Distances and the Fibonacci Sequence

The Fibonacci Sequence is made up of numbers that are the sum of the previous two numbers in the sequence, starting with 0 and 1.

It goes like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, et cetera.

Observe how 1 equals 0+1; whereas 2 equals 1+1; 3 is 1+12; 5 is 2+3; and 8 is 3+5; and so on. The number after 144 is 233, or 89+144.

The Fibonacci number describes the golden spiral, an ideal form much beloved by artists, architects and designers through the ages. Interestingly, it also neatly matches the relationship between kilometers and miles.

Observe that three miles is roughly five kilometers, five miles is roughly eight kilometers, eight miles is roughly 13 kilometers. It’s not absolutely perfect because eight miles is actually 12.875 kilometers, but as top-off-the-head conversions go, it is immensely accurate.

“Thus the metric system did not really catch on in the States, unless you count the increasing popularity of the nine-millimeter bullet.”
– Dave Barry

If you need to convert a number that’s not on the Fibonacci sequence, you can just break out the Fibonacci numbers, convert, and add the answers. For instance, 100 can be broken down into 89 + 8 + 3, all Fibonacci numbers. The next numbers are 144, 13, and 5, which add up to 162. Note that 100 miles is actually equal to 160.934. Again, close enough.

16/x mmxiv

Earthworms turn over the earth’s top 6 inches of soil about every 20 years, maintaining the fertile humus that covers most land areas and supports agricultural production.

Lena Horne (1917-2010), the highest paid African American actor of her time, signed a contract guaranteeing she didn’t have to play cooks, maids, or other stereotypes.

One in four Canadians ate fast food in the past 24 hours.

Caribou make a clicking sound when they walk. This is caused by tendons slipping over bones in their feet.

Designed for efficiency of construction, the Empire State Building was erected in just 11 months from the setting of the first steel columns on April 7, 1930 to the fully enclosed structure on March 31, 1931. At the peak of construction the tower rose at the rate of a storey a day.

See other: Quite Interesting Facts

Thanks to the Ancient Greeks

α. Democracy. The cherished idea flowered in the 6th century BC when power was first passed to the people. Or rather, the ten per cent of Athens’ population who classified as citizens – women, slaves and foreigners did not make the grade. The remainder, men of 18 years and over, were divided into local groups – the demoi – who were then represented on the city’s major council or parliament of 500 called the boule. In addition, 40 times each year the people (the ekklesia) met in their thousands to vote on issues of both foreign and domestic policy.

β. Love. Indeed, love is actually Greek. The Goddess of Love in Greek mythology is Aphrodite who was born in the sea off Cyprus, conjured from the foam produced by the severed genitals of Zeus’s grand-father, hurled there by his son. Not the most auspicious of starts in life, though Aphrodite subsequently thrived, going on to become not only the Goddess of Love but, in the eyes of the Spartans, also a Goddess of War. So, by default, she was the Goddess of Married Couples.

A Greek and an Italian argue over who has the superior culture.
The Greek: We built the Parthenon.
The Italian: We built the Coliseum.
The Greek: We gave birth to advanced mathematics.
The Italian: Yes, but we built the Roman Empire.
The Greek: We invented sex!
The Italian: That’s true, but we thought of having it with women.

γ. Philosophy. The entire canon of western thought: Marx, Descartes, Derrida are all balanced on the shoulders of three men. Aristotle (384-322 BCE), who stood on the shoulders of his master Plato (429-347 BCE) who was propped up by Socrates (469-399 BCE). The latter was a veteran of the Peloponnesian War and when he was sentenced to death by drinking hemlock, he showed future generations how to die with dignity.

δ. Geometry. The ancient phrase “Beware Greeks bearing gifts” was coined to describe the dubious present of the Trojan horse, and any school pupil baffled by Pythagoras’ theorem is likely to turn up his nose at their gift of geometry. The word is derived from the term geometria, meaning the measurement of the earth, a discipline which Euclid, who was actually from Alexandria in Egypt, first studied at Plato’s Academy. In case our weary pupil wishes to nurse any further grudges, mathematics was also the Greeks’ fault.

ε. The Secret Police. The Gestapo in Nazi Germany and the KGB of Stalin’s Russia can trace their antecedents to Sparta, the original military state. The heroics of the ancient warrior race are justly celebrated at Thermopylae. It was here that 300 Spartans held off a Persian army of 150,000, and left the moving inscription: “Go tell the Spartans, stranger passing by/That here obedient to their words we lie”. But their society was one of great cruelty. A percentage of the Helots – their slaves – were slaughtered each year and at night an elite group known as the kryptia would seek out and execute trouble-makers.

ς. Ostracism. To be ostracised from a group or, as it is described today, “sent to Coventry”, is a Greek invention, which required the spurned party to depart from the city walls and live in exile. This law of banishment was first introduced in 508 BC but was first used almost 20 years later in 487 BC. The term is derived from ostraka the fragments of pottery on which the unfortunate nominee’s name was inscribed.

ζ. The Marathon. What a poor Athenian messenger tackled out of necessity, millions now do for fun. Following the battle of Marathon at which the Greeks broke the Persian army, losing just 192 men to their opponents 6,000, a messenger was dispatched to run the 25 miles back to Athens to announce the city’s salvation and their success. It was a feat he doggedly achieved before dying of exhaustion.

η. Alexander the Great. Before Colin Farrell dyed his hair blond and slipped into a leather skirt, a young Greek (well Macedonian actually) led an army of 40,000 infantry and 5,000 cavalry on a mission to conquer the known world. Today the battle tactics and wisdom of Alexander, who was a pupil of Aristotle, are studied by business leaders who adore his direct approach. When faced with the riddle of the Gordian Knot, which prophecy decreed could only be unravelled by he who would rule Asia, Alexander paused for a second before slicing it in two.

θ. The Olympic Games. The first gathering of top athletes drawn from the Hellenic world was in 776 BC and afterwards, every four years, they would return to Olympia to compete for crowns of wild olives in events such as chariot racing, wrestling, boxing and the pentathlon. There is no mention in recorded history of a competitor being banned for steroid abuse.

ι. The Muse. Artistic success in ancient Greece was not 1 per cent inspiration and 99 per cent perspiration. Instead, creative ideas were bounteous gifts bestowed by the Muses, a group of nine goddesses who were each responsible for a particular endeavour. They were Calliope (epic poetry), Clio (history), Erato (lyric poetry), Euterpe (the flute), Melpomene (tragedy), Polyhymnia (hymns), Terpsichore (dance), Thalia (comedy) and Urania (astronomy).

“Two thousand summers have imparted to the monuments of Grecian literature, as to her marbles, only a maturer golden and autumnal tint, for they have carried their own serene and celestial atmosphere into all lands to protect them against the corrosion of time.”

– Henry David Thoreau

ια. Bawdy Comedy. Some 2,500 years before Frankie Howerd and Sid James, the Greek world had its own ‘Carry On’ star in the shape of Aristophanes (450-388 BC). The playwright attacked the establishment’s sacred bulls with scandalously sexual plays that mocked both politicians and the city’s elite with a merciless wit.

ιβ. Public Jury. The system of trying the accused by a jury of his peers first took place in Athens. A full participation in the political and criminal process began after male citizens turned 30, at which point they were eligible to serve on juries or stand for election as a magistrate. Each day potential jurors would gather at the Athenian Agora, the market place, where they were picked by lot.

ιγ. History. As in the fields of philosophy, medicine and mathematics, the father of history was also a Greek. Herodotus of Halicarnassus wrote his famous “histories” in the 5th century BC. This recounts the expansion of the Achaemenid empire under its kings Cyrus the Great, Cambyses and Darius I the Great. It also records for posterity the actions of the Spartans at Thermopylae.

ιδ. The Hippocratic Oath. The pledge made by modern physicians that they will, in essence, ‘do no harm’ is attributed to the Greek physician, from the island of Kos, who in the 5th and 4th century BC laid the foundation stones of scientific medicine by freeing medical study from the constraints of philosophical speculation and superstition.

ιε. Hades/Hell. Hades was controller of the Kingdom of the Dead and the brother of Zeus. In many ways the Christian concept of Hell, as an underground lair ruled by the Devil, is an appropriation of the Greek myth.

ις. Argos. Richard E Grant’s spaced out rock star may pronounce it as ‘Argoose’ and believe the catalogue shop in which your Tefal kettle comes sliding down a conveyer belt is, in fact, a chic designer outlet, but he’s wrong. The real Argos was a settlement to which Greeks in the ‘Golden Age’ choose to transplant the court of Agamemnon, the King of Mycenae and the destroyer of Troy.

ιζ. Doric Architecture. The evolution of Greek architectural styles took place during the 7th and 6th centuries BC. At the time the majority of buildings in Ancient Greece used limestone but the nearest quarry to Athens produced marble, and the result was the temple to Athena Parthenos or Athena the Virgin. This is known today as the Parthenon, on which the architects, Callicrates and Ictinus, perfected the Doric columns.

ιη. The Screw. The little metal device beloved by B&Q devotees across Britain was invented by Archimedes, a Greek mathematician, who lived during the 2nd century BC. The author of Archimedes’s Principle, a law of physics relating to buoyancy and specific gravity, discovered the screw while developing a device that would draw water from ships to prevent them sinking.

ιθ. Homer. The finest poet of ancient times was said to have been born in Izmir, in what is now Turkey. He has traditionally been credited with writing two of the greatest works of world literature, The Iliad and The Odyssey, which together tell the story of the Trojan war. Latterly it has been suggested that he did not even exist, and that these epics were the combined work of various oral storytellers.

κ. Tragedy. The Greeks were not the first society bedevilled by foul deeds and atrocities; they were, however, the first to shout about it. The theatrical Greek tragedy began with the dithyramb, a choral song in honour of Dionysus, but it was Aeschylus (525-456 BC) who developed the form from one actor to two. Sophocles (496-406 BC) added a third player and scripted the Theban plays that tells the story of Oedipus who murdered his father, married his mother, and went on to became the first soap opera star.

“So again we have learned something, instead of making a cheap joke about the Greek civilisation upon which everything around us depends, from electricity to clothes to democracy and logic and philosophy and everything we take for granted and is so dear to us.”

– Stephen Fry


In architecture, entasis is the slight convex (curved or bowed outward like the outside of a bowl or sphere or circle) curvature introduced into the shaft of a column for aesthetic reasons, or to compensate for the illusion of concavity.

'Ba'lbek, Collones du Grande Temple, vue du su...

Example of ancient columns

Its best-known use is in certain orders of Classical columns that curve slightly as their diameter is decreased from the bottom upwards.

In the Hellenistic period some columns with entasis are cylindrical in their lower parts.

Some Roman columns are ‘cigar-shaped’, with the widest point some distance above the foot, but this is unusual and to most eyes unattractive.

On a more contemporary note, Rolls-Royce cars all made use of the entasis principle in their radiator grilles to give an illusion of greater solidity.

Vitruvian Man

The Vitruvian Man is a world-renowned drawing created by Leonardo da Vinci around the year 1487. It is accompanied by notes based on the work of the famed architect, Vitruvius. The drawing, which is in pen and ink on paper, depicts a male figure in two superimposed positions with his arms and legs apart and simultaneously inscribed in a circle and square. The drawing and text are sometimes called the Canon of Proportions or, less often, Proportions of Man. It is stored in the Gallerie dell’Accademia in Venice, Italy, and, like most works on paper, is displayed only occasionally.

da Vinci’s Vitruvian Man

The drawing is based on the correlations of ideal human proportions with geometry described by the ancient Roman architect Vitruvius in Book III of his treatise De Architectura. Vitruvius described the human figure as being the principal source of proportion among the Classical orders of architecture. Leonardo’s drawing is traditionally named in honour of the architect.

Encyclopaedia Britannica online states, “Leonardo envisaged the great picture chart of the human body he had produced through his anatomical drawings and Vitruvian Man as a cosmografia del minor mondo – cosmography of the microcosm. He believed the workings of the human body to be an analogy for the workings of the universe.”

The multiple viewpoint that set in with Romanticism has convinced us that there is no such thing as a universal set of proportions for the human body. The field of anthropometry was created in order to describe individual variations. Vitruvius’ statements may be interpreted as statements about average proportions. Vitruvius takes pains to give a precise mathematical definition of what he means by saying that the navel is the center of the body, but other definitions lead to different results; for example, the center of mass of the human body depends on the position of the limbs, and in a standing posture is typically about 10 cm lower than the navel, near the top of the hip bones.

Leonardo’s drawing combines a careful reading of the ancient text with his own observation of actual human bodies. In drawing the circle and square he correctly observes that the square cannot have the same center as the circle, the navel, but is somewhat lower in the anatomy. This adjustment is the innovative part of Leonardo’s drawing and what distinguishes it from earlier illustrations. He also departs from Vitruvius by drawing the arms raised to a position in which the fingertips are level with the top of the head, rather than Vitruvius’s much lower angle, in which the arms form lines passing through the navel.

The drawing itself is often used as an implied symbol of the essential symmetry of the human body, and by extension, of the universe as a whole.

It may be noticed by examining the drawing that the combination of arm and leg positions actually creates sixteen different poses. The pose with the arms straight out and the feet together is seen to be inscribed in the superimposed square. On the other hand, the ‘spread-eagle’ pose is seen to be inscribed in the superimposed circle.

Leonardo most probably tried to illustrate Vitruvius’ De architectura 3.1.2-3 which reads:

‘For the human body is so designed by nature that the face, from the chin to the top of the forehead and the lowest roots of the hair, is a tenth part of the whole height; the open hand from the wrist to the tip of the middle finger is just the same; the head from the chin to the crown is an eighth, and with the neck and shoulder from the top of the breast to the lowest roots of the hair is a sixth; from the middle of the breast to the summit of the crown is a fourth. If we take the height of the face itself, the distance from the bottom of the chin to the under side of the nostrils is one third of it; the nose from the under side of the nostrils to a line between the eyebrows is the same; from there to the lowest roots of the hair is also a third, comprising the forehead. The length of the foot is one sixth of the height of the body; of the forearm, one fourth; and the breadth of the breast is also one fourth. The other members, too, have their own symmetrical proportions, and it was by employing them that the famous painters and sculptors of antiquity attained to great and endless renown. Similarly, in the members of a temple there ought to be the greatest harmony in the symmetrical relations of the different parts to the general magnitude of the whole. Then again, in the human body the central point is naturally the navel. For if a man be placed flat on his back, with his hands and feet extended, and a pair of compasses centred at his navel, the fingers and toes of his two hands and feet will touch the circumference of a circle described therefrom. And just as the human body yields a circular outline, so too a square figure may be found from it. For if we measure the distance from the soles of the feet to the top of the head, and then apply that measure to the outstretched arms, the breadth will be found to be the same as the height, as in the case of plane surfaces which are perfectly square.’

– Vitruvius. 1914. The Ten Books On Architecture (Translated by Morris Hicky Morgan) London, Great Britain: Humphrey Milford Oxford University Press (1914) p. 72-73