118 Fun Facts About Geometry

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

180 degrees, but only on a flat surface: draw the triangle on a globe and its three angles always add up to more.

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

Equilateral: equal sides force equal angles, so every corner is exactly 60 degrees, which is why six of them fit around a single point.

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

Scalene: with no two sides equal, no two angles are equal either, so every scalene triangle is lopsided in its own way.

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

The hypotenuse faces the right angle and is always the longest side; its name is Greek for 'stretching under', because it stretches under that angle.

How many degrees are in a right angle?

90 degrees is a quarter of a full turn, so two lines that cross at a right angle make four equal corners.

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

Obtuse comes from the Latin obtusus, 'blunted': the wide corner looks dull beside a pointed one, and the same word is used for a slow-witted remark.

Which polygon has exactly six sides?

Hexagon comes from the Greek hex, 'six', and gonia, 'angle', so the name actually counts the shape's corners rather than its sides.

The interior angles of which shape always add up to 540 degrees?

Pentagon: two lines drawn from one corner split it into three triangles, which is where the total comes from; in a regular pentagon every corner measures 108 degrees.

What is the name for a four-sided shape whose four sides are all the same length?

Rhombus: lean a square over and its sides stay equal while the angles change, giving the diamond of playing cards; the two diagonals still cross at right angles.

What do Americans call a four-sided shape with exactly one pair of parallel sides?

In American usage a trapezoid is a quadrilateral with one pair of parallel sides, which the British call a trapezium; the parallel sides are its bases.

Which four-sided shape has two pairs of equal-length sides that are adjacent rather than opposite?

A kite: the shape is named after the flying toy, and the toy in turn took its name from the kite, a bird of prey that hangs in the wind.

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

The exterior angles of any convex polygon sum to 360°: walk once around it, turning at each corner, and you make exactly one full turn.

How many sides does a chiliagon have?

A chiliagon has 1,000 sides, from the Greek chilioi; Descartes used it in his Meditations as a shape one can reason about but not picture.

Canada's loonie coin has 11 sides, making it which shape?

Hendecagon, from the Greek hendeka, 'eleven': the coin replaced Canada's one-dollar bill in 1987 and got its nickname from the loon pictured on it.

Which regular polygon has the fewest sides of any that cannot be drawn with compass and straightedge?

The regular heptagon is the first that cannot be drawn with compass and straightedge alone; it can be built with a marked ruler by neusis.

Which teenager proved compass and straightedge can draw a regular 17-sided polygon?

Carl Friedrich Gauss made the discovery in 1796, the first advance of its kind since the ancient Greeks, by turning the drawing problem into one about solving equations.

Which president taught himself six books of Euclid?

Abraham Lincoln wrote in an 1860 campaign autobiography that he 'studied and nearly mastered' them after serving in Congress, regretting his want of education and doing what he could to supply it; the Elements has 13 books in all.

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

The fifth postulate is the parallel postulate; two thousand years of failed attempts to prove it from the other four ended with the discovery of non-Euclidean geometry.

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

Euclid worked in Alexandria around 300 BC under Ptolemy I; almost nothing else is known of his life, most of it from Proclus and Pappus centuries later.

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

Book I ends on it: the Pythagorean theorem is its second-to-last proposition, and the last one turns it around to prove the converse.

Which civilisation's clay tablets show the Pythagorean rule in use over a thousand years before Pythagoras?

The Babylonians: their scribes listed whole-number right triangles on the tablet Plimpton 322 and worked out a square's diagonal to within one part in a million on another, both over 3,500 years ago.

Which future physicist proved the Pythagorean theorem for himself at about age 12?

Albert Einstein recalled that an uncle told him the theorem before he had seen a geometry book, and that 'after much effort' he proved it with similar triangles; over 300 other proofs are known.

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

James A. Garfield, then an Ohio congressman, found his proof by working out one figure's area in two different ways and published it in the New-England Journal of Education five years before he became president.

In one legend, Hippasus of Metapontum drowned at sea for revealing the existence of what?

Irrational numbers: the tale goes that he let slip that the diagonal of a square shares no common measure with its edge; other ancient versions name a different secret.

How many Platonic solids are there?

Five: at least three faces must meet at every corner and their angles must total less than a full turn, a squeeze only five shapes survive; Euclid ends the Elements by proving it.

Which Platonic solid has twelve pentagonal faces?

The dodecahedron has twelve pentagonal faces; in Plato's Timaeus the other regular solids each stand for an element, while this one is used for the universe as a whole.

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

Plato matched the cube with earth in the Timaeus because of its stability; fire got the tetrahedron, air the octahedron and water the icosahedron.

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

Regular icosahedron: 'icosa' is Greek for twenty, and because all twenty triangular faces are identical, every number has the same chance of landing face up.

A cube has twelve of which feature?

Edges: each of a cube's six square faces has four of them, and every edge is shared by two faces, so the count is 6 × 4 ÷ 2 = 12.

Whose widely read books made π the standard symbol for the circle ratio, decades after William Jones first used it?

Leonhard Euler adopted the symbol in the 1730s, and his books and constant letter-writing spread it across Europe; William Jones, a Welsh maths teacher, had introduced it in 1706.

Which ancient Greek trapped pi between 223/71 and 22/7 by squeezing a circle between 96-sided polygons?

Archimedes started with six-sided polygons inside and outside the circle and doubled their sides four times; the method stayed the standard way to pin down pi for over a thousand years.

In the Comte de Buffon's 18th-century puzzle, pi can be estimated by dropping what onto a floor of parallel lines?

Needles: the chance that a dropped needle lands across a line works out to a formula containing pi, so tallying enough drops gives an estimate of it.

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

14 March is written 3/14 in US date order, matching pi's opening digits; the holiday began in 1988 at San Francisco's Exploratorium, where staff paraded in a circle and then ate fruit pies.

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

Transcendental: Ferdinand von Lindemann showed that pi solves no polynomial equation with rational coefficients, while compass and straightedge can only build lengths that do, so the task is impossible.

Which classic task did Pierre Wantzel prove impossible with compass and straightedge in 1837?

Trisecting an angle: Wantzel showed that splitting a general angle into three means solving a cubic equation, which compass and straightedge cannot do; the same paper ruled out doubling the cube.

In legend, an oracle told the people of which Greek island to double the size of Apollo's cube-shaped altar?

Delos: doubling a cube's volume is still called the Delian problem; in one telling the new altar was meant to end a plague, and Plato read the oracle as a push to study geometry.

General relativity describes gravity as curved spacetime using the geometry of which German mathematician?

Bernhard Riemann set out his geometry of curved spaces in an 1854 lecture; six decades later it gave general relativity the mathematics it needed.

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

Hyperbolic geometry drops Euclid's rule that only one such line exists, the very assumption mathematicians had spent two thousand years trying to prove, and still holds together.

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

Grigori Perelman posted his proof on arXiv in 2002-03 and refused the 2010 Millennium Prize, saying Richard 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?

Appel and Haken's 1976 proof checked nearly 2,000 map configurations by computer, the first major theorem to rely on one; many mathematicians balked because no human could verify the case analysis.

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

Francis Guthrie noticed that the counties of England could be coloured with four colours; his brother Frederick took the puzzle to Augustus De Morgan, who could not settle it.

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

Triangle, square, hexagon: only these three tile the plane alone, because their corner angles (60°, 90°, 120°) divide 360° evenly.

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

M. C. Escher spent days copying the Alhambra's Moorish tilings on a 1936 visit, and those sketches became the raw material for the interlocking-animal prints that followed.

Which regular grid divides a plane into equal areas with the least total perimeter?

Hexagonal: ancient writers already suspected that the honeycomb's cells were the most economical shape, but the honeycomb conjecture went unproved until Thomas Hales settled it in 1999.

Who coined the word 'fractal' in 1975?

Benoît Mandelbrot built the word from the Latin fractus, 'broken', for shapes such as coastlines that are too irregular and fragmented for ordinary geometry to describe.

Which fractal, made by adding ever-smaller triangles to each side of a triangle, has an infinite perimeter around a finite area?

The Koch snowflake's edge grows by a third at every step, so its length has no limit, yet the area it encloses stays finite, as each round of new triangles adds less than the last.

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

The Sierpiński triangle emerges because the even numbers cluster in upside-down triangles of every size, the same holes the fractal has; the more rows you colour, the closer the match.

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

Its side: the diagonals also cut one another in the same ratio, so the five-pointed star they draw inside the pentagon is packed with golden ratios.

Which artist drew the geometric solids that illustrate Luca Pacioli's 1509 book Divina Proportione?

Leonardo da Vinci drew them in Milan, where both men worked at the Sforza court; the two fled the city together when French troops took it in 1499.

Which one-sided surface can be made by giving a paper band a half-twist and joining its ends?

The Möbius strip was described independently in 1858 by August Möbius and Johann Benedict Listing; drive belts were later given the same half-twist so that the whole surface wears evenly.

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

The Klein bottle has only one side, so an ant could walk from its 'inside' to its 'outside' without crossing an edge; it would need four dimensions to exist without passing through itself.

Topologically, a coffee mug with a handle is the same as which shape?

A coffee mug with a handle is topologically a doughnut: both have genus one, 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?

Right: the angle comes out the same wherever the third point sits on the circle; Thales is said to have sacrificed an ox in thanks for the discovery.

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

A sphere and a cylinder: he had proved that the ball fills exactly two-thirds of the tightest can around it, and Cicero later recognised the overgrown tomb by the carving.

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

A Reuleaux triangle turning inside a square stays in contact with all four sides, so a drill bit of that shape sweeps out an almost square hole, missing only the very corners.

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

The sphere encloses a given volume with the least surface area, which is why surface tension pulls a soap bubble into a sphere.

Which angle unit, first printed in 1873 by Lord Kelvin's brother, fits about 6.28 times into a full circle?

The radian is the angle whose arc is as long as the circle's radius, so 2π of them, about 6.28, make a full turn; James Thomson first printed the word in Belfast exam papers.

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

A planetarium projector: Zeiss engineer Walther Bauersfeld built the dome on the factory roof in the early 1920s to test it, decades before Buckminster Fuller made geodesic domes famous.

One theory links the circle's 360 degrees to early astronomers' rough count of what?

Days in a year: the Sun shifts about one degree against the stars each day, and some ancient calendars rounded the year to 360 days, a number that also suited base-60 arithmetic.

The Sun sits off-centre in each planet's elliptical orbit, at which point of the ellipse?

Kepler's first law puts the Sun at one focus of each planet's elliptical orbit; strictly the focus is the Sun-planet barycentre, which lies inside the Sun for most planets.

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

René Descartes (La Géométrie, 1637) and Pierre de Fermat independently created analytic geometry, turning curves into equations so algebra and geometry stopped being separate.

Which country's coast is named in the title of a famous 1967 paper on why coastlines have no fixed length?

Britain: the paper showed that a shorter ruler follows more bays and inlets, so the measured coast keeps growing, an early case for giving rough shapes a dimension between 1 and 2.

What name is given to a polygon whose vertices all lie on a single circle?

Cyclic: any triangle qualifies, since a circle can always be drawn through its three corners, but a four-sided shape makes the cut only when each pair of opposite angles adds up to a straight line.

Thales of Miletus is said to have used geometry in the 6th century BC to work out the height of what?

The pyramids: ancient accounts say Thales worked out their height from the length of their shadows, by comparing them with the shadow of something whose height he knew.

Which ancient people's scribes found a circle's area by squaring eight-ninths of its diameter, putting pi near 3.16?

The Egyptians: the scribe Ahmes copied the rule into the Rhind Papyrus over 3,500 years ago, and it gives pi as 256/81, closer than the plain 3 implied by the Bible's account of Solomon's temple.

Apollonius of Perga's Conics named three curves cut from a cone: the ellipse, the hyperbola and which other?

The parabola: it reflects incoming parallel rays to a single point, which is why satellite dishes are built in its shape and car headlamps run the same trick backwards.

Which Persian mathematician found geometric solutions to cubic equations using intersecting conics?

Omar Khayyam solved cubic equations geometrically by intersecting conic sections in his 11th-century Treatise on Algebra; his work on quadrilaterals fed into later studies of the parallel postulate.

Which Greek scholar famously measured the Earth's circumference using the angle of a midsummer noon shadow in Egypt?

Eratosthenes compared midsummer noon shadows at two Egyptian towns a known distance apart and scaled the small difference in the Sun's angle up to a full circle.

Which country's flag, two stacked pennants, is the world's only national flag that is not four-sided?

Nepal joined the pennons of two rival branches of its ruling dynasty into a single flag. It is also the only national flag that is taller than it is wide.

Which everyday name for a rhombus is also given to a baseball infield, really a 90-foot square?

Diamond is the name the playing-card suit takes from the same shape. A baseball diamond's four bases mark the corners of a square with 90-foot sides.

Which Chinese puzzle cuts a square into seven flat pieces, one of them a parallelogram?

Tangram pieces are called tans, and the parallelogram is the only one that must be flipped over to make its mirror image. Trading ships carried the puzzle from China to Europe and America.

The word trapezoid traces back to the Greek trapeza, the name of which piece of furniture?

Table in Greek was trapeza, built from words for 'four' and 'foot', and the shape was a trapezion, a 'little table'. Classical Greek tables are said to have had one side shorter than the one opposite.

Which other word for a quadrilateral also names the enclosed courtyard of an Oxford college?

Quadrangle literally means 'four-angle', and the courtyard is often shortened to 'quad'. At Cambridge the same enclosed spaces are always called courts instead.

A parallelogram whose two diagonals are exactly the same length as each other must be which shape?

A rectangle is the only kind of parallelogram with equal diagonals. Its name comes from the Latin rectangulus, built from rectus, 'right', and angulus, 'angle'.

Which word describes a four-right-angled shape, such as a banknote, that is longer than it is wide?

Oblong comes from the Latin oblongus, 'more long than broad', so a perfect square can never be one. Euclid's Elements listed the two as separate kinds of quadrilateral.

Copies of any uncrossed quadrilateral, however lopsided, can always be arranged to do what?

Every such quadrilateral can tile an endless floor with no gaps, however irregular its sides and angles. The trick is to keep rotating copies about the midpoints of their edges.

A quadrilateral whose sides cross each other is nicknamed after a butterfly or which item of neckwear?

A bow tie quadrilateral is also called crossed or self-intersecting. Traced around its outline, its four 'interior' angles add up to 720° rather than the usual 360°.

The two sides of a trapezoid that are not its bases share their name with which parts of the body?

Legs is also the name for the two shorter sides of a right-angled triangle. On a trapezoid they are sometimes called the lateral sides instead.

Roger Penrose's non-repeating tiling uses two quadrilaterals: a kite and which partner, named after a missile?

A dart is the arrowhead-shaped partner of the kite, and the two tiles can be joined to make a rhombus. Laid by Penrose's rules, they cover a surface without the pattern ever repeating.

Which word describes a quadrilateral with one angle over 180°, so that a diagonal lies outside it?

Concave quadrilaterals look dented, like an arrowhead, yet their four interior angles still add up to 360° just as in any other simple quadrilateral.

In every parallelogram, from a leaning one to a perfect square, what do the two diagonals always do?

The diagonals bisect each other in every parallelogram, so their crossing point is the midpoint of both. Only special cases such as the rhombus add a right-angled crossing.

A trapezoid's midsegment runs midway between its two bases; its length is what of the two base lengths?

Their average is the answer, so the midsegment sits between the two bases in length as well as in position. It is also known as the median of the trapezoid.

Which quadrilateral has equal diagonals and one line of symmetry that cuts through two opposite sides?

Isosceles trapezoid and kite are the only two uncrossed quadrilaterals with exactly one line of symmetry. The kite's mirror line runs through two corners instead of two sides.

Joining the midpoints of a convex quadrilateral's sides makes a parallelogram with what share of its area?

One half of the original area always ends up inside. The new shape is called the Varignon parallelogram after Pierre Varignon, whose proof was only published after his death, in 1731.

Which rarely used Greek-derived word is a counterpart to the Latin 'quadrilateral' for any four-sided flat shape?

Tetragon joins the Greek tetra, 'four', with gon, 'corner' or 'angle'. Quadrilateral, the usual name, is built from Latin instead and counts sides rather than corners.

A square is the only quadrilateral with all its sides and all its angles equal, earning it which label?

Regular polygons have every side and every angle equal, and the square is the only four-sided one. That makes it the quadrilateral with the least perimeter for a given area.

A leaning parallelogram's area is not base times slanted side but base times which other measurement?

Its height, measured at right angles to the base, is what counts. Slice a corner off one end and slide it to the other, and the shape squares up with the same base and area.

A square has more lines of symmetry than any other quadrilateral; counting its diagonals, how many has it?

Four lines work: the two diagonals plus the two lines through the midpoints of opposite sides. Add its rotations and the square has eight symmetries in all.

Which carmaker's logo of three rhombuses is a picture of what its own Japanese name means?

Mitsubishi joins mitsu, 'three', with hishi, the Japanese word for a water caltrop and hence for a rhombus. Hishi turns into 'bishi' when the two words are run together.

Which circus apparatus takes its name, by way of French, from the Late Latin word for a four-sided shape?

Trapeze probably got the name because its bar, its ropes and the ceiling outline a trapezium. Frenchman Jules Léotard, who performed the first flying trapeze routine in 1859, also gave his name to the leotard.

A classic quilt pattern made only of 60° rhombuses tricks the eye into seeing a stack of which solids?

Cubes appear because the pattern matches a corner-on view of stacked boxes. The eye can read the stack in two different ways, so it is also called the reversible cubes illusion.

Which South American civilisation built doorways, windows and wall niches in a distinctive trapezoid shape?

The Inca gave most wall openings sloping sides, and some were dressed up with double or triple jambs. A doorway of this kind is one of the familiar sights of Machu Picchu.

The four corners of a quadrilateral are properly called what, from a Latin word for a summit or whirlpool?

Vertices is the plural of vertex, Latin for 'highest point' and originally a whirling column, from vertere, 'to turn'. Each diagonal of a quadrilateral joins an opposite pair of them.

Which muscles between the spine and shoulder blades are named after a slanted, unequal-sided parallelogram?

Rhomboids keep a name mathematicians have dropped: a rhomboid was a parallelogram with unequal neighbouring sides and no right angles. The muscles pull the shoulder blades together.

The name rhombus comes from the Greek rhombos, used for objects such as a bullroarer that do what?

Spin is the root idea: rhombos could mean a spinning top, a bullroarer or a sorcerer's magic wheel. The Greeks also gave the same name to a kind of flatfish.

Which term describes any quadrilateral, such as a kite or a rhombus, whose diagonals cross at right angles?

Orthodiagonal quadrilaterals share a handy area rule: multiply the two diagonals together and divide by two. No other convex quadrilateral with the same diagonals encloses more area.

Which Greek astronomer's theorem on the diagonals of cyclic quadrilaterals helped him build a table of chords?

Ptolemy needed the chord table for the astronomy in his Almagest, and it became the earliest trigonometric table extensive enough for practical use.

Which 7th-century Indian mathematician's formula gives a cyclic quadrilateral's area from its side lengths?

Brahmagupta's formula extends Heron's rule for a triangle: shrink one side of the quadrilateral to nothing and the two formulas become the same.

In the Elements, Euclid defined a point, the simplest object in geometry, as 'that which has' what?

No part means a point cannot be cut any smaller: it marks an exact position but has no length, width or thickness of its own.

The first postulate in Euclid's Elements says a straight line can be drawn between any how many points?

Two points are enough, and Euclid's wording was 'to draw a straight line from any point to any point'. His second postulate lets that line be extended as far as you like.

Which geometric figure starts at a fixed point and runs on forever in one direction, like a beam of light?

A ray has one fixed end, known as its initial point. Any point on a line splits that line into two of them, heading in opposite directions.

Three points always fix exactly one plane unless all three sit in one straight row, a condition called what?

Collinear points turn up in surprising places: in any triangle the centroid, circumcentre and orthocentre always sit on one straight line, known as the Euler line.

Which word describes points or lines that all lie in one flat surface, like dots on a single sheet of paper?

Coplanar lines must either cross or run parallel. Only out in space, where two lines need not share a plane, can a pair manage to do neither.

Which word names the point, line or curve that two geometric figures have in common, as where two roads cross?

Intersection can also come up empty: two parallel lines share no points at all. Two different straight lines that do cross can only ever share a single point.

In geometry notation, the letters AB under a plain bar with no arrowheads stand for which figure?

A segment has two endpoints and a length that can be measured. The bar drawn above the two letters is known as an overline or vinculum.

Euclid's Elements defines a line as a length that has none of which other measurement?

Breadth is what a line lacks: Euclid called it 'breadthless length'. For him a line could curve, and a straight one was a line that 'lies evenly with the points on itself'.

Which word describes two lines that never meet yet are not parallel, like opposite edges of a tetrahedron?

Skew lines cannot exist on a flat sheet, where two lines that never meet are always parallel. Out in space they can miss each other while pointing different ways.

Which Latin word for 'place' names the set of all points that obey one rule, as a circle's points do?

Locus is how older textbooks defined shapes: a circle was the locus of a point moving at a fixed distance from a centre. Modern mathematics prefers to speak of sets of points.

In a perspective drawing, parallel lines such as straight railway tracks appear to meet at what?

The vanishing point is where the drawn images of parallel lines come together, although the real lines never meet. Lines at right angles to the picture give one-point perspective.

On a coordinate grid, what name is given to the point (0, 0), where the x-axis and y-axis cross?

Origin is the fixed reference for every other position on the grid and is usually labelled with the letter O. It splits each axis into a positive side and a negative side.

A point has none, a line has one and a plane has two of which basic geometric property?

Dimensions count the coordinates needed to pin down a position: one number is enough along a line, but a surface such as a globe needs two, like latitude and longitude.

Which word names a starting statement, such as 'all right angles are equal', that is accepted without proof?

Axiom and postulate now mean the same thing: a statement taken to be true as a starting point for reasoning. Euclid believed his were self-evident facts about physical reality.

Which 1884 novella by Edwin Abbott is narrated by a square whose entire world is a single plane?

Flatland's men are polygons with various numbers of sides, and its narrator signs himself 'A Square'. Abbott, a schoolmaster, used the flat world to satirise Victorian class and gender hierarchies.

Which word describes three or more lines that all pass through one single point, as a triangle's medians do?

Concurrent lines are everywhere in a triangle: its three medians meet at the centroid, the spot on which a cut-out triangle of even thickness would balance.

On a globe, which curves, such as the equator, play the part that straight lines play on a flat plane?

Great circles contain the shortest path between any two places on a globe, which is why they are used to plan routes for air and sea navigation.

In modern geometry, point, line and plane are deliberately left undefined, earning them which label?

Primitive notions are the handful of starting words used without explaining their meanings, because every definition has to rest on earlier terms and reasoning must begin somewhere.

Which German mathematician said points, lines and planes might as well be called chairs and beer mugs?

David Hilbert meant that only the stated relationships between the objects matter, not what they are. His 1899 book rebuilt Euclid's geometry on a short list of carefully chosen assumptions.

Which branch of geometry, born from perspective art, adds 'points at infinity' where parallel lines meet?

Projective geometry lets parallel lines meet, like railway tracks at the horizon, and has no use for measuring angles. Kepler and Desargues each came up with the point at infinity independently.