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Conic Sections | World of Mathematics

From Orbits to Pi

Ptolemaic Cosmos

The Ptolemaic Cosmos (1660)

For as long as human beings have existed, they have looked to the sky and tried to explain life on earth using the motion of stars, planets and the moon. The Greeks were among the first to discover that all celestial objects move on certain paths which can be described precisely using mathematics. These paths are called orbits.

Greek astronomers like Claudius Ptolemy (c. 90 – 168 AD) believed that the Earth is at the centre of the universe (the geocentric model), and that the sun and all planets move around it on the most perfect of all shapes: circles.

Circles are symmetric in every direction, and all points on a circle have the same distance from its centre – this distance is called the radius. The ratio between the circumference and the diameter is the same for circles of any size: it is a mysterious number called Pi.

circle radius

Circles and Pi are at the very heart of nature. This is because the circle is very often the most efficient shape possible. For example, given a fixed circumference, the circle is the shape which encloses the largest area. Other shapes, like a square, always have a smaller area than a circle with the same circumference. Objects in nature can save energy by becoming circular or spherical – and nature always wants to minimise the energy used.

flower earth water ripples fruit storm
Flowers Stars and Planets Water Ripples Fruit Storms

Calculating Pi

Pi, written as π, equals 3.1415926… and its digits goes on forever, without any repeating pattern. Numbers with this property are called irrational numbers. Many ancient mathematicians – including the famous Pythagoras – were horrified when they discovered that such bizarre and ‘impure’ numbers exist.

Today many mathematicians believe that Pi has an even more curious property: that it is a normal number. This would mean that the digits from 0 to 9 appear completely at random, as if nature had rolled a 10-sided dice, again and again, to find the next digit. It also means that if you think of any string of digits, like 123456789, it has to appear somewhere in the digits of Pi – but you might have to calculate millions of digits.

We could even convert an entire book, like the works of Shakespeare, into a very long string of digits (a = 01, b = 02, and so on). If Pi is normal, this string must also appear somewhere in its digits. But even if we used all computers on Earth to calculate more and more digits of Pi, we would probably have to look for longer than the age of the universe…

The First 50 000 Digits of Pi

3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852968995773622599413891249721775283479131515574857242454150695950829533116861727855889075098381754637464939319255060400927701671139009848824012858361603563707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473263914199272604269922796782354781636009341721641219924586315030286182974555706749838505494588586926995690927210797509302955321165344987202755960236480665499119881834797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548161361157352552133475741849468438523323907394143334547762416862518983569485562099219222184272550254256887671790494601653466804988627232791786085784383827967976681454100953883786360950680064225125205117392984896084128488626945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645995813390478027590099465764078951269468398352595709825822620522489407726719478268482601476990902640136394437455305068203496252451749399651431429809190659250937221696461515709858387410597885959772975498930161753928468138268683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244136549762780797715691435997700129616089441694868555848406353422072225828488648158456028506016842739452267467678895252138522549954666727823986456596116354886230577456498035593634568174324112515076069479451096596094025228879710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821682998948722658804857564014270477555132379641451523746234364542858444795265867821051141354735739523113427166102135969536231442952484937187110145765403590279934403742007310578539062198387447808478489683321445713868751943506430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675142691239748940907186494231961567945208095146550225231603881930142093762137855956638937787083039069792077346722182562599661501421503068038447734549202605414665925201497442850732518666002132434088190710486331734649651453905796268561005508106658796998163574736384052571459102897064140110971206280439039759515677157700420337869936007230558763176359421873125147120532928191826186125867321579198414848829164470609575270695722091756711672291098169091528017350671274858322287183520935396572512108357915136988209144421006751033467110314126711136990865851639831501970165151168517143765761835155650884909989859982387345528331635507647918535893226185489632132933089857064204675259070915481416549859461637180270981994309924488957571282890592323326097299712084433573265489382391193259746366730583604142813883032038249037589852437441702913276561809377344403070746921120191302033038019762110110044929321516084244485963766983895228684783123552658213144957685726243344189303968642624341077322697802807318915441101044682325271620105265227211166039666557309254711055785376346682065310989652691862056476931257058635662018558100729360659876486117910453348850346113657686753249441668039626579787718556084552965412665408530614344431858676975145661406800700237877659134401712749470420562230538994561314071127000407854733269939081454664645880797270826683063432858785698305235808933065757406795457163775254202114955761581400250126228594130216471550979259230990796547376125517656751357517829666454779174501129961489030463994713296210734043751895735961458901938971311179042978285647503203198691514028708085990480109412147221317947647772622414254854540332157185306142288137585043063321751829798662237172159160771669254748738986654949450114654062843366393790039769265672146385306736096571209180763832716641627488880078692560290228472104031721186082041900042296617119637792133757511495950156604963186294726547364252308177036751590673502350728354056704038674351362222477158915049530984448933309634087807693259939780541934144737744184263129860809988868741326047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4…

Search for a string of digits: Not found in the first 50 000 digits of Pi…

There are many different ways to calculate Pi, some of which use sequences or series of numbers. One example is the following series discovered by Gottfried Wilhelm Leibniz (1646 – 1716). As you add more and more terms, following the same pattern, the result will get closer to Pi:

π  =  4143 + 4547 + 49411 + …

Another sequence, published by Nilakantha Somayaji (1444 – 1544), is even better since it gets closer to Pi with fewer terms:

π  =  3 + 42 × 3 × 444 × 5 × 6 + 46 × 7 × 848 × 9 × 10 + …

This formula was published by John Wallis in 1655:

π  =  2 × 21 × 23 × 43 × 45 × 65 × 67 × 87 × 89 × …

Using powerful computers, Pi has been calculated up to 10 trillion digits (that’s a 1 with 13 zeros)! Because Pi is so easy to understand, yet important in many areas of mathematics, it enjoys an unusual popularity in our culture (unusual, at least, for areas of mathematics). There even is a Pi Day on 14 March or 22 July, since 227 is a close approximation to Pi.

Conic Sections

The circle is one of four different shapes which can be created using slices through a cone. This can be demonstrated using the light cone of a torch: if it points vertically downwards, we see a circle of light. If we tilt the cone, we get an ellipse. If we tilt it even further, we first get a parabola and then a hyperbola. These four shapes are called conic sections.

conic sections

On first sight, the four conic sections look very different – circle and ellipse are closed, for example, while parabola and hyperbola are open and go on to infinity. But all four shapes can be created using the same equation:

x2 + y2  =  l + ex

Here x and y are the usual cartesian coordinates, and e and l are certain constants which are different for each conic section. (We call e the “eccentricity” and l the “semi-latus rectum”.) For example, if e = 0, we get the equation of a circle with radius l.

The ellipse is a particularly interesting shape. We could think about it as a circle with “two centres” – these are called the foci. If both foci are in the same place, we get a circle. If they move further apart, the shape becomes more and more elliptical.

Just like the distance between points on a circle and its centre is constant, the sum of the distances from points on an ellipse to its two foci is constant. This means that we can easily create an ellipe using a piece of string:

Created by Pavel Boytchev, programmed in Elica Logo and published with permission of the Elica team.

Created by Pavel Boytchev, programmed in Elica Logo and published with permission of the Elica team.

Created by Pavel Boytchev, programmed in Elica Logo and published with permission of the Elica team.

Created by Pavel Boytchev, programmed in Elica Logo and published with permission of the Elica team.

The Scientific Revolution

epicycles

The motion of planets using Epicycles

Ptolemy believed that the Earth is at the centre of the universe and that planets and the sun move around the Earth in circles (the Geocentric Model). Unfortunately, astronomical observation showed that the sun appeared larger during some parts of the year and smaller during others – while, in a circle, the sun should always have the same distance from the Earth. Therefore astronomers added Epicycles to the orbits: all objects in the sky move on a large circle around the Earth, but at the same time they move along a smaller circle around themselves. And, though complicated, this was the most widely accepted model of our universe for more than 1000 years.

But in 1543, Nicolaus Copernicus (1473 – 1543) published “On the Revolutions of the Celestial Spheres”, a book in which he explained what many astronomers had suspected for some time: that the sun is at the centre of the universe and that the Earth moves around the sun on a circle, like all the other planets. This is called the Heliocentric Model.

At that time, the heliocentric model still had many errors, and it still couldn’t explain why the sun sometimes appears larger and sometimes smaller. Furthermore, the fact that the Earth is flying through space at a rapid speed seemed impossible – as well as contradictory to the Bible.

geocentric model heliocentric model
The Geocentric (Ptolemaic) Model – Illustration from Cosmographia by Bartolomeu Velho The Heliocentric Model – Illustration from Harmonia Macrocosmica by Andreas Cellarius

During the following years, the Danish astronomer Tycho Brahe (1546 – 1601) made very accurate observations of the movements of the planets, as well as developing his own model of the solar system. After his (sudden and unexplained) death, his assistant Johannes Kepler (1571 – 1630) succeeded him as imperial mathematician. And many years of research in a variety of areas culminated in Kepler’s three Laws of Planetary Motion, which essentially describe the solar system as we know it to be true today.

Nicolaus Copernicus Tycho Brahe Johannes Kepler Galileo Galilei
Nicolaus Copernicus
1473 – 1543
Tycho Brahe
1546 – 1601
Johannes Kepler
1571 – 1630
Galileo Galilei
1564 – 1642

At the same time in Italy, Galileo Galilei (1564 – 1642) made astronomical observations using newly developed telescopes. He discovered the phases of Venus, four moons of Jupiter, as well as the sunspots.

Galileo was a strong advocate of the new heliocentric model. He was a friend of Pope Urban VIII and tried to unify this new model with the Bible. The controversy that arose from his book “Dialogue Concerning the Two Chief World Systems” was primarily founded on politics, not science or religion. Galileo was forced to recant and set under house arrest for the rest of his life.

But here are the celebrated laws of Kepler, which describe the motion of all planets:

  1. The planets move on elliptical orbits, not circles, and the sun is in one of the two foci. This explains why the sun appears larger during some parts of the year and smaller during others.
  2. A line joining the planet and the Sun sweeps out equal areas during equal times. This means that a planet speeds up as it moves closer to the sun, and slows down as it moves further away. In the animation, all blue segments have the same area, and the planet moves across all segments in equal times.
  3. The square of the orbital period is proportional to the cube of the average radius of the orbit. This law allows us to find the time a planet needs for one full rotation, if we know the size of the orbit.
Isaac Newton

Isaac Newton (1642 – 1727)

In 1687, Sir Isaac Newton (1642 – 1727) published the “Principia Mathematica”, where he develops the mathematical foundations that give rise to Kepler’s laws. Between any two masses in the universe there is a force, similar to the attraction between two magnets. This force is called gravity. Gravity is what makes everything fall to the ground and gravity is what makes the planets move around the sun. It is only the great speed at which planets move, that prevents them from falling into the sun.

Using Newton’s laws, one can derive an equation that describes the motion of objects moving under the force of gravity. This equation happens to be the same as the equation for conic sections (see above). Planets move on ellipses, but other objects, such as comets, can travel on parabolic or hyperbolic paths: they come close to the sun before turning around and shooting off into the universe, never to come back.

Newton lived and worked at Trinity College Cambridge, where a falling apple is said to have inspired him to think about gravity. His description of classical mechanics, among many other discoveries, makes Newton the greatest and most influential scientist of all times. His ideas shaped our understanding of the world for nearly 300 years, until Albert Einstein discovered the laws of relativity in 1905.