70 Fun Facts About Geometry

The interior angles of any triangle add up to how many degrees?

That is why a triangle can have at most one right angle or one obtuse angle: the other two must share what is left.

A triangle with all three sides the same length is called what?

You see one every day: the yield sign at a road junction is one of these.

A triangle with all sides of different lengths is called what?

A triangle with exactly two equal sides is isosceles, which is the shape of most gables and pediments.

What is the name for the longest side of a right-angled triangle?

The word comes from Greek for 'stretching under', because it stretches beneath the right angle.

How many degrees are in a right angle?

The word 'right' comes from the Latin rectus, meaning upright, describing a vertical line meeting a horizontal base.

A triangle in which one angle is larger than 90 degrees is called what?

If every angle is smaller than 90 degrees, the triangle is acute instead.

How many sides does a hexagon have?

The name comes from Greek hex, meaning six, and each interior angle of a regular one is 120 degrees.

What is the sum of the interior angles of a pentagon?

Each corner of a regular pentagon is 108 degrees, which is why five of them will not tile a floor without gaps.

Which four-sided shape has all four sides the same length but does not need right angles?

The playing-card diamond suit is named after this shape; other names include lozenge and calisson.

What is the American term for what the British call a trapezium?

The parallel sides are called the bases and the other two the legs.

A quadrilateral with two pairs of adjacent equal-length sides is named after which toy?

When it is not convex, mathematicians call the same shape a dart.

What is the sum of the exterior angles of any convex polygon?

Walk once around the shape turning at each corner and you end up facing the way you started, one full turn.

How many sides does a chiliagon have?

Descartes and Kant both used it as an example of a shape you can reason about but cannot picture.

Beyond decagons and dodecagons, how do mathematicians usually name polygons?

A regular 17-gon is famous because a 19-year-old Gauss proved in 1796 that it can be built with compass and straightedge.

Which regular polygon is the simplest one that cannot be constructed with compass and straightedge?

The seven-sided shape can be built with a marked ruler using a neusis construction, but not with the classical tools.

Gauss proved a regular polygon with how many sides can be built with compass and straightedge?

He did it in 1796 at age 19, and it was the first such discovery since the ancient Greeks.

How many books make up Euclid's Elements?

It covers plane and solid geometry, number theory and incommensurable magnitudes, and dominated teaching for two thousand years.

Which of Euclid's five postulates is known as the parallel postulate?

Centuries of failed attempts to prove it from the other four eventually led to hyperbolic and elliptic geometry.

Euclid, the 'father of geometry', spent his career in which ancient city?

Almost nothing is known of his life; most of it comes from Proclus and Pappus, writing many centuries later.

The Pythagorean theorem is Proposition 47 in which part of Euclid's Elements?

Proposition 48, right after it, proves the converse: if the squares add up, the angle is right.

Which civilisation used the Pythagorean rule over a thousand years before Pythagoras?

The clay tablet Plimpton 322, from about 1800 BC, lists what look like fifteen Pythagorean triples.

The book The Pythagorean Proposition collects how many different proofs of the theorem?

That may be more proofs than any other theorem has, though quadratic reciprocity is a rival for the title.

Which future US president published his own proof of the Pythagorean theorem in 1876?

He was still a member of the House of Representatives when it appeared in a weekly mathematics column.

According to legend, Hippasus of Metapontum was drowned at sea for revealing the existence of what?

The offending discovery was probably that the diagonal of a square cannot be measured exactly against its side.

How many Platonic solids are there?

Theaetetus, a contemporary of Plato, may have written the first proof that no others exist.

Which Platonic solid has twelve pentagonal faces?

Plato said the god used this one for arranging the constellations across the whole heaven.

In Plato's Timaeus, which classical element was matched with the cube?

Fire got the spiky tetrahedron, air the octahedron and water the icosahedron.

The twenty-sided die used in tabletop role-playing games is which solid?

Nature got there first: the adenovirus has the same shape.

How many edges does a cube have?

It has eight corners and six faces; edges are the number people forget.

Who first used the Greek letter π for the ratio of a circle's circumference to its diameter, in 1706?

Euler's later adoption of the symbol is what made it universal.

Around 250 BC, Archimedes bounded pi using polygons with how many sides?

He started with hexagons inside and outside a circle and kept doubling the number of sides.

William Shanks calculated pi to 707 digits, but his figures were wrong from which digit onward?

The wrong digits were even painted on the wall of the pi room in the Palais de la Découverte in Paris.

Pi Day is celebrated on which date in the United States?

The date 3/14/15 at 9:26:53 in 2015 matched the first ten digits, and Tau Day fans prefer 28 June.

Squaring the circle was proved impossible in 1882 once pi was shown to be what?

The Lindemann–Weierstrass theorem did it; the phrase 'squaring the circle' now means attempting the impossible.

Who proved in 1837 that an arbitrary angle cannot be trisected with compass and straightedge?

He published before Galois's work appeared, without using the field theory now taught alongside the result.

The ancient problem of doubling the cube is named after the citizens of which Greek island?

An oracle told them to double the size of Apollo's cubic altar to end a plague; Plato read it as a maths lesson.

Which Hungarian published non-Euclidean geometry in the early 1830s, independently of Lobachevsky?

Gauss told the young man's father he had worked it all out years earlier but never published.

In which geometry are there infinitely many lines through a point that never meet a given line?

In elliptic geometry the opposite happens: every line through the point eventually crosses the given one.

Who proved the Poincaré conjecture in 2002–03, then turned down a $1 million prize?

He said Hamilton, whose Ricci flow programme he completed, deserved equal credit.

The four colour theorem, proved in 1976, was the first major theorem proved with the help of what?

Many mathematicians balked because no human could check the case analysis by hand.

Francis Guthrie first raised the four colour problem in 1852 while trying to colour a map of what?

His brother took the puzzle to their teacher Augustus De Morgan, who admitted he did not know whether it was true.

Which three regular polygons are the only ones that can tile a flat surface by themselves?

Periodic tilings of any shape fall into just 17 wallpaper groups.

Which Dutch artist, inspired by the Alhambra's tiles, became famous for tessellations of interlocking animals?

He visited Spain in 1936 and later drew tessellations in hyperbolic geometry as well as ordinary flat ones.

Per the honeycomb theorem, which grid divides a plane into equal areas with the least total perimeter?

The claim was first written down around 36 BC by the Roman scholar Varro, long before bees got the credit.

Who coined the word 'fractal' in 1975?

He built it from the Latin fractus, meaning broken or fractured.

The Koch snowflake curve has a finite area but what kind of perimeter?

Swedish mathematician Helge von Koch described the curve in 1904 as a continuous curve with no tangents anywhere.

Colouring the odd numbers black in Pascal's famous number pyramid approximates which fractal?

Sierpiński published it in 1915, but the same pattern appears in decorative art centuries older.

The golden ratio, roughly 1.618, is the ratio of a regular pentagon's diagonal to what?

That link is why the number turns up in the construction of the dodecahedron and icosahedron.

Luca Pacioli's 1509 book about the golden ratio, illustrated by Leonardo da Vinci, was called what?

Leonardo called the ratio the sectio aurea, the golden section, and the name stuck.

The Möbius strip was discovered as a mathematical object in 1858 by August Möbius and which other German?

Roman mosaics from the third century show the same twisted band, and it is hidden in the recycling symbol.

Which surface, first described in 1882, has no distinct inside or outside and self-intersects in 3D space?

Unlike the Möbius strip, it has no boundary edge at all, like a sphere or a doughnut.

Topologically, the surface of a coffee cup is the same as which shape?

Both have genus one, meaning a single hole, so one can be stretched into the other without cutting.

In Thales's theorem, joining both ends of a diameter to any third point on the circle makes what kind of angle?

Euclid recorded it as Proposition 31 of Book III of the Elements.

Archimedes asked for which two shapes to be carved on his tomb, marking his proudest discovery?

He had shown that a sphere has two-thirds the volume of the cylinder that just encloses it; Cicero later found the neglected tomb.

The Reuleaux triangle, a shape of constant width, is used in drill bits that bore holes of what shape?

It also shapes guitar picks, fire hydrant nuts and some pencils, and its cousins appear on coins.

Which shape has the smallest surface area of any surface enclosing a given volume?

That is why soap bubbles are round: surface tension pulls them into the shape with the least skin.

James Thomson, who first printed the word 'radian' in 1873, was the brother of which physicist?

A full circle is 2π radians, so one radian is a shade under 57.3 degrees.

The first geodesic dome was built after World War I in Jena, Germany, to house what?

Buckminster Fuller later popularised the domes in the US and got a patent in 1954; Epcot's Spaceship Earth is one.

According to one theory, why does a full circle have 360 degrees?

Babylonian and Persian calendars used 360-day years, which fits their base-60 counting system.

The orbit of each planet is approximately an ellipse with the Sun located where?

Strictly it is the barycentre of the Sun-planet pair, but the Sun is so massive that it hardly matters.

Who created analytic geometry, describing curves with coordinates and equations, in the 17th century?

Once shapes became equations, algebra and geometry stopped being separate subjects.

The Mandelbrot set was first defined and drawn in 1978 by Robert Brooks and which collaborator?

Mandelbrot produced the famous high-quality pictures two years later at IBM's research centre.

The Pentagon building near Washington has how many ring corridors on each floor?

Ground was broken on 11 September 1941, exactly 60 years before the 2001 attack on the building.

Thales of Miletus, in the 7th century BC, used geometry to calculate the height of what?

He also worked out the distance of ships from the shore, geometry as a practical tool.

The Egyptian Moscow Papyrus of about 1890 BC gives a formula for the volume of which solid?

Together with the Rhind Papyrus and Babylonian tablets like Plimpton 322, it is among the oldest known geometry texts.

Plimpton 322, a geometry text from about 1900 BC, is what kind of artefact?

Later Babylonian tablets show astronomers using trapezoid procedures to track Jupiter, anticipating the Oxford Calculators by 14 centuries.

Which ancient people south of Egypt built a geometry system that included early sun clocks?

Thales of Miletus, a century or two later, is credited with the first deductive reasoning in geometry.

Which Persian poet-mathematician found geometric solutions to cubic equations?

His work on quadrilaterals, with Alhazen and al-Tusi, fed into the later study of non-Euclidean geometry.

Gauss's Theorema Egregium says which property of a surface is independent of how it sits in space?

The 'remarkable theorem' means surfaces can be studied intrinsically, the seed of Riemannian geometry.

Which mathematician's Erlangen programme made symmetry the central idea uniting geometries?

The programme generalised both Euclidean and non-Euclidean geometry, and groups have been seen as geometric objects ever since.