kutamaya

While virginity may seem a bit outdated and old-fashioned, a number of women in their 20s and 30s are exactly that -- virginal. If you’re dating a "future sex kitten" and she’s decided you’re the one, you’ll want to keep several things in mind as you rock her world. Unlike sexually experienced women, if she's a virgin, she doesn’t really know what to expect, apart from what her girlfriends have told her -- which means she’s either going to be anticipating lots of pain or lots of ecstasy, depending who she’s talked to.

Regardless, she’s definitely going to be nervous, and that’s where you come in. The following tips will help to ensure that if she's a virgin, her first time will be anything but awkward.

Avoid ultimatums

No one likes being under the gun. It’s absolutely OK to let her know you’re ready to take things to the next level, but you don’t want her thinking that’s all you’re after. Women who are still virgins at 25 and 30 are unlikely to cave to the pressure, anyway -- they’re obviously pretty skilled at saying no! If she's a virgin and hasn’t given the goods up yet, that’s an indication she’s waiting for someone special. Laying the pressure on will make her feel cornered, trapped and objectified -- not exactly the hallmark of Prince Charming!

Take control

oing to need your firm, but gentle, guidance. In fact, she’s going to expect it, so don’t be afraid to lead with confidence; women love men who take charge in the bedroom. Let her determine the pace, but never lose sight of who’s teaching whom. She may be screaming with delight, begging you to shag her silly -- but that doesn’t mean she’s ready.

You’re the expert; it’s your job to gauge her level of arousal. You want her incredibly horny (and as wet as you can get her) before you go the whole nine yards. She’s going to remember this for the rest of her life. You want her reflecting on this moment with a saucy grin, not crossing her legs in pain at the memory.

Relax her

You’re the experienced one here, and she’s g It’s natural for her to feel a bit jittery, but you can ease much of that anxiety with a few simple tricks. She’s likely to be feeling self-conscious, so start with a few confidence boosters. Tell her she’s sexy and let her know she looks fantastic in your bed. Even if she already knows this, hearing you say it will do wonders.

Spend extra time kissing, letting her know you're focused is on her, and not just her body. Talk to her as you touch her body, find out what she likes and what she doesn’t. It may sound one-sided, but this should be all about her and what she’s offering you -- as opposed to what you’re taking. Let her know this, and she’ll relax completely.

Turn her on

No doubt it sounds like an obvious tip, but this one is more overlooked than you might realize. You want her so hot she’s purring with lust and begging for it -- and you want to keep her that way until the moment of truth. Unfortunately, many men break the mojo just before penetration. It’s not intentional, they just get a bit tense knowing they’re about to cause their partner a minor amount of pain. Relax, she knows what's coming and she’s not going to hold it against you. Go slowly, but don’t forget to kiss and caress her as you do. This gives her something else to focus on and keeps her aroused, which will actually decrease her sensitivity to pain.

Be willing to slow down and stop

She may be ready and willing at the start of the evening, but be prepared for that to change before the deed is done. Rest assured, it doesn’t mean she doesn’t want you, it just means she’s not ready. Some women are more sexually liberated than others -- her virginity might be something she can’t cope with losing right now. On the other hand, she might just need a little reassurance. She may need evidence that you really care about her and are willing to wait. A willingness to stop and take a step back is often all the proof a woman needs. It may even be enough for her to decide to keep going. Either way, a little compassion in this area will earn you major points.

Istilah Kota hilang adalah nyata adanya, kebanyakan kota-kota hilang telah di temukan dan masih banyak di teliti oleh para ilmuwan, untuk mengetahui banyak hal tentang kota tersebut.

Here's our list of the 10 most amazing lost cities in the world.

1. ANGKOR (Cambodia): Contains the world's largest religious monument _{[More info at Wiki]}

Angkor served as the seat of the Khmer empire that flourished from approximately the 9th century to the 15th century A.D. More precisely, the Angkorian period may be defined as the period from 802 A.D., when the Khmer Hindu monarch Jayavarman II declared himself the "universal monarch" and "god-king" of Cambodia, until 1431 A.D., when Thai invaders sacked the Khmer capital, causing its population to migrate south to the area of Phnom Penh.

The ruins of Angkor are located amid forests and farmland to the north of the Great Lake (Tonle Sap) and south of the Kulen Hills, near modern day Siem Reap (13°24'N, 103°51'E), and are a UNESCO World Heritage Site.

The temples of the Angkor area number over one thousand, ranging in scale from nondescript piles of brick rubble scattered through rice fields to the magnificent Angkor Wat, said to be the world's largest single religious monument. Many of the temples at Angkor have been restored, and together they comprise the most significant site of Khmer architecture. Visitor numbers approach one million annually.

In 2007 an international team of researchers using satellite photographs and other modern techniques concluded that Angkor had been the largest preindustrial city in the world with an urban sprawl of 1,150 square miles. The closest rival to Angkor, the Mayan city of Tikal in Guatemala, was roughly 50 square miles in total size.

2. MACHU PICCHU (Peru): The Lost City of the Incas _{[More info at Wiki]}

Machu Picchu ("Old Peak") is a pre-Columbian Inca city located at 2,430 m (7,970 ft) altitude on a mountain ridge above the Urubamba Valley in Peru, near Cusco. Machu Picchu is probably the most familiar symbol of the Inca Empire. It is often referred to as "The Lost City of the Incas".

The site was designated as a World Heritage Site in 1983 when it was described as "an absolute masterpiece of architecture and a unique testimony to the Inca civilization".

Machu Picchu was constructed around 1450, at the height of the Inca empire, and was abandoned less than 100 years later, as the empire collapsed under Spanish conquest.

Although the citadel is located only about 50 miles from Cusco, the Inca capital, it was never found and destroyed by the Spanish, as were many other Inca sites.

Over the centuries, the surrounding jungle grew to enshroud the site, and few knew of its existence.

In 1911, Yale historian and explorer Hiram Bingham brought the "lost" city to the world's attention. Bingham and others hypothesized that the citadel was the traditional birthplace of the Inca people or the spiritual center of the "virgins of the sun," while curators of a recent exhibit have speculated that Machu Picchu was a royal retreat.

3. MEMPHIS (Egypt): ancient capital of Egypt _{[More info at Wiki]}

Memphis was the ancient capital of the first nome of Lower Egypt, and of the Old Kingdom of Egypt from its foundation until around 2200 BC and later for shorter periods during the New Kingdom, and an administrative centre throughout ancient history.

Its Ancient Egyptian name was Ineb Hedj ("The White Walls"). The name "Memphis" is the Greek deformation of the Egyptian name of Pepi I's (6th dynasty) pyramid, Men-nefer, which became Menfe in Coptic. According to Herodotus, the city was founded around 3100 BC by Menes, who united the two kingdoms of Egypt.

Estimates of population size differ widely. According to T. Chandlerm, Memphis had some 30,000 inhabitants and was by far the largest settlement worldwide from the time of its foundation until around 2250 BC and from 1557 to 1400 BC.

Memphis reached a peak of prestige under the 6th Dynasty as a centre of the cult of Ptah. It declined briefly after the 18th Dynasty with the rise of Thebes and was revived under the Persian satraps before falling firmly into second place following the foundation of Alexandria.

Under the Roman Empire, Alexandria remained the most important city and Memphis remained the second city of Egypt until the establishment of Fustat (or Fostat) in 641. It was then largely abandoned and became a source of stone for the surrounding settlements. It was still an imposing set of ruins in the 12th century but soon became little more than an expanse of low ruins and scattered stone.

The remains of the temple of Ptah and of Apis have been uncovered at the site as well as a few statues, including two four-metre ones in alabaster of Ramesses II. The Saqqara necropolis is close to Memphis.

4. PALMYRA (Syria): the Bride of the Desert _{[More info at Wiki]}

Palmyra was in the ancient times an important city of central Syria. It has long been a vital caravan city for travellers crossing the Syrian desert and was known as the Bride of the Desert.

The earliest documented mention of the city by its pre-Semitic name Tadmor, Tadmur or Tudmur, is recorded in Babylonian tablets found in Mari. Though the ancient site fell into disuse after the 16th century, it is still known as Tadmor and there is a small newer settlement next to the ruins of the same name.

n the mid-first century, Palmyra, a wealthy and elegant city located along the caravan routes linking Persia with the Mediterranean ports of Roman Syria and Phoenicia, came under Roman control.

During the following period of great prosperity, the Arab citizens of Palmyra adopted customs and modes of dress from both the Iranian Parthian world to the east and the Graeco-Roman west.

Tadmor is mentioned in the Hebrew Bible (Second Book of Chronicles 8:4) as a desert city built by the King Solomon of Judea, the son of David.

Palmyra was made part of the Roman province of Syria during the reign of Tiberius (14–37). It steadily grew in importance as a trade route linking Persia, India, China, and the Roman empire.

In 634 the first Muslims arrived in Palmyra. The city was taken by the Muslim Arabs under Khalid ibn Walid in 636. In the 6th century, Fakhreddine al Maany castle was built on top of a mountain overlooking the oasis.

The castle was surrounded by a moat, with access only available through a drawbridge. The city of Palmyra was kept intact. After year 800, people started abandoning the city.

5. PETRA: stone structures carved into rocks _{[More info at Wiki]}

Petra ("Rock") lies on the slope of Mount Hor (Jordan) in a basin among the mountains which form the eastern flank of Arabah (Wadi Araba), the large valley running from the Dead Sea to the Gulf of Aqaba.

It is famous for having many stone structures carved into the rock. The long-hidden site was revealed to the Western world by the Swiss explorer Johann Ludwig Burckhardt in 1812

It was famously described as "a rose-red city half as old as time" in a Newdigate prize-winning sonnet by John William Burgon. Burgon had not actually visited Petra, which remained accessible only to Europeans accompanied by local guides with armed escorts until after World War I. The site was designated as a UNESCO World Heritage Site in 1985 when it was described as "one of the most precious cultural properties of man's cultural heritage."

Enclosed by towering rocks and watered by a perennial stream, Petra not only possessed the advantages of a fortress but controlled the main commercial routes which passed through it to Gaza in the west, to Bosra and Damascus in the north, to Aqaba and Leuce Come on the Red Sea, and across the desert to the Persian Gulf. Petra's decline came rapidly under Roman rule, in large part due to the revision of sea-based trade routes. In 363 an earthquake destroyed buildings and crippled the vital water management system.

The ruins of Petra were an object of curiosity in the Middle Ages and were visited by the Sultan Baibars of Egypt towards the close of the 13th century. The first European to describe them was Johann Ludwig Burckhardt in 1812.

6. POMPEII (Italy): buried by the volcano _{[More info at Wiki]}

Along with Herculaneum, this Roman city near modern Naples was destroyed and completely buried during a catastrophic eruption of the volcano Mount Vesuvius spanning two days on 24 August year 79.

The volcano collapsed higher roof-lines and buried Pompeii under many meters of ash and pumice, and it was lost for nearly 1700 years before its accidental rediscovery in 1748.

Since then, its excavation has provided an extraordinarily detailed insight into the life of a city at the height of the Roman Empire.

Today, it is one of the most popular tourist attractions of Italy and a UNESCO World Heritage Site.

7. PALENQUE (Mexico): one of Mayan's most exquisite cities _{[More info at Wiki]}

Palenque is an ancient Maya city near the Usumacinta River in the Mexican state of Chiapas.

It contains some of the finest architecture, sculpture, roof comb and bas-relief carvings the Maya produced.

The ancient Mayan city of Palenque, with its superb jungle setting and exquisite architecture and decoration, is one of the marvels of Mexico.

First occupied around 100 BC, it flourished from about 600 to 700 AD, and what a glorious century that was. The city rose to prominence under Pakal, a club-footed king who reigned from 615 to 683 AD, represented by hieroglyphs of sun and shield, he is also referred to as Sun Shield or White Macaw.

During Pakal's reign, many plazas and buildings, including the superlative Templo de las Inscripciones (his Mausoleum), were constructed in Palenque, characterized by very fine stucco bas-reliefs.

Pakal's son Chan-Bahlum II continued Palenque's political and economic expansion and the development of its art and architecture and presided over the construction of the Grupo de la Cruz temples, placing sizable narrative stone stelae within each.

One can see the influence of Palenque's architecture in the Mayan city of Tikal. The rival Mayan city of Toniná's hostility was perhaps the major factor in Palenque's precipitous decline after Chan-Bahlum II's death in 702. Sources speak of a devastating Toniná attack on Palenque in 730

After the 10th century Palenque was largely abandoned. In an area that receives the heaviest rainfall in Mexico, the ruins lay undiscovered until the 18th century. Frans Blom, an early-to-mid-20th century investigator remarked: 'The first visit to Palenque is immensely impressive. When one has lived there for some time this ruined city becomes an obsession.'

8. VIJAYANAGAR (India): capital of one of the largest Hindu empires _{[More info at Wiki]}

Vijayanagar, the capital of one of the largest Hindu empires ever, was founded by Sangama dynasty princes Harihara and Bukka in 1336. Its power peaked under Krishnadevaraya (1509-29), when it controlled nearly the whole of the peninsula south of the Krishna and Tungabhadra rivers.

Comparable to Delhi in the 14th century, the city, with an estimated population of half a million, covered 33 sq km and was surrounded by several concentric lines of fortification.

Its wealth derived from the control of spice trade and the cotton industry. Its busy bazaars, described by travelers such as Portuguese Nunez and Paes, were centers of international commerce.

The empire collapsed after the battle of Talikota in 1565 when the city was ransacked by the confederacy of Deccan sultans (Bidar, Bijapur, Golconda, Ahmednagar and Berar), thus opening up southern India for Muslim conquest.

The ruins are set in a strange and beautiful boulder strewn landscape with an almost magical quality.

The undisputed highlight, the 16th century Vittala Temple, is a World Heritage Monument. Started by Krishnadevaraya, it was never finished or consecrated; its incredible sculptural work is the pinnacle of Vijayanagar art. The outer pillars are known as musical pillars as they reverberate when tapped.

An ornate stone chariot in the temple courtyard containing an image of Garuda.

9. EPHESUS (Turkey): one of the most important cities of early Christianity _{[More info at Wiki]}

Ephesus was an Ionian Greek city in ancient Anatolia, founded by colonists from Athens in the 10th century BC. The city was located in Ionia, where the Cayster River (Küçük Menderes) flows into the Aegean Sea, and was part of the Panionian League.

Ephesus hosted one of the seven churches of Asia, addressed in the Book of Revelation (2:1–7). It is also the site of a large Gladiator graveyard.

Ephesus was an important center for early Christianity. Paul used it as a base. He became embroiled in a dispute with artisans, whose livelihood depended on the Temple of Artemis there (Acts 19:23–41), and wrote 1 Corinthians from Ephesus.

Later Paul wrote to the Christian community at Ephesus.

10. SANCHI (India): the best-preserved group of Buddhist monuments _{[More info at Wiki]}

A UNESCO world heritage site in central India near the Betwa River, Sanchi is on a flat-topped sandstone hill, 90m above the countryside, and stands the best-preserved group of Buddhist monuments in India.

Most noteworthy is the Great Stupa, discovered in 1818. It was probably begun by the emperor Asoka in the mid-3rd century BCE and later enlarged. Solid throughout, it is enclosed by a massive stone railing pierced by four gateways on which are elaborate carvings depicting the life of the Buddha.

The stupa itself consists of a base bearing a hemispherical dome representing the dome of heaven enclosing the Earth; it is surmounted by a squared rail unit, the world mountain, from which rises a mast to symbolize the cosmic axis.

The mast bears umbrellas that represent the various heavens. Other remains include several smaller stupas, an assembly hall (caitya), an Asokan pillar with inscription, and several monasteries (4th–11th cent. CE). Several relic baskets and more than 400 epigraphical records have also been discovered.

Number 5

Meyer Lansky (1902 - 1983)

Born Maier Suchowljansky in Russia to Jewish parents in 1902, Lansky moved to New York when he was 9. He met Charles Luciano when they were just schoolboys. Luciano demanded protection money from Lansky, and when he refused to pay, the two boys fought. Impressed by Lansky's toughness, Luciano befriended the younger boy and the two remained lifelong friends. Lansky also met Bugsy Siegel when he was a teenager, and the three formed a powerful partnership. Lansky and Siegel formed the Bug and Meyer Mob, which became Murder, Inc.

Lansky's primary order of business was money and gambling, and he had operations in Florida, Cuba and New Orleans. He was an investor in Siegel's Las Vegas casino, and he even bought an offshore bank in Switzerland that was used for money laundering. A financial genius, he codeveloped the National Crime Syndicate and the Commission. But business is never personal, and he approved the murder of his best friend Bugsy Siegel when Siegel was unable to produce profits for the Syndicate. Even with a gambling racket in operation across the planet, Lansky never spent a day in jail.

Number 4

Frank Costello (1891 - 1973)

Francesco Castiglia was born in 1891 in Italy and moved to the United States with his family when he was 4. He changed his name to Frank Costello when he joined a street gang at age 13. After numerous petty crimes landed him in prison, he became best friends with Charlie Luciano; together, they dealt in bootlegging and gambling. Costello's strength was his position as a link between the Mob and politicians, especially the Democratic Party's Tammany Hall in New York, which enabled him and his associates to pay off certain officials.

Following Luciano's arrest, Costello became the man in charge, and he solidified and expanded the operation during this time. A power struggle between him and Vito Genovese (who served as Underboss) erupted in the '50s, and Vincent Gigante tried to kill Costello. Eventually, Costello grew tired of the gangster life and retired, but not before framing Genovese and Gigante for a drug bust. He died peacefully in 1973

Number 3

Carlo Gambino (1902 - 1976)

Carlo Gambino came from a family that had been part of the Mafia for centuries in Italy. He started carrying out murders when he was a teenager and became a made guy in 1921 at the age of 19. With Mussolini gaining power, he immigrated to America, where his cousin Paul Castellano lived. He became a thug for different New York families until he joined Lucky Luciano's crew.

After Luciano was extradited in the '40s, Albert Anastasia took over. But Gambino thought it was his time to shine and had Anastasia killed in 1957. He appointed himself Boss of the family and reigned with an iron fist over New York until his natural death in 1976.

Number 2

Charlie "Lucky" Luciano (1897 - 1962)

Salvatore Lucania was born in Sicily in 1897, but his family moved to New York nine years later. At a young age, he became a member of the Five Points gang, in which Al Capone also received his education. Five years after establishing an empire based mostly on prostitution, Luciano controlled the racket all over Manhattan. After a failed but brutal attack on his life in 1929 , Luciano started planning the National Crime Syndicate, an extension of Salvatore Maranzano's Commission, with Meyer Lansky.

They eliminated the competition, and by 1935, Lucky Luciano was known as the Boss of Bosses -- not just of New York City, but of the whole country. He was arrested and sentenced to 30 to 50 years in 1936, but was let out on parole in 1946 on the condition that he be deported to Italy. He had so much power that U.S. Navy intelligence sought his help when the Allies were set to invade Italy during World War II. He died of a heart attack in 1962.

Number 1

Al Capone (1899 - 1947)

If there ever was a gangster who earned the No. 1 spot, it is Al Capone. Alphonse Capone was born in 1899 to Italian immigrants in Brooklyn, New York, where he got his start in street gangs. He then joined the Five Points gang and became a bouncer. It was during these days that a series of facial wounds earned him the "Scarface" nickname. Capone moved to Chicago in 1919 and quickly moved up the Mafia hierarchy while working for Johnny Torrio (Capone became Torrio's protege).

It was the time of the Prohibition, and Capone ran prostitution, gambling and bootlegging rings. In 1925, at the age of 26, Capone took over after Torrio was wounded in a gang war. Known for his intelligence, flamboyance and love of public attention, Capone was also known to be very violent; his role in the orchestration of the St. Valentine's Day Massacre in 1929, in which key rival gangsters were murdered, proves this. In 1931, Federal Treasury agent Eliot Ness arrested him for tax evasion. [askmen]

If you browse around your local video store, you'll notice dozens of films about the Mafia. Witness the popularity of Goodfellas, The Godfather, Casino, and Bugsy. Why have so many films been made about these tough-guy hooligans? Because men have a fascination with gangster culture and organized crime. But who are some of the most notorious gangsters of all time?

To make the list, gangsters must have had a significant impact on the Mob thanks to the way they did business. They must have done most of their business in America, their legacy must have stood the test of time, and they must have had a significant impact on pop culture. Unfortunately, our own Mr. Mafioso didn't make the cut.

Number 10

Vincent "The Chin" Gigante (1928 - 2005)

Born in New York in 1928, Vincent Gigante was quite a character. He dropped out

Gigante's first significant act as a gangster and member of the Genovese family was an attempt to kill the powerful Frank Costello, but Gigante's bullet missed the target. Nevertheless, he continued to climb the ranks within New York's Genovese organization, eventually becoming a capo and consigliere in the early '80s.

Then, when Mob boss Tony Salerno was convicted, Gigante became the main man. What makes Gigante so memorable is his 30-year ploy of acting insane. After he successfully averted prison in the late '60s by employing psychiatrists to testify to his insanity, he took it upon himself to continue the act; throughout his career, he was often seen walking around the streets of New York wearing a bathrobe. For this reason, he was nicknamed the "Oddfather" and the "Pajama King." Imprisoned for racketeering, he finally admitted in 2003 that he was not crazy.

Gigante died in prison on December 19, 2005 due to heart complications. The Gigante family and his lawyer, Flora Edwards, filed a federal lawsuit regarding the lack of health care that Vincent received while in prison. Vincent was scheduled for release in 2010.

Number 9

of high school in the ninth grade and started boxing, winning 21 of 25 light-heavyweight bouts. By the time he was 17, he had turned to crime to support himself, which resulted in seven arrests before he was 25.

Albert Anastasia (1903 - 1957)

Born in Tropea, Italy in 1903, Albert Anastasia was still a teenager when he came to America. Involved in the docks operations in Brooklyn, Anastasia was sent to Sing Sing Prison for 18 months for the murder of a longshoreman; the mysterious deaths of witnesses led to his early release. Albert Anastasia (aka "Lord High Executioner" and "Mad Hatter") was known as a killer, a reputation that led Joe Masseria's gang to recruit him. Anastasia was also extremely loyal to Charles "Lucky" Luciano, who had plans to rule America's crime world. Anastasia had no problem betraying Masseria -- by being one of four people sent to kill him in 1931 -- when approached by Lucky Luciano.

At this time, Anastasia started taking on hits for the Murder, Incorporated outfit in New York, and in 1944, he became the leader of the murder squad. Although Anastasia was never prosecuted for any killings, Murder, Inc. was responsible for between 400 and 700 murders. In the '50s, he became the leader of the Luciano family, but Carlo Gambino wanted the job. Though the murder is officially unsolved, many believe that Gambino had Anastasia killed in a barbershop in 1957.

Number 8

Joseph Bonanno (1905 - 2002)

Born in 1905, Joe Bonanno grew up in his native Sicily and became an orphan at the age of 15. He left Italy due to the fascist power of the Mussolini regime and made a brief stopover in Cuba before moving to the United States when he was 19. Joe joined the Mafia as a way to prevent Mussolini from taking over Sicily. Nicknamed "Joey Bananas," he joined forces with Salvatore Maranzano. Before Luciano killed him, Maranzano created The Commission, the ruling body over Mafia families in the entire country.

Bonanno stepped up and took over one of these families. He became powerful in New York with cheese factories, clothing businesses and funeral homes, which were a terrific way to dispose of bodies. But plans to eliminate all the rival families turned against him and Bonanno was kidnapped for 19 days until he agreed to retire. In 1965, he initiated the Banana War to settle scores, but he retired for good soon thereafter due to bad health. Never in his life was he convicted of a serious offense.

Number 7

Dutch Schultz (1902 - 1935)

Arthur Flegenheimer, later known as Dutch Schultz, was born in the Bronx in 1902. As a teenager, he held up crap games to impress his boss and mentor, Marcel Poffo. At the age of 17, he did some time at Blackwell's Island (now known as Roosevelt Island) for theft. With prohibition in full swing in the 1920s, he realized that money was in bootlegging. A ruthless man, he would kill whenever his temper flared, which helped keep his competition in line.

He had a part in the founding of the Syndicate, but soon Luciano and Capone became his enemies. In 1933, the law wanted to shut down Schultz, so he went into hiding in New Jersey, which left his New York territory free for a takeover; Luciano seized the opportunity. Schultz made a comeback in 1935, but members of Albert Anastasia's crew killed him in a restaurant men's room before he could do any damage.

Number 6

John Gotti (1940 - 2002)

In the wake of the great gangsters who ruled New York, John Gotti had his work cut out for him. Born in Brooklyn in 1940, he was always quick with his fists and it was his life's dream to become a wiseguy. By the age of 16, he had joined a local street gang known as the Fulton-Rockaway Boys. He quickly became their leader, stealing cars and fencing stolen goods. In the '60s, he began associating with Mafia hoods and hijacking trucks. In the early '70s, he became a capo for the Bergin crew, a part of the Gambino family. Extremely ambitious, Gotti started to deal drugs, which was forbidden by family rules.

As a result, Paul Castellano, the Boss, wished to expel Gotti from the organization. In 1985, Gotti and his guys killed Castellano outside a steakhouse and Gotti took over the Gambino family. No matter how many times the authorities tried to indict him for being the most powerful criminal in New York, the charges were always dropped. Because of this -- and the fact that he dressed well and loved media attention -- he was nicknamed "The Dapper Don" and "The Teflon Don." He was finally convicted for murder in 1992 and died of cancer in prison in 2002.

The golden ratio, also known as the divine proportion, golden mean, or golden section, is a number often encountered when taking the ratios of distances in simple geometric figures such as the pentagon, pentagram, decagon and dodecahedron. It is denoted , or sometimes .

The designations “phi” (for the golden ratio conjugate ) and “Phi” (for the larger quantity ) are sometimes also used (Knott), although this usage is not necessarily recommended.

The term “golden section” (in German, goldener Schnitt or der goldene Schnitt) seems to first have been used by Martin Ohm in the 1835 2nd edition of his textbook Die Reine Elementar-Mathematik (Livio 2002, p. 6). The first known use of this term in English is in James Sulley’s 1875 article on aesthetics in the 9th edition of the Encyclopedia Britannica. is an abbreviation of the Greek tome, meaning “to cut.” (”phi”) was apparently first used by Mark Barr at the beginning of the 20th century in commemoration of the Greek sculptor Phidias (ca. 490-430 BC), who a number of art historians claim made extensive use of the golden ratio in his works (Livio 2002, pp. 5-6). Similarly, the alternate notation The symbol

In the Season 1 episode “Sabotage” (2005) of the television crime drama NUMB3RS, math genius Charlie Eppes mentions that the golden ratio is found in the pyramids of Giza and the Parthenon at Athens. Similarly, the character Robert Langdon in the novel The Da Vinci Code makes similar such statements (Brown 2003, pp. 93-95). However, claims of the significance of the golden ratio appearing prominently in art, architecture, sculpture, anatomy, etc., tend to be greatly exaggerated.

has surprising connections with continued fractions and the Euclidean algorithm for computing the greatest common divisor of two integers.

Given a rectangle having sides in the ratio , is defined as the unique number (i.e., such that the yellow rectangles shown above are similar). Such a rectangle is called a golden rectangle, and successive points dividing a golden rectangle into squares lie on a logarithmic spiral, giving a figure known as a whirling square. such that partitioning the original rectangle into a square and new rectangle as illustrated above results in a new rectangle which also has sides in the ratio

Based on the above definition, it can immediately be seen that

(1)

giving

(2)

Euclid ca. 300 BC gave an equivalent definition of by defining it in terms of the so-called “extreme and mean ratios” on a line segment, i.e., such that

(3)

for the line segment illustrated above (Livio 2002, pp. 3-4). Plugging in,

(4)

and clearing denominators gives

(5)

which is exactly the same formula obtained above (and incidentally means that is a algebraic number of degree 2.) Using the quadratic equation and taking the positive sign (since the figure is defined so that 1" border="0" height="14" width="31">) gives the exact value of , namely

			(6)
			(7)
			(8)

(Sloane’s A001622).

In an apparent blatant misunderstanding of the difference between an exact quantity and an approximation, the character Robert Langdon in the novel The Da Vinci Code incorrectly defines the golden ratio to be exactly 1.618 (Brown 2003, pp. 93-95).

The legs of a golden triangle (an isosceles triangle with a vertex angle of ) are in a golden ratio to its base and, in fact, this was the method used by Pythagoras to construct . The ratio of the circumradius to the length of the side of a decagon is also ,

(9)

Bisecting a (schematic) Gaullist cross also gives a golden ratio (Gardner 1961, p. 102).

Exact trigonometric formulas for include

			(10)
			(11)
			(12)

The golden ratio is given by the infinite series

(13)

(B. Roselle). Another fascinating connection with the Fibonacci numbers is given by the infinite series

(14)

A representation in terms of a nested radical is

(15)

(Livio 2002, p. 83).

is the “most” irrational number because it has a continued fraction representation

(16)

(Sloane’s A000012; Williams 1979, p. 52; Steinhaus 1999, p. 45; Livio 2002, p. 84). This means that the convergents are given by the quadratic recurrence equation

(17)

with , which has solution

(18)

where is the th Fibonacci number. This gives the first few convergents as 1, 2, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, … (Sloane’s A000045 and A000045), which are good to 0, 0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 5, … (Sloane’s A114540) decimal digits, respectively.

As a result,

infty)x_n=lim_(n->infty)(F_n)/(F_(n-1)), " border="0" height="37" width="136">

(19)

as first proved by Scottish mathematician Robert Simson in 1753 (Wells 1986, p. 62; Livio 2002, p. 101).

The golden ratio also satisfies the recurrence relation

(20)

Taking gives the special case

(21)

Treating (20) as a linear recurrence equation

(22)

in , setting and , and solving gives

(23)

as expected. The powers of the golden ratio also satisfy

(24)

where is a Fibonacci number (Wells 1986, p. 39).

The sine of certain complex numbers involving gives particularly simple answers, for example

			(25)
			(26)

(D. Hoey, pers. comm.). A curious (although not particularly useful) approximation due to D. Barron is given by

(27)

where is Catalan’s constant and is the Euler-Mascheroni constant, which is good to two digits.

In the figure above, three triangles can be inscribed in the rectangle of arbitrary aspect ratio such that the three right triangles have equal areas by dividing and in the golden ratio. Then

			(28)
			(29)
			(30)

which are all equal.

The substitution map

	" border="0" height="14" width="12">		(31)
	" border="0" height="14" width="12">		(32)

gives

01->010->01001->..., " border="0" height="14" width="171">

(33)

giving rise to the sequence

(34)

(Sloane’s A003849). Here, the zeros occur at positions 1, 3, 4, 6, 8, 9, 11, 12, … (Sloane’s A000201), and the ones occur at positions 2, 5, 7, 10, 13, 15, 18, … (Sloane’s A001950). These are complementary Beatty sequences generated by . This sequence also has many connections with the Fibonacci numbers. It is plotted above (mod 2) as a recurrence plot. and

Let the continued fraction of be denoted and let the denominators of the convergents be denoted , , …, . As can be seen from the plots above, the regularity in the continued fraction of means that is one of a set of numbers of measure 0 whose continued fraction sequences do not converge to the Khinchin constant or the Khinchin-Lévy constant.

The golden ratio has Engel expansion 1, 2, 5, 6, 13, 16, 16, 38, 48, 58, 104, … (Sloane’s A028259).

Steinhaus (1983, pp. 48-49) considers the distribution of the fractional parts of in the intervals bounded by 0, , , …, , 1, and notes that they are much more uniformly distributed than would be expected due to chance (i.e., is close to an equidistributed sequence). In particular, the number of empty intervals for , 2, …, are a mere 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 2, 2, … (Sloane’s A036414). The values of for which no bins are left blank are then given by 1, 2, 3, 4, 5, 6, 8, 10, 13, 16, 21, 34, 55, 89, 144, … (Sloane’s A036415). Steinhaus (1983) remarks that the highly uniform distribution has its roots in the continued fraction for .

The sequence , of power fractional parts, where is the fractional part, is equidistributed for almost all real numbers 1" border="0" height="14" width="30">, with the golden ratio being one exception.

Salem showed that the set of Pisot numbers is closed, with the smallest accumulation point of the set (Le Lionnais 1983). [ dagdigdug ]