Transcript d (p1, p2)
SM2222 - Class 06 What is Space? Starting from number • A number, 7 • A sequence of numbers, 1, 2, 3, 5, 8, 13, 21, … • A collection of numbers. From number to space • Can this collection of numbers give you a sense of space? 1 8 5 9 2 7 6 3 4 From number to space • Can this collection of numbers give you a sense of space? 1 2 3 4 5 6 7 8 9 From symbol to space • Can this collection of dots give you a sense of space? From symbol to space • Can this collection of dots give you a sense of space? Characteristics • A collection of dots may not give you a sense of space. • Their relationship among themselves about the togetherness can give you a sense of space. The connection • The use of lines more emphasizes the togetherness. The connection • If we label the dots, we can easily notice the configuration. 1 2 3 6 7 4 5 The connection • The connectivity information can be written down in symbol form like, (1,2,4,5,6), (3,7) Note that it is not a formal Mathematical notation. Neighbour • We can say that 1, 2, 4, 5, 6 are neighbours and 3, 7 are also neighbours. (1,2,4,5,6), (3,7) Topology • The connectivity and neighborhood give the concept of topology. • Note that it is a very simplified description. Möbius Strip Escher Klein Bottle Characteristics • Shape, size, orientation are not important in this context. • The most important one is the connectivity. Applications London Tube Map A Quiz Space in real life • Shape and size may be important in some applications. Metric Space • Need the concept of distance. • That is, a function (d) which takes two points in the space and returns a real number. d (p1, p2) -> r Distance • • • • The distance function (d) obeys the following, d (p1, p1) = 0 d (p1, p2) = d (p2, p1) d (p1, p2) + d (p2, p3) >= d (p1, p3) Neighborhood • By using the concept of distance, we can redefine the neighborhood. • The neighborhood for a point p with a distance D are all the point x, such that • d (p, x) <= D Distance in 1D The distance function d = x - p in the real number line. Distance in 2D The distance function d = Math.sqrt((qx-px)*(qxpx) + (qy-py)*(qy-py)) in the plane. Neighbour in Digital 1 2 3 4 5 6 7 8 9 We can say the neighborhood of 5 is the numbers 1, 2, 3, 4, 6, 7, 8, 9. Neighbour in Digital 1 2 3 4 5 6 7 8 9 Or we can only include 2, 4, 6, 8 as the neighborhood of 5. What is the Creativity? 1 2 3 4 5 6 7 8 9 Think of how the neighbours of 5 affect the value of 5 itself. Neighbour in Action Convolution Matrix Convolution Matrix Scale : 1 0 -1 0 -1 5 -1 0 -1 0 Convolution Matrix Scale : 9 1 1 1 1 1 1 1 1 1 Convolution Matrix Scale : 1 0 0 0 -1 2 0 0 0 0 Convolution Matrix Scale : 1 0 1 0 1 -4 1 0 1 0 Game of Life 1 2 3 4 5 6 7 8 9 By John Conway, http://www.math.com/students/wonders/life/life.html Game of Life - The Rules 1 2 3 4 5 6 7 8 9 For a space that is 'populated': Each cell with one or no neighbors dies, as if by loneliness. Each cell with four or more neighbors dies, as if by overpopulation. Each cell with two or three neighbors survives. For a space that is 'empty' or 'unpopulated' Each cell with three neighbors becomes populated. Getting Virtual • Cyberspace • Virtual reality • Virtual community Cyberspace • Exists in the computer network, the Internet. • Can we apply the same spatial concept in this type of space? Internet • Connects computers together to form a network. • Each computer (node) has an address (IP address). • The lowest level of connectivity is real, e.g. Ethernet cable, etc. IP Address • Computers and other devices (like router, printer, webcam, etc.) have unique IP addresses, though some of them are private. • The format of IP address is, • aaa.bbb.ccc.ddd • where each of the aaa, bbb, ccc, ddd is a number between 0 to 255, though some numbers are reserved and cannot be used. IP Address Space • Each IP address can be treated as a point in a space. • Two IP addresses can be treated as neighbours if they are similar. • For example, 144.214.26.1 and 144.214.26.2. • It is very likely that they are in the same area and directly connected together. Network Topology Network Metrics • Since IP address does not correspond to physical location, we have to use different ways to measure distance. • Time (e.g. ping command) PING www.netvigator.com (218.102.23.36): 56 data bytes 64 bytes from 218.102.23.36: icmp_seq=0 ttl=252 time=20.355 ms 64 bytes from 218.102.23.36: icmp_seq=1 ttl=252 time=19.894 ms 64 bytes from 218.102.23.36: icmp_seq=2 ttl=252 time=20.117 ms 64 bytes from 218.102.23.36: icmp_seq=3 ttl=252 time=17.994 ms 64 bytes from 218.102.23.36: icmp_seq=4 ttl=252 time=18.682 ms 64 bytes from 218.102.23.36: icmp_seq=5 ttl=252 time=18.263 ms 64 bytes from 218.102.23.36: icmp_seq=6 ttl=252 time=18.311 ms 64 bytes from 218.102.23.36: icmp_seq=7 ttl=252 time=18.213 ms 64 bytes from 218.102.23.36: icmp_seq=8 ttl=252 time=18.636 ms 64 bytes from 218.102.23.36: icmp_seq=9 ttl=252 time=18.768 ms 64 bytes from 218.102.23.36: icmp_seq=10 ttl=252 time=17.535 ms Network Metrics • Between 2 computers in the Internet, data needs to travel through a number of devices, like the routers. • Distance can be measured in the number of Hops. Virtual and Real • Since IP address is registered with real world information like geographical details, it is possible to combine both the virtual and real world together. • http://www.visualroute.com/ • http://www.linkwan.com/vrhk/ Creativity Potential Creativity Potential Creativity Potential Creativity Potential Creativity Potential Creative Applications • A piece of text can be transformed into a space. • Valence by Ben Fry. • http://acg.media.mit.e du/people/fry/valence/ applet/ Creative Applications • A word and its related topics can also be visualized in a space. • Visual Thesaurus • http://www.visualthesa urus.com/ Creative Applications • P2P network visualization • Mini-tasking • http://www.sw.ofcd.com/ Creative Applications • Newsgroup discussion visualization • WebSOM • http://websom.hut.fi/web som/ Creative Applications • Every website • Lisa Jevbratt • http://128.111.69.4/~jevbr att/1_to_1/index_ng.html Creative Applications • Babel • By Simon Biggs • http://hosted.simonb iggs.easynet.co.uk/b abel/babel.htm