人教版初中七年级上册数学：两点之间线段最短-英文版课件.ppt

1、人教版初中七年级上册数学：两点之间线段最短_英文版课件Mathematical Ideas that Shaped the WorldNon-Euclidean geometryPlan for this classnWho was Euclid?What did he do?nFind out how your teachers lied to younCan parallel lines ever meet?nWhy did the answer to this question change the philosophy of centuries?nCan you imagine a w

2、orld in which there is no left and right?nWhat shape is our universe?Your teachers lied to you!nYour teachers lied to you about at least one of the following statements.qThe sum of the angles in a triangle is 180 degrees.qThe ratio of the circumference to the diameter of a circle is always.qPythagor

3、as Theorem qGiven a line L and a point P not on the line,there is precisely one line through P in the plane determined by L and P that does not intersect L.EuclidnBorn in about 300BC,though little is known about his life.nWrote a book called the Elements,which was the most comprehensive book on geom

4、etry for about 2000 years.nOne of the first to use rigorous mathematical proofs,and to do pure mathematics.The ElementsnA treatise of 13 books covering geometry and number theory.nThe most influential book ever written?nSecond only to the Bible in the number of editions published.(Over 1000!)nWas a

5、part of a university curriculum until the 20th century,when it started being taught in schools.nPartly a collection of earlier work,including Pythagoras,Eudoxus,Hippocrates and Plato.DefinitionsnA straight line segment is the shortest path between two points.nTwo lines are called parallel if they ne

6、ver meet.The axioms of geometrynThe Elements starts with a set of axioms from which all other results are derived.1.A straight line segment can be drawn joining any two points.The axioms of geometry2.Any straight line segment can be extended indefinitely in a straight line.The axioms of geometry3.Gi

7、ven any straight line segment,a circle can be drawn having the segment as radius and one endpoint as centre.The axioms of geometry4.All right angles are congruent.The axioms of geometry5.If a straight line N intersects two straight lines L and M,and if the interior angles on one side of N add up to

8、less than 180 degrees,then the lines L and M intersect on that side of N.LMNAxiom 5nAxiom 5 is equivalent to the statement that,given a line L and a point P not on the line,there is a unique line through P parallel to L.nThis is usually called the parallel postulate.LPAngles in a trianglenFrom Axiom

9、 5 we can deduce that the angles in a triangle add up to 180 degrees.Hidden infinitiesnPeople were not comfortable with Axiom 5,including Euclid himself.nThere was somehow an infinity lurking in the statement:to check if two lines were parallel,you had to look infinitely far along them to see if the

10、y ever met.nCould this axiom be deduced from the earlier,simpler,axioms?Constructing parallel linesnFor example,we could construct a line parallel to a given line L by joining all points on the same side of L at a certain distance.ProblemnThe problem is:how do we prove that the line we have construc

11、ted is a straight line?Could parallel lines not exist?nEver since Euclid wrote his Elements,people tried to prove the existence and uniqueness of parallel lines.nThey all failed.nBut if it was impossible to prove that Axiom 5 was true,could it therefore be possible to find a situation in which it wa

12、s false?Exhibit 1:The EarthCan we make Euclids axioms work on a sphere?Shortest distances?nQuestion:What is a straight line on Earth?nAnswer:The shortest distance between two points is the arc of a great circle.What is a great circle?nA great circles centre must be the same as that of the sphere.Not

13、 a straight lineShortest distances on a mapnIf we take a map of the world and draw straight lines with a ruler,these are not the shortest distances between points.The curve depends on the distancenBut straight lines on a map are a good approximation to the shortest distance if you arent travelling f

14、ar.nThe further you travel,the more curved the path you will travel.Axiom 5 on a spherenAmazing fact:there are no parallel lines on a sphere.nProof:all great circles intersect,so no two of them can be parallel.Why does our construction fail?nStudying the sphere shows why our previous attempt to cons

15、truct parallel lines went wrong.nIf we take all points equidistant from a great circle,the resulting line is a small circle and is thus not straight.Angles in a trianglenBut if the parallel axiom fails,then what about angles in a triangle?nIt turns out that if you draw triangles on a sphere,the angl

16、es will always add up to more than 180 degrees.A triangle with 3 right angles!nFor example,draw a triangle with angles of 270 degrees by starting at the North Pole,going down to the equator,walking a quarter of the way round the equator,then back to the North Pole.Even the value of changes!nThe rati

17、o of the circumference of a circle to its diameter is no longer fixed at 3.14159nIt is always less than and varies with every different circle drawn on the sphere.=2 =2.8284Do spheres contradict Euclid then?nGeometry on a sphere clearly violates Euclids 5th axiom.nBut people were not entirely satisf

18、ied with this counterexample,since spherical geometry also didnt satisfy axioms 2 and 3.n(That is,straight lines cannot be extended indefinitely,and circles cannot be drawn with any radius.)What if axiom 5 were not true?nIs it possible to construct a kind of geometry that does satisfy Euclids axioms

19、 1-4 and only contradicts axiom 5?nFor a long time,people didnt even think to try.nAnd when they did try,they were unable to overcome the force of their intuition.What if axiom 5 were not true?nThe Italian mathematician Saccheri was unable to find a contradiction when he assumed the parallel postula

20、te to be false.nYet he rejected his own logic,sayingit is repugnant to the nature of straight lines Kants philosophynIn 1781 the philosopher Immanuel Kant wrote his Critique of Pure Reason.nIn it,Euclidean geometry was held up as a shining example of a priori knowledge.nThat is,it does not come from

21、 experience of the natural world.The players in our storySome people were willing to change the status quo,or at least to think about itnCarl Friedrich GaussnFarkas BolyainJnos Bolyai nNikolai Ivanovich LobachevskyCarl Friedrich Gauss(1777 1855)nBorn in Braunschweig,Germany,to poor working class par

22、ents.nA child prodigy,completing his magnum opus by the age of 21.nOften made discoveries years before his contemporaries but didnt publish because he was too much of a perfectionist.Father and sonnGauss discussed the theory of parallels with his friend,Farkas Bolyai,a Hungarian mathematician,who tr

23、ied in vain to prove Axiom 5.nFarkas in turn taught his son Jnos about the theory of parallels,but warned himnot to waste one hours time on that problemnHe went onAn imploring letterI know this way to the very end.I have traversed this bottomless night,which extinguished all light and joy in my life

24、 It can deprive you of your leisure,your health,your peace of mind,and your entire happiness I turned back when I saw that no man can reach the bottom of this night.I turned back unconsoled,pitying myself and all mankind.Learn from my example An imploring letter“For Gods sake,please give it up.Fear

25、it no less than the sensual passion,because it,too,may take up all your time and deprive you of your health,peace of mind and happiness in life.”An imploring letterHis son ignored him.Jnos Bolyai(1802 1860)nBorn in Kolozsvr(Cluj),Transylvania.nCould speak 9 languages and play the violin.nMastered ca

26、lculus by the age of 13 and became obsessed with the parallel postulate.Jnos BolyainIn 1823 he wrote to his father sayingI have discovered things so wonderful that I was astounded.out of nothing I have created a strange new world.nHis work was published in an appendix to a book written by his father

27、.A reply from GaussnBolyai was excited to tell the great mathematician Gauss about his discoveries.nImagine his dismay,then,at receiving the following reply:To praise it would amount to praising myself.For the entire content of the work.coincides almost exactly with my own meditations which have occ

28、upied my mind for the past thirty or thirty-five years.At the same time,there was yet another rival to the claim of the first non-Euclidean geometry.Lobachevsky(1792 1856)nBorn in Nizhny Novgorod,Russia.nWas said to have had 18 children.nWas the first person to officially publish work on non-Euclide

29、an geometry.nSome people have accused him of stealing ideas from Gauss,but there is no evidence for this.Hyperbolic geometrynIn hyperbolic geometry,there are many lines parallel to a given line and going through a given point.nIn fact,parallel lines diverge from one another.nAngles in triangles add

30、up to less than 180 degrees.n is bigger than 3.14159The Poincar disk modelnDistances in a hyperbolic circle get larger the closer you are to the edge.nImagine a field which gets more muddy at the boundary.nA straight line segment is one which meets the boundary at right angles.Hyperbolic geometry in

31、 real lifenAlthough hyperbolic geometry was completely invented by pure mathematicians,we now find it crops up surprisingly often in the real worldPlantsMushroomsCoral reefsThe hyperbolic crochet coral reefBrainsArt(especially Escher!)Your teachers lied to you!nYour teachers lied to you about at lea

32、st one of the following statements.Which one(s)?qThe sum of the angles in a triangle is 180 degrees.qThe ratio of the circumference to the diameter of a circle is always.qPythagoras Theorem qGiven a line L and a point P not on the line,there is precisely one line through P in the plane determined by

33、 L and P that does not intersect L.Your teachers lied to you!nAnswer:all of them can sometimes be false!It depends on the space that you are drawing lines on.nAny space in which these statements are false is called a non-Euclidean geometry.SummarynParallel lines do different things in different geom

34、etries:qIn flat space,there are unique parallel linesqIn a spherical geometry,there are no parallel linesqIn a hyperbolic geometry,there are infinitely many lines parallel to a given line going through a particular point.Euclids 5th axiom of geometry is not always trueReaction of the philosophersnMa

35、thematics,and in particular Euclid,had always been examples of perfect truth.nThe concept of mathematical proof meant that we could know things absolutely.nThe fact that Euclid had got one of his basic axioms wrong meant that a large part of philosophy needed to be re-written.nWas there such a thing

36、 as absolute truth?Another dimensionnAll the geometry weve looked at so far has been in 2 dimensions what happens in 3D?nThere are also 3 kinds of geometry:qFlat(180 degree triangles)qSpherical(180 degree triangles)qHyperbolic(1 then space is spherical.(Positive curvature)qIf 1 then space is hyperbo

37、lic.(Negative curvature)Why does it matter?nOur universe is currently expanding.nThe shape of our universe will determine its eventual fate.qIf space is spherical,it will eventually stop expanding,and contract again in a big crunch.qIf space is hyperbolic,the expansion will continue to get faster,re

38、sulting in the big freeze or big rip.qIf space is flat,the expansion will gradually slow down to a fixed rate.Current estimatesnOur current best guess is that is about 1,so that space is flat.nBut finding relies on being able to tell how much dark matter there is.nNew question:if space is flat,what

39、does it look like?TopologynTo answer the question about what space looks like,we will need some topology.nBut this area of maths is not concerned with measuring distances on objects.nTopology is about the properties of a shape that get preserved when a shape is wiggled and stretched like a sheet of

40、rubber.=Wiggling and stretchingnFor example:qHow many holes does the object have?qDoes the object have an edge?qHow many sides does the object have?qHow does the object sit inside another object?qCan you get from one point to any other point?ExamplesnA square has no holes and 1 edgenA sphere has no

41、holes and no edgesExamplesnA cylinder has 1 hole and 2 edgesnA torus has 1 hole and no edges2-dimensional planetsnImagine you are an ant living on some kind of surface.nHow would you decide what the surface was?nAnswer:you have to travel around it.All surfaces look the same locally.nThink how long i

42、t took people to discover that we live on a sphere!Weirder topologiesnCan you imagine a shape with 1 hole and 1 edge?nIt is called a Mbius strip.The Mbius stripnHow to make one:qTake a strip of paper and join the ends together with a half-twist.nHow to destroy one!qCut the strip in half along the lo

43、ng edge.What do you think will happen?qWhat if you cut it in half again?qCut the strip in thirds.Should the answer be different from before?HomeworknRepeat the experiments with a Mbius strip which has extra twists.nCan you spot any patterns?Non-orientabilitynThe Mbius strip is the simplest example o

44、f a shape which is non-orientable.nThis means that the concepts of left and right make no sense to a creature living on the surface.nAn alien living on the surface could travel around it and come home to find that everything had been reversed.Non-orientabilityThe Klein bottlenHere is another example

45、 of a non-orientable surface,called the Klein bottle.nUnlike the Mbius strip,it has no edge.nIt is a surface which has no inside or outside.What shape is our universe?nJust like the ant on the surface,we cannot tell the shape of our universe without travelling around it.nOrby seeing what happens to

46、light that travels around it.nDoes the universe have holes in it?Does it have an edge?Could it be non-orientable?The microwave backgroundnScientists study radiation from the very earliest universe:the Cosmic Microwave Background Radiation.nIn the beginning the universe was much smaller,so this light

47、 has had time to go all around it.nIf the universe were toroidal or Mbius-shaped,we would see repeating patterns.The Cosmic Microwave BackgroundCan you spot any patterns?The evidencenThere is currently no conclusive evidence for anything other than a flat universe.nSome scientists believe that the u

48、niverse is a strange kind of 3D dodecahedron.Lessons to take homenDont believe everything your teachers tell you!nThat straight lines can have different meanings depending on where you draw them.nThat crazy ideas in pure maths often turn out to be useful.nThat our universe could be much more complicated than we ever imagined.