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(同名2434)

1、1.两点之间线段最短2.两点决定一条直线3.直线可以无限延长；以线段的长为半径，线段的一个端点为圆心可作一个圆Mathematical Ideas that Shaped the WorldNon-Euclidean geometryPlan 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 change the philosophy of centu

2、ries?Can you imagine a world in which there is no left and right?What shape is our universe?Your 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.The ratio of the circumference to the diameter of a circle

3、 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.EuclidBorn in about 300BC, though little is known about his life.Wrote a book called the Elements, which was the most compre

4、hensive book on geometry for about 2000 years.One of the first to use rigorous mathematical proofs, and to do pure mathematics.The ElementsA treatise of 13 books covering geometry and number theory.The most influential book ever written?Second only to the Bible in the number of editions published. (

5、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, Hippocrates and Plato.DefinitionsA straight line segment is the shortest path between two points.Two lines are called p

6、arallel if they never meet.The axioms of geometryThe Elements starts with a set of axioms from which all other results are derived.A straight line segment can be drawn joining any two points. The axioms of geometryAny straight line segment can be extended indefinitely in a straight line. The axioms

7、of geometryGiven any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre. The axioms of geometryAll right angles are congruent. The axioms of geometryIf a straight line N intersects two straight lines L and M, and if the interior angles on one side of

8、 N add up to less than 180 degrees, then the lines L and M intersect on that side of N.LMNAxiom 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 postulate.LPAngles in a trian

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

10、 to see if they ever met.Could this axiom be deduced from the earlier, simpler, axioms?Constructing parallel linesFor 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.ProblemThe problem is: how do we prove that the line w

11、e have constructed is a straight line?Could parallel lines not exist?Ever since Euclid wrote his Elements, people tried to 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

12、in which it was false?Exhibit 1: The EarthCan we make Euclids axioms work on a sphere?Shortest distances?Question: What is a straight line on Earth?Answer: The shortest distance between two points is the arc of a great circle. What is a great circle?A great circles centre must be the same as that of

13、 the sphere.Not a straight lineShortest 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.The curve depends on the distanceBut straight lines on a map are a good approximation to the shortest distance if you aren

14、t travelling far.The further you travel, the more curved the path you will travel.Axiom 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.Why does our construction fail?Studying the sphere shows why our previous a

15、ttempt 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.Angles in a triangleBut if the parallel axiom fails, then what about angles in a triangle?It turns out that if you draw triangles on a sp

16、here, the angles will always add up to more than 180 degrees.A triangle with 3 right angles!For 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

17、changes!The ratio of the circumference of a circle to its diameter is no longer fixed at 3.14159It is always less than and varies with every different circle drawn on the sphere. = 2 = 2.8284Do spheres contradict Euclid then?Geometry on a sphere clearly violates Euclids 5th axiom.But people were not

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

19、fy Euclids axioms 1-4 and only contradicts axiom 5?For a long time, 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?The Italian mathematician Saccheri was unable to find a contradiction when he assumed the

20、parallel postulate to be false.Yet he rejected his own logic, sayingit is repugnant to the nature of straight lines 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 doe

21、s not come from experience of the natural world.The 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 LobachevskyCarl Friedrich Gauss (1777 1855)Born in Braunschweig, Germany, to poor wor

22、king 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.Father and sonGauss discussed the theory of parallels with his friend, Farkas Bolyai, a Hungarian mathem

23、atician, 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 onAn imploring letterI know this way to the very end. I have traversed this bottomless night, which extinguished all light an

24、d 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 turned back unconsoled, pitying myself and all mankind. Learn from my example An imploring letter“For Gods sake,

25、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 happiness in life.”An imploring letterHis son ignored him.Jnos Bolyai (1802 1860) Born in Kolozsvr (Cluj), Transylvania.Could speak 9 languages an

26、d play the violin.Mastered calculus by the age of 13 and became obsessed with the parallel postulate.Jnos 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 t

27、o a book written by his father.A reply from GaussBolyai 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

28、my own meditations which have occupied 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)Born in Nizhny Novgorod, Russia.Was said to have had 18 children.Was the first person to off

29、icially publish work on non-Euclidean geometry.Some people have accused him of stealing ideas from Gauss, but there is no evidence for this.Hyperbolic geometryIn hyperbolic geometry, there are many lines parallel to a given line and going through a given point.In fact, parallel lines diverge from on

30、e another.Angles in triangles add up to less than 180 degrees. is bigger than 3.14159The 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.A straight line segment is one which meets the boundary at right

31、 angles.Hyperbolic geometry in real lifeAlthough 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!Your tea

32、chers lied to you about at least one of the following statements. Which one(s)?The sum of 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

33、 P in the plane determined by L and P that does not intersect L.Your 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.SummaryParallel lines do differe

34、nt things in different geometries: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 trueReac

35、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

36、en.Was there such a thing as absolute truth?Another 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 spherical. (Positive curvature)If

37、 1 then space is hyperbolic. (Negative curvature)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 expanding, and contract again in a big crunch.If space is hyperbolic, the expansion will c

38、ontinue 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.Current estimatesOur current best guess is that is about 1, so that space is flat.But finding relies on being able to tell how much dark matter there is.New question

39、: if space is flat, what does it look like?TopologyTo answer the question about what space looks like, we will need some topology.But this area of maths 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 str

40、etched like a sheet of rubber.=Wiggling and stretchingFor example:How many holes does the object have?Does the object have 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?ExamplesA square has no holes and 1 edgeA

41、 sphere has no holes and no edgesExamplesA cylinder has 1 hole and 2 edgesA torus has 1 hole and no edges2-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.Thin

42、k how long it took people to discover that we live on a sphere!Weirder topologiesCan you imagine a shape with 1 hole and 1 edge?It is called a Mbius strip.The Mbius stripHow to make one:Take a strip of paper and join the ends together with a half-twist.How to destroy one! Cut the strip in half along

43、 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?HomeworkRepeat the experiments with a Mbius strip which has extra twists.Can you spot any patterns?Non-orientabilityThe Mbius strip is the simplest example of a shape which is non-orientable.This means that the concepts of left and r