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

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 geometryMathematical Ideas that ShapedPlan for this classWho was Euclid? What did he do?Find out how your teachers lied to youCan parallel lines ever meet?Why did the answer to this question c

2、hange the philosophy of centuries?Can you imagine a world in which there is no left and right?What shape is our universe?Plan for this classWho was EuYour teachers lied to you!Your teachers lied to you about at least one of the following statements.The sum of the angles in a triangle is 180 degrees.

3、The ratio of the circumference to the diameter of a circle is always .Pythagoras Theorem Given 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.Your teachers lied to you!YourEuclidBorn in about 300BC, though li

4、ttle is known about his life.Wrote a book called the Elements, which was the most comprehensive book on geometry for about 2000 years.One of the first to use rigorous mathematical proofs, and to do pure mathematics.EuclidBorn in about 300BC, thoThe ElementsA treatise of 13 books covering geometry an

5、d number theory.The most influential book ever written?Second only to the Bible in the number of editions published. (Over 1000!)Was a part of a university curriculum until the 20th century, when it started being taught in schools.Partly a collection of earlier work, including Pythagoras, Eudoxus, H

6、ippocrates and Plato.The ElementsA treatise of 13 bDefinitionsA straight line segment is the shortest path between two points.Two lines are called parallel if they never meet.DefinitionsA straight line segThe axioms of geometryThe Elements starts with a set of axioms from which all other results are

7、 derived.A straight line segment can be drawn joining any two points. The axioms of geometryThe ElemThe axioms of geometryAny straight line segment can be extended indefinitely in a straight line. The axioms of geometryAny straThe axioms of geometryGiven any straight line segment, a circle can be dr

8、awn having the segment as radius and one endpoint as centre. The axioms of geometryGiven anThe axioms of geometryAll right angles are congruent. The axioms of geometryAll righThe axioms of geometryIf a straight line N intersects two straight lines L and M, and if the interior angles on one side of N

9、 add up to less than 180 degrees, then the lines L and M intersect on that side of N.LMNThe axioms of geometryIf a strAxiom 5Axiom 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.This is usually called the parallel p

10、ostulate.LPAxiom 5Axiom 5 is equivalent tAngles in a triangleFrom Axiom 5 we can deduce that the angles in a triangle add up to 180 degrees.Angles in a triangleFrom AxiomHidden infinitiesPeople were not comfortable with Axiom 5, including Euclid himself.There was somehow an infinity lurking in the s

11、tatement: to check if two lines were parallel, you had to look infinitely far along them to see if they ever met.Could this axiom be deduced from the earlier, simpler, axioms?Hidden infinitiesPeople were nConstructing parallel linesFor example, we could construct a line parallel to a given line L by

12、 joining all points on the same side of L at a certain distance.Constructing parallel linesForProblemThe problem is: how do we prove that the line we have constructed is a straight line?ProblemThe problem is: how do Could parallel lines not exist?Ever since Euclid wrote his Elements, people tried to

13、 prove the existence and uniqueness of parallel lines.They all failed.But if it was impossible to prove that Axiom 5 was true, could it therefore be possible to find a situation in which it was false?Could parallel lines not existExhibit 1: The EarthCan we make Euclids axioms work on a sphere?Exhibi

14、t 1: The EarthCan we makShortest distances?Question: What is a straight line on Earth?Answer: The shortest distance between two points is the arc of a great circle. Shortest distances?Question: WWhat is a great circle?A great circles centre must be the same as that of the sphere.Not a straight lineW

15、hat is a great circle?A greatShortest distances on a mapIf we take a map of the world and draw straight lines with a ruler, these are not the shortest distances between points.Shortest distances on a mapIf The curve depends on the distanceBut straight lines on a map are a good approximation to the s

16、hortest distance if you arent travelling far.The further you travel, the more curved the path you will travel.The curve depends on the distaAxiom 5 on a sphereAmazing fact: there are no parallel lines on a sphere.Proof: all great circles intersect, so no two of them can be parallel.Axiom 5 on a sphe

17、reAmazing facWhy does our construction fail?Studying the sphere shows why our previous attempt to construct parallel lines went wrong.If we take all points equidistant from a great circle, the resulting line is a small circle and is thus not straight.Why does our construction failAngles in a triangl

18、eBut if the parallel axiom fails, then what about angles in a triangle?It turns out that if you draw triangles on a sphere, the angles will always add up to more than 180 degrees.Angles in a triangleBut if theA triangle with 3 right angles!For example, draw a triangle with angles of 270 degrees by s

19、tarting at the North Pole, going down to the equator, walking a quarter of the way round the equator, then back to the North Pole.A triangle with 3 right anglesEven the value of changes!The ratio of the circumference of a circle to its diameter is no longer fixed at 3.14159It is always less than and

20、 varies with every different circle drawn on the sphere. = 2 = 2.8284Even the value of changes!ThDo spheres contradict Euclid then?Geometry on a sphere clearly violates Euclids 5th axiom.But people were not entirely satisfied with this counterexample, since spherical geometry also didnt satisfy axio

21、ms 2 and 3.(That is, straight lines cannot be extended indefinitely, and circles cannot be drawn with any radius.)Do spheres contradict Euclid tWhat if axiom 5 were not true?Is it possible to construct a kind of geometry that does satisfy Euclids axioms 1-4 and only contradicts axiom 5?For a long ti

22、me, people didnt even think to try.And when they did try, they were unable to overcome the force of their intuition.What if axiom 5 were not true?What if axiom 5 were not true?The Italian mathematician Saccheri was unable to find a contradiction when he assumed the parallel postulate to be false.Yet

23、 he rejected his own logic, sayingit is repugnant to the nature of straight lines What if axiom 5 were not true?Kants philosophyIn 1781 the philosopher Immanuel Kant wrote his Critique of Pure Reason.In it, Euclidean geometry was held up as a shining example of a priori knowledge.That is, it does no

24、t come from experience of the natural world.Kants philosophyIn 1781 the pThe players in our storySome people were willing to change the status quo, or at least to think about itCarl Friedrich GaussFarkas BolyaiJnos Bolyai Nikolai Ivanovich LobachevskyThe players in our storySome pCarl Friedrich Gaus

25、s (1777 1855)Born in Braunschweig, Germany, to poor working class parents.A child prodigy, completing his magnum opus by the age of 21.Often made discoveries years before his contemporaries but didnt publish because he was too much of a perfectionist.Carl Friedrich Gauss (1777 1Father and sonGauss d

26、iscussed the theory of parallels with his friend, Farkas Bolyai, a Hungarian mathematician, who tried in vain to prove Axiom 5.Farkas in turn taught his son Jnos about the theory of parallels, but warned himnot to waste one hours time on that problemHe went onFather and sonGauss discussed An implori

27、ng letterI know this way to the very end. I have traversed this bottomless night, which extinguished all light and joy in my life 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

28、turned back unconsoled, pitying myself and all mankind. Learn from my example An imploring letterI know thiAn imploring letter“For Gods sake, please give it up. Fear 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 happ

29、iness in life.”An imploring letterAn imploring letterHis son ignored him.An imploring letterHis son ignJnos Bolyai (1802 1860) Born in Kolozsvr (Cluj), Transylvania.Could speak 9 languages and play the violin.Mastered calculus by the age of 13 and became obsessed with the parallel postulate.Jnos Bol

30、yai (1802 1860) BorJnos BolyaiIn 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. His work was published in an appendix to a book written by his father.Jnos BolyaiIn 1823 he wrote tA reply from GaussBoly

31、ai was excited to tell the great mathematician Gauss about his discoveries.Imagine 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 occupied my mind for th

32、e past thirty or thirty-five years . A reply from GaussBolyai was eAt the same time, there was yet another rival to the claim of the first non-Euclidean geometry.At the same time, there was yeLobachevsky (1792 1856)Born in Nizhny Novgorod, Russia.Was said to have had 18 children.Was the first person

33、 to officially publish work on non-Euclidean geometry.Some people have accused him of stealing ideas from Gauss, but there is no evidence for this.Lobachevsky (1792 1856)Born Hyperbolic geometryIn hyperbolic geometry, there are many lines parallel to a given line and going through a given point.In f

34、act, parallel lines diverge from one another.Angles in triangles add up to less than 180 degrees. is bigger than 3.14159Hyperbolic geometryIn hyperbolThe Poincar disk modelDistances in a hyperbolic circle get larger the closer you are to the edge.Imagine a field which gets more muddy at the boundary

35、.A straight line segment is one which meets the boundary at right angles.The Poincar disk modelDistancHyperbolic geometry in real lifeAlthough hyperbolic geometry was completely invented by pure mathematicians, we now find it crops up surprisingly often in the real worldHyperbolic geometry in real l

36、iPlantsPlantsMushroomsMushroomsCoral reefsCoral reefsThe hyperbolic crochet coral reefThe hyperbolic crochet coral rBrainsBrainsArt (especially Escher!)Art (especially Escher!)Your teachers lied to you!Your teachers lied to you about at least one of the following statements. Which one(s)?The sum of

37、the angles in a triangle is 180 degrees.The ratio of the circumference to the diameter of a circle is always .Pythagoras Theorem Given 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.Your teachers lied to you!

38、YourYour teachers lied to you!Answer: all of them can sometimes be false!It depends on the space that you are drawing lines on.Any space in which these statements are false is called a non-Euclidean geometry.Your teachers lied to you!AnswSummaryParallel lines do different things in different geometr

39、ies:In flat space, there are unique parallel linesIn a spherical geometry, there are no parallel linesIn 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 trueSummaryParallel lines do diffeReac

40、tion of the philosophersMathematics, and in particular Euclid, had always been examples of perfect truth.The concept of mathematical proof meant that we could know things absolutely.The fact that Euclid had got one of his basic axioms wrong meant that a large part of philosophy needed to be re-writt

41、en.Was there such a thing as absolute truth?Reaction of the philosophersMaAnother dimensionAll the geometry weve looked at so far has been in 2 dimensions what happens in 3D?There are also 3 kinds of geometry:Flat (180 degree triangles)Spherical ( 180 degree triangles)Hyperbolic (1 then space is sph

42、erical. (Positive curvature)If 1 then space is hyperbolic. (Negative curvature)Matter changes geometryThe mor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Why does it matter?Our universe is currently expanding.The shape of our universe will determine its eventual fate.If space is spherical, it will eventually stop exp

43、anding, and contract again in a big crunch.If space is hyperbolic, the expansion will continue to get faster, resulting in the big freeze or big rip.If space is flat, the expansion will gradually slow down to a fixed rate.Why does it matter?Our universCurrent estimatesOur current best guess is that

44、is about 1, so that space is flat.But finding relies on being able to tell how much dark matter there is.New question: if space is flat, what does it look like?Current estimatesOur current bTopologyTo answer the question about what space looks like, we will need some topology.But this area of maths

45、is not concerned with measuring distances on objects.Topology is about the properties of a shape that get preserved when a shape is wiggled and stretched like a sheet of rubber.=TopologyTo answer the questionWiggling and stretchingFor example:How many holes does the object have?Does the object have

46、an edge?How many sides does the object have?How does the object sit inside another object?Can you get from one point to any other point?Wiggling and stretchingFor exaExamplesA square has no holes and 1 edgeA sphere has no holes and no edgesExamplesA square has no holes ExamplesA cylinder has 1 hole

47、and 2 edgesA torus has 1 hole and no edgesExamplesA cylinder has 1 hole 2-dimensional planetsImagine you are an ant living on some kind of surface. How would you decide what the surface was?Answer: you have to travel around it. All surfaces look the same locally.Think how long it took people to disc

48、over that we live on a sphere!2-dimensional planetsImagine yWeirder topologiesCan you imagine a shape with 1 hole and 1 edge?It is called a Mbius strip.Weirder topologiesCan you imagThe Mbius stripHow to make one:Take a strip of paper and join the ends together with a half-twist.How to destroy one!

49、Cut the strip in half along the long edge. What do you think will happen?What if you cut it in half again?Cut the strip in thirds. Should the answer be different from before?The Mbius stripHow to make onHomeworkRepeat the experiments with a Mbius strip which has extra twists.Can you spot any patterns?HomeworkRepeat the experimentsNon-orientabilityThe Mbius strip is the simplest example of a shape which is non-orientable.This means that the concepts of left and right make no sense to