet". Italy produced a wave of low-budget films, often space operas or comedies, including Omicron (1963) and Il disco volante (1964). By the end of the 1960s, Japan, Italy, and Britain largely ceased producing saucer films. Disc-shaped spacecraft fell out of favor in straight science fiction but continued to be used ironically in comedies. The image is often invoked retrofuturistically to produce a nostalgic feel in period works. For example, Mars Attacks! (1996) draws on the flying saucer as part of the larger satire of 1950s B movie tropes.
The sleek, silver flying saucer is widely regarded as a symbol of 1950s culture. The motif is common in Googie architecture and Atomic Age décor. Notable flying saucer structures include Seattle's Space Needle and Los Angeles International Airport's Theme Building. Googie architecture in California, such as the Chemosphere home, influenced the futuristic structures in the 1960s cartoon The Jetsons. The cartoon popularized the style to such an extent, that it is often referred to as the "Jetsons look". Architect Frank Lloyd Wright, who collaborated on the design of the flying saucer in The Day The Earth Stood Still, went on to use the flying saucer as an architectural motif. Wright's circular Annunciation Greek Orthodox Church in Wauwatosa, Wisconsin, United States, is capped by a flattened dome over a hundred feet across.
Exhibition model of a flying saucer (2022)
Spaceships are one of the subjects of novelty architecture.
Also known as mimetic architecture, novelty architecture is the
practice of creating structures shaped like other existing objects. The Communist-era Kielce Bus Station in Kielce, Poland, was designed by architect Edward Modrzejewski [pl] to resemble a UFO. The historic landmark arena in Katowice, Poland, is called Spodek (Polish for "saucer") based on its resemblance to the saucers of 1960s science fiction. Other modernist and brutalist UFO structures include the Ukrainian Institute of Scientific, Technical and Economic Information, Bulgaria's concrete Buzludzha monument, the Most SNP in Bratislava, Slovakia, and The Flying Saucer in Sharjah, United Arab Emirates. The Westall UFO was commemorated with the Grange Reserve UFO Park, featuring a UFO with red slides modeled after the reported sighting. Roswell, New Mexico,
is a UFO tourist destination in the Southwestern United States. Many
structures in Roswell, including the streetlights and the McDonald's,
are designed around alien themes. Moonbeam, Ontario, Canada, has an alien for its mascot and a prominent roadside flying saucer at its welcome center. UFO-shaped homes include the Futuro pods designed by Matti Suuronen, the former Sanzhi UFO houses from the Sanzhi District, New Taipei, Taiwan, and artist Harry Visser's iconic home in Roodepoort, Johannesburg, South Africa.
Broader pop culture
Battery-operated tin UFO
Flying saucers were a ubiquitous part of pop culture from 1947 into the mid-1970s. Flying disc motifs were used in toys and other novelties soon after the earliest reports. The frisbee was released in 1948 and initially branded the "flying saucer". Flying saucer candy was introduced in the 1950s when a Belgian producer of communion wafers had a dip in sales. Along with other vintage candies, they have since seen renewed interest from customers as "retro". In the 1950s and early 1960s, Japan was a major manufacturer of tin toys often with space themes such as robots, rockets, and flying discs. Throughout the 1950s, musicians such as Billy Lee Riley, Jesse Lee Turner, and Betty Johnson released novelty songs about flying discs and alien invaders. Bill Buchanan and Dickie Goodman released the first break-in record, "The Flying Saucer", which took the form of a mock news broadcast covering an alien invasion. Disneyland opened Flying Saucers, an attraction where guests could pilot a hovering disc by tilting their own body.
Video games have a long history of depicting flying saucers, typically as antagonists. In the arcades, the popular early shooting games Asteroids (1979) and Space Invaders (1978) featured flying saucers as "bonus" enemies that only emerged briefly. Super Mario Land, one of Nintendo's launch titles for the original Game Boy, contained spaceships modeled after photographs by George Adamski and set among various monuments falsely attributed to ancient astronauts, such as the Egyptian pyramids and the monolithic Moai of Easter Island. The XCOM
series tasks players with countering an invasion of aliens landing on
Earth in flying discs. Saucers have appeared as a craft that players can
control in Fortnite, Destroy All Humans, and Spore.
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