80 Fun Facts About Pythagoras

On which Greek island was Pythagoras born?

Herodotus and Isocrates agree he was the son of Mnesarchus, said to be a gem-engraver or wealthy merchant.

What does the Pythagorean theorem state about a right-angled triangle?

Babylonians and Indians knew and used the relationship centuries before Pythagoras was born.

In which southern Italian city did Pythagoras found his school around 530 BC?

Today's Crotone in Calabria was then part of Magna Graecia, the Greek colonies of southern Italy.

Pythagoras was reputedly the first man to describe himself with which Greek word?

It means 'lover of wisdom'; Augustine later praised his humility in not calling himself a sage.

Which teaching is most securely identified with Pythagoras himself?

Metempsychosis holds that every soul is immortal and enters a new body at death.

What is the doctrine of musica universalis that Pythagoras may have devised?

Kepler's search for the mathematics behind this 'harmony of the spheres' led to his laws of planetary motion.

How old was Pythagoras when he left Samos, according to Aristoxenus?

He left during the reign of the tyrant Polycrates, giving a birth date around 570 BC.

Which Egyptian pharaoh supposedly taught Pythagoras the Egyptian language, according to a later legend?

He was then said to be the only foreigner allowed to join the priests' worship at Thebes.

Which Greek sage is most often named as Pythagoras's teacher?

One story says Pythagoras returned to Delos to nurse the dying Pherecydes.

What was the name of the school Pythagoras supposedly founded on Samos for public debate?

He himself was said to study in a secret cave.

Which famous wrestler was said to have saved Pythagoras's life when a roof was about to collapse?

The story may confuse the philosopher with an athletics trainer of the same name.

The city's victory over which neighbouring colony in 510 BC was credited to Pythagorean discipline?

After the victory, supporters of democracy turned on the Pythagoreans and burned their meeting house.

Who led the democratic faction that roused the city against Pythagoras's brotherhood?

Cylon was said to be irritated at being excluded from the brotherhood.

In one legend, Pythagoras was caught and killed because he refused to run through what?

Doing so would have violated his teachings; the story was probably originally told about later Pythagoreans.

In Dicaearchus's account, where did Pythagoras and his followers die of starvation after forty days?

They had first pleaded for sanctuary at Locris and been refused.

In Xenophanes' satirical poem, Pythagoras stops a man beating a dog because he recognises what?

It is the earliest source for his belief in reincarnation.

Which minor hero of the Trojan War did Pythagoras claim to have been in a former life?

He said he could remember four previous lives, including one as a fisherman from Delos.

What is the tetractys, which Pythagoras's students swore oaths by?

The Pythagoreans regarded ten as the 'perfect' number.

Which later Pythagorean do scholars usually credit with the number philosophy pinned on Pythagoras?

Walter Burkert argued Pythagoras himself never dealt with numbers at all.

What was the Pythagorean maxim 'koina ta philon'?

Members shared all their possessions and ate communal meals in the Spartan manner.

What were the two groups within early Pythagoreanism called?

'Learners' and 'listeners', traditionally seen as the rationalist and mystical wings.

How long did new initiates supposedly have to stay silent before being allowed to meet Pythagoras?

Those who broke the community's laws were expelled and had tombstones erected for them as if dead.

Which of these was one of the Pythagoreans' surviving 'oral sayings'?

Others forbade breaking bread, poking fires with swords or picking up crumbs.

How many of the 235 famous Pythagoreans listed by Iamblichus were women?

Pythagoras was said to be unusually progressive towards women, who played an active role in the school.

Which food is Pythagoras most famously said to have forbidden his followers to eat?

One theory links the ban to favism, an enzyme-deficiency anaemia common around the Mediterranean.

What did Aristotle say Pythagoras had that he exhibited at the Olympic Games?

He showed it to Abaris the Hyperborean as proof he was the 'Hyperborean Apollo'.

What was Pythagoras supposedly given by the priest of Apollo, which let him fly over long distances?

He was also reportedly seen in two cities at the same time.

According to a fragment of Aristotle, what did Pythagoras do when a deadly snake bit him?

He also reportedly persuaded a bear to swear never to harm a living thing again.

What did the river Kosas supposedly do when Pythagoras crossed it?

'Several witnesses' reported it, according to the legend.

What was Pythagoras said to have sacrificed to the gods after discovering his theorem?

Cicero rejected the story since Pythagoras supposedly forbade blood sacrifice; Porphyry said the ox was made of dough.

What sound supposedly led Pythagoras to realise that music was mathematical?

He concluded the pitch was proportional to the size of the hammer.

Which philosopher has a stronger claim than Pythagoras to first teaching that the Earth is spherical?

Both were also credited with identifying the morning and evening stars as the same object, Venus.

Which philosopher mocked Pythagoras as a clever charlatan practising 'artful knavery'?

He was born a few miles across the sea from Pythagoras's island and may have lived in his lifetime.

Which Pythagorean philosopher and friend of Plato was elected general of Tarentum seven times?

Others were forbidden from serving more than a year; he was also a noted mathematician and musician.

Which of Plato's dialogues most clearly exhibits Pythagorean ideas?

Cicero repeated the saying that 'Plato learned all things Pythagorean'.

The oldest known Pythagorean building, a Rome basilica, dates from which emperor's reign?

The Porta Maggiore Basilica has an interior almost entirely white, the colour Pythagoreans held sacred.

Which astronomer called Pythagoras the 'grandfather' of all Copernicans?

He titled his book on planetary harmony Harmonices Mundi.

Which Roman poet's Metamorphoses gives Pythagoras a speech urging his followers to eat no meat?

Until the word 'vegetarianism' was coined in the 1840s, English vegetarians were simply called Pythagoreans.

Which fraternal society deliberately modelled itself on Pythagoras's community in southern Italy?

Rosicrucians and John Dee also drew on Pythagorean symbolism.

Which Babylonian clay tablet from about 1800 BC lists what can be read as 15 Pythagorean triples?

The rule was in widespread use in the Old Babylonian period, over a thousand years before Pythagoras.

Per Plutarch, the Egyptians identified the sides of the 3:4:5 triangle with which three gods?

The Berlin Papyrus of about 1800 BC includes the triple 6:8:10 without mentioning a triangle.

Which future US president published a proof of the Pythagorean theorem while serving as a congressman?

His proof uses a trapezoid instead of a square.

How many proofs of the theorem does the book The Pythagorean Proposition contain?

The theorem may have more known proofs than any other, rivalled only by quadratic reciprocity.

Where in Euclid's Elements is the theorem proved?

Euclid's construction of squares on each side depends on the parallel postulate.

What happens to a Pythagorean cup when it is filled beyond a certain point?

Also called a greedy cup, its hidden siphon was described by Hero of Alexandria; the link to Pythagoras is a recent folk tradition.

What is the meaning of the maxim 'Ex pede Herculem', linked to a story about Pythagoras?

He supposedly estimated the hero's height from the length of a racecourse at Pisae.

What trade is Pythagoras's father Mnesarchus said to have followed?

Other sources call him a wealthy merchant; his mother Pythaïs was claimed to descend from Ancaeus, Samos's mythical founder.

Which Roman building, with its oculus symbolising the monad, was designed on Pythagorean numerology?

Its twenty-eight ribs stand for the moon, matching the months of the Pythagorean lunar calendar.

Which Samian astronomer did Copernicus pointedly omit when crediting Pythagoreans for his heliocentric model?

That non-Pythagorean had worked out a fully heliocentric model in the fourth century BC, which spoiled the story.

By what name is the right-triangle relation known in China?

The Zhoubi Suanjing, with contents known long before its surviving 1st-century BC texts, reasons it out for the 3-4-5 triangle.

In The Wizard of Oz, which character garbles the Pythagorean theorem right after receiving a diploma?

The Scarecrow reels off a muddled version as soon as he gets his 'brain', an honorary diploma. The 1939 film's joke is that the Wizard only gave the friends confidence in qualities they already had.

Which physicist taught himself geometry and found his own proof of the Pythagorean theorem before turning 13?

Albert Einstein began teaching himself algebra, calculus and geometry at twelve. One proof sometimes credited to him simply drops a line from the right angle to the longest side.

A baseball diamond is a 90-foot square; what is the rough distance from home plate to second base?

127 feet is the diagonal of that square: 90² + 90² = 16,200, and its root is 127.3. The rulebook's exact figure is 127 feet 3⅜ inches.

TV screens are sold by diagonal: what size is a screen 16 inches wide and 12 inches tall?

20 inches: 16² + 12² = 400, and 20 squared is 400. Corner-to-corner sizing was inherited from the first televisions, which used circular picture tubes.

A 13-foot ladder leans on a wall with its base 5 feet out; what height, in feet, does its top reach?

12 feet, because 13² − 5² = 169 − 25 = 144, which is 12 squared. The 5-12-13 triangle is the next whole-number right triangle after 3-4-5 that is not just a scaled-up copy.

Which New York borough names the distance a taxi drives along a right triangle's legs, not its hypotenuse?

Manhattan distance, also called taxicab distance, simply adds the two legs, because a cab on a street grid cannot cut the corner the way Pythagoras's straight line does.

In 2023 two New Orleans students made headlines by proving the Pythagorean theorem with which branch of maths?

Trigonometry was the tool Calcea Johnson and Ne'Kiya Jackson each used, working independently. The pair were featured on 60 Minutes and their proofs were later published in The American Mathematical Monthly.

On a coordinate grid, what is the straight-line distance from the origin (0, 0) to the point (6, 8)?

10 units: the point sits 6 across and 8 up, and 36 + 64 = 100, whose root is 10. The distance formula taught in school is simply the Pythagorean theorem written in coordinates.

Which 17th-century lawyer claimed in a margin that Pythagoras's equation has no whole-number solutions using cubes?

Pierre de Fermat added that the margin was too narrow to hold his proof. The claim, which covers every power above two, went unproven for more than 350 years until Andrew Wiles cracked it in 1994.

Sailors use the Pythagorean theorem with Earth's radius and their eye height to find the distance to what?

The horizon is where a line of sight just grazes the Earth, meeting the radius at a right angle. For someone standing on a beach it is only about 5 km away.

In one legend, what befell the Pythagorean Hippasus for revealing that some lengths cannot be written as fractions?

Drowned at sea is the best-known version; such lengths clashed with the school's view of numbers as whole numbers only. Other tellings blame a leaked secret about the dodecahedron.

A right triangle has a hypotenuse of 17 and one leg of 8; what is the length of the other leg?

15, since 17² − 8² = 289 − 64 = 225, which is 15 squared. To find a leg you subtract the squares instead of adding them; 8-15-17 is one of just 16 such triangles under 100 that are not scaled copies of smaller ones.

A rectangle is 7 cm wide and 24 cm long; what is the length of its diagonal in centimetres?

25 cm: 49 + 576 = 625, which is 25 squared. A rectangle's diagonal is the hypotenuse of the right triangle made by two adjacent sides, so here it is barely longer than the 24 cm side.

The squares on a right triangle's legs cover 20 cm² and 30 cm²; what area is the square on the hypotenuse?

50 cm², found by simple addition. The theorem is a statement about areas, so the two smaller squares together exactly fill the largest one, with no squaring or rooting needed.

Halving an equilateral triangle makes a right triangle whose hypotenuse is what multiple of its shortest side?

2 times: the cut halves one side while the hypotenuse stays a full side. The result is the 30-60-90 triangle of drafting set squares, with sides in the ratio 1 : √3 : 2.

If a triangle's longest side c makes c² bigger than a² + b², which kind of angle lies opposite that side?

Obtuse: the two smaller squares fall short of the largest because the angle has opened wider than a right angle. When they overshoot instead, every angle in the triangle is acute.

Only one of these sets of three side lengths obeys the Pythagorean theorem; which set makes a right triangle?

9, 40, 41 works because 81 + 1,600 = 1,681, which is 41 squared. A rule credited to Pythagoras himself produces triples like this one, whose two largest numbers differ by one.

Which whole number completes the Pythagorean triple whose two smaller sides, 20 and 21, differ by just one?

29 completes it, since 400 + 441 = 841, the square of 29. Right triangles with legs one apart are rare: after 3-4-5 and this one, the next is 119-120-169.

A box measures 2 by 3 by 6; using the Pythagorean theorem twice, what length is the diagonal through its middle?

7 exactly, because 2² + 3² + 6² = 4 + 9 + 36 = 49. Boxes whose sides and inner diagonal are all whole numbers are known as Pythagorean boxes, the three-dimensional cousins of triples such as 3-4-5.

Which term describes geometries, such as a globe's surface, where the Pythagorean theorem no longer holds?

Non-Euclidean geometry breaks the rule: a triangle running from the North Pole to two points on the equator can have three right angles and three equal sides.

Hypotenuse comes from a Greek word describing a right triangle's longest side as doing what to the right angle?

Stretching under is the literal sense of the Greek hypoteinousa: the side was pictured as stretched beneath the right angle. The word reached English in the 1570s and was long misspelt 'hypothenuse'.

The Babylonian clay tablet YBC 7289 gives a six-digit estimate of which number, the diagonal of a one-by-one tile?

The square root of 2 is that diagonal, and the tablet, thought to be a student's work from between 1800 and 1600 BC, pins it down to better than one part in two million.

The Baudhayana Shulba Sutra, a guide to building fire altars, states the Pythagorean theorem; where was it composed?

India produced it, perhaps as early as the 8th century BC. Its rule that a rectangle's diagonal yields both areas made by the two sides is the oldest surviving statement of the theorem in words.

Which rule for any triangle shrinks to the Pythagorean theorem when the angle between two sides is a right angle?

The law of cosines adds a correction term for the angle between the sides; at a right angle that term is zero, leaving a² + b² = c². It is sometimes called the generalized Pythagorean theorem.

In the Pythagorean identity, the squares of an angle's sine and cosine always add up to which number?

1 is always the total, because sine and cosine are the two legs of a right triangle whose hypotenuse is 1. The identity is the Pythagorean theorem in disguise.

Which French thinker's coordinate grid turns the Pythagorean theorem into a formula for the distance between points?

René Descartes published the idea in 1637, which is why x-and-y axes are called Cartesian. Measure the horizontal and vertical gaps between two points and the theorem gives the straight-line distance.

A Pythagorean triple such as 12, 35, 37, whose three numbers share no common factor, is given which label?

Primitive triples are the building blocks: every other triple, such as 6-8-10, is one of them multiplied up, and a formula recorded by Euclid generates them all.

When the three whole numbers of a Pythagorean triple share no common factor, the largest of them is always what?

Odd, every time: one of the two smaller numbers is even and the other is not, which forces the largest to be odd. The even one, meanwhile, is always a multiple of 4.

The Pythagorean theorem still works with semicircles or pentagons instead of squares, provided the shapes are what?

Similar shapes grow in area with the square of their side, so the two smaller areas still add up to the largest. Euclid put this wider version in the Elements, and Hippocrates of Chios knew it in the 5th century BC.

The 'Pythagoras tree', which sprouts ever-smaller squares from right triangles, is which kind of mathematical figure?

A fractal, it was invented in 1942 by the Dutch maths teacher Albert Bosman. Each trio of touching squares frames a right triangle, just like the classic diagram of the theorem.