(1.1)--Chapter1_OnDistance.pdf
How Long are Your Home Stairs?On DistanceLife on MathematicsThe Story of Staircase LengthDistance is not unfamiliar to people,because we frequently use this concept in daily life.When we have a closer look at how the concept is used,it tends to be so explicit and also implicit.A soldier in the watchtower reports to the head,“The situation is critical.The enemys attacking units are only 50 meters away from us!”Thus,how should we understand the distance between one man and a group of people?Furthermore,how to describe the distance between one group of people and the other group?A Chinese saying goes,“A distant relative is not helpful as a near neighbor.”The concepts of being far and near spatially map emotional factors.Then,does distance describe affection?Distance in mathematics cannot be as casual as people use.It is a concept,but at the same time it has to be diversified.Length of Approach BridgeAn overpass on Harbin Hexing RoadThe lengths of the stairs at both sides are different,in addition,the difference in slope also causes the calculation of the length of the stairs to be different.baThree Ways of WalkingA.Ants walk：walk along the stairs covering every single inchLengths of the stairs:()n abb+n stairs,each stairheight is a,length is bbaa2aThree Ways of WalkingB.Mens walk：small step,then walk to the endLengths of the stairs:(2)nabab+ab-abaThree Ways of WalkingC.Stride way：to finish each stair by only one strideLengths of the stairs:22n abb+22ab+For a flight of stairs,we get three lengths.Is this reasonable?Can we set a formula at will to calculate the distance between two points?What is the law of identity in the formal logic?Can You Give A Formula by The Concept of Distance?Aristotle was an ancient Greek philosopher,scientist,andeducationalist.He made great contributions to nearly everydiscipline,exerting profound influence on the mankind afterhim.He was considered an encyclopedia of a scientist.Helaid the foundation for formal logic,in which discipline hetook mathematics,especially geometry as examples.Aristotles Identity LawIdentity LawAristotles formal logic：The law of Identity,the principles of contradiction，the law of excluded middle The concept right：different from left.The law of Identity:speaking about the uniqueness of a concept.The childs eyes are really two springs of water.Water is a liquid.So,the childs eyes are liquid.The concept of wateris not uniform.Distance ConceptThe stairs have three lengths,however,are they against the the law of Identity？ba2a()n abb+A.Ants walk:B.Mens walk：C.Stride way：(2)nabab+22n abb+Wikipedia Description of DistanceDistance is defined as a scalar quantitywithoutdirection.Itcannotbenegative.In physics,distance refers tothe length of journeys covered by aperson,animal,vehicle or object.ABEuclidean DistanceStraight-line distance:AB11(,)x y22(,)xy221212(,)()()d A Bxxyy=+628.48Distance formula from point B to point A1 11 12Manhattan DistanceThe actual moving way of a taxiAB12 1126 2=Manhattan distance:The distance from point B to point AManhattan distance?Euclidean distance?1The Vector NormDefinition.Let be real number space and be n-dimensional real vector space.Then the mappingnR:nRR3.XYXY+for anynXR and nYR.1.0X for any nXR and 0X=if and only if 0X=;2.XX=for any scalar R and nXR;is a norm in n-dimensional vector space if it satisfiesthe following three properties:M1M2FFxFy3.XYXY+for anynXR and nYR.1.0X for any nXR and 0X=if and only if 0X=;2.XX=for any scalar R and nXR;The Vector Norm3.xyxy+for anyxR andyR.1.0 x for any xR and x0=if and only if x=;2.xx=for any scalar R and xR;Absolute Value Is a NormSubtraction of Vector11(,)A x y22(,)B xyxoy1212(,)xxyy1122,OAx yOBxy=Use coordinates to represent vectors：Subtraction of Vector：1212,OAOBBAxxyy=Distance on the basis of normThe distance between point A and point B：11(,)A x y22(,)B xyxoy1212(,)xxyyFor a vector，2-norm:,Xx y=222Xxy=+121222(,),d A BBAxxyy=221212()()xxyy=+What are the real meanings of the three properties of norm?Why do we say the shortest distance between two points is a straight line?Why does the concept of distance describe an optimized feature?Meanings of The Three Properties of NormThree Properties of NormNonnegativity measureZero-metric nature(1)For any real number ,;if and only if x0 x0=x0=xThe Meaning of Zero MetricIf the two points coincide,the distance is zero.Distance between two points00 xyxyxy=0 x y A B(,)d A Bxy=0 xy=xy=Point A coincides with point B,If the distance is zero,do the two points coincide?No problem.Three Properties of NormDescribe the linear characteristics of spaceXX=；(2)For any number ,Triangle InequalityoyxXY+YX222XYXY+FGEThe straight line is always the shortest path.Distance Shows the Shortest FeatureI swim 1000 meters in the swimming pool.HEU500 metersImplies a shortest lengthWhy is distance diversified?Is the Manhattan distance the cabby covers a real distance?Is The Length Which A Taxi Covers A Sort of Distance?Diversity of DistanceBased on normEuclidean distance formula from A to B(,)d A BBA=121222(,),d A BBAxx yy=221212()()xxyy=+Minkowski gave a set of distances whose construction is based on a-normpLMinkowskiMinkowski(1864-1909)was a German mathematician,whomade excellent contributions to number theory,algebra,andmathematical physics.Minkowski space provides a frameworkfor the construction of general relativity.1-Norm of Vector pLDefinitionFor n-dimensional vectorp=112(,)nXx xx=121nXxxx=+112()ppppnpXxxx=+NormManhattan Ddistance Is 1-NormManhattan distance from B to A:AB11(,)x y22(,)x y112121(,)xxyy=1212xxyy=+12=1(,)d A BBA=i).ii).Xxy1110=+=+=110,0 xy=0X=1-Norm of 2-dimentional real vector :Xxy11(,)=Xxy111=+=+xy110,0?(1).X10 for any XR2 and X10=if and only if X=;Xxy1110=+=+Manhattan Distance is a Norm?(,)(,)Xxyxy1111111=()1111xyxy=+=+=+=+(2).XX11=for any scalar R and XR2;?1X=Manhattan Distance is a Norm?1-Norm of 2-dimentional real vector :Xxy11(,)=Xxy111=+=+1121211212(,)XYxxyyxxyy+=+=+=+=+121211xxyyXY+=+=+(3).111XYXY+for anyXR2 and YR2.?For 2-dimentional vectors and ,we have Xxy11(,)=Yxy22(,)=Manhattan Distance is a Norm?1-Norm of 2-dimentional real vector :Xxy11(,)=Xxy111=+=+The 1-norm is truly a type of norm,so the Manhattan distance on the basis of 1-norm is also a kind of distance.Can the norm describe the distance between one man and a group of people?What is the mathematical difference between a confidant and a friend?The Difference between Confidants And FriendsDistance from the Enemy(,)min,2d X AX YYA=The distance between the soldier and a group of enemies:Min:minimumA Represents enemy setThe distance between the soldier and this enemy:2(,)d X YXY=Soldiers position:X,some enemys position:YDistance between the Two ArmiesDistance between the two armies2(,)min,d A BXYXA YB=A armyB armyX,Y respectively represent some military soldier of both sidesHuman temper and behaviourTimeThe concept of distance is related to the feelings.Distance Applications in Other FieldsConfucius once said:If a man is not farsighted,he is bound to encounter difficulties in the near future.An idiom in China goes that Be beneath the human character”Chinese people often say:a near neighbor is better than a distant cousin.Close RelationshipWorking relationship friendFriends of neighborhood relationshipChildhood partnerCoorpertion at workMutual care of living at homeLife experience is similarThe requirements of the confidant are among the highest,so the companion is most difficult to meetYue Fei wrote in the poemXiaozhongshan：I want to show what I thought in Yao Qin,however,I ve got too few friends.Even if you break the string,who will listen to you?Distance of The ConfidantWe give a character vectorDefinition of-norm:Then for two persons:12(,)nRr rr=12(,),nXx xx=12(,)nYy yy=1,2,maxiinRr=1,2,maxiiinxy=1122(,)(,)nnd X YXYxy xyxy=max:Maximumri：spirit,emotion,personality,worldview and so on.Confidant?To examine the distanceWhat kind of psychological hallucination can the beauty produced by distance make?Why we say the ultimate temptation of beauty produced by distance is to pursue“none-distance”?Mathematical Explanation of“Distance Produces Beauty”Distance Produces BeautyPainting Color paste beautyOverall beautyVan Goghs painting Seascape near LesDistance Produces BeautyPhotographyBare loess and sandstone Trees were knocked down by lightningDistance covers everythingWe can observe the strong and light touches and traces of brushwork in a close distance,which is the beauty of color paste from a professional view.How Much Distance Will Produce Beauty?Mathematical implications:the diversity of distance concepts.It seems to imply that dont go too far to pursue the beauty of the distance.Who knows?!Different distances will bring you different feelings,and different people have their own“appreciation distance”.Distance of the MindThe painter was working with the mind when he was creating.Appreciation of PaintingsThe appreciation of the painting is wonderful the processthe appreciator and the painter approaching the soul together.Distance OptimizationAppreciating art is looking for the shortest distance with the creator in emotional or spiritual.Pursuit of Zero DistanceThe beauty of the distance makes you discover the beauty,and will eventually induce you to approach the beauty,even close to the beauty without distance.Su Shis“Xi Lin Wall”From the side,a whole range;from the end,a single peak,Far,near,high,low,no two parts alike.Why cant I tell the true shape of Lu-shan?Because I myself am in the mountain.Su contacted directly with nature,and the distance between them is“zero”.Dont you think it is beautiful?Thank you!
