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.

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 5 9

8 62

73 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

4

65

2 3

7

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)*(qx-px) + (qy-py)*(qy-py)) in the plane.

Neighbour in Digital

We can say the neighborhood of 5 is the numbers 1, 2, 3, 4, 6, 7, 8, 9.

1 2 3

4 5 6

7 8 9

Neighbour in Digital

Or we can only include 2, 4, 6, 8 as the neighborhood of 5.

1 2 3

4 5 6

7 8 9

What is the Creativity?

Think of how the neighbours of 5 affect the value of 5 itself.

1 2 3

4 5 6

7 8 9

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

By John Conway,

http://www.math.com/students/wonders/life/life.html

1 2 3

4 5 6

7 8 9

Game of Life - The Rules

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.

1 2 3

4 5 6

7 8 9

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

PING www.netvigator.com (218.102.23.36): 56 data bytes64 bytes from 218.102.23.36: icmp_seq=0 ttl=252 time=20.355 ms64 bytes from 218.102.23.36: icmp_seq=1 ttl=252 time=19.894 ms64 bytes from 218.102.23.36: icmp_seq=2 ttl=252 time=20.117 ms64 bytes from 218.102.23.36: icmp_seq=3 ttl=252 time=17.994 ms64 bytes from 218.102.23.36: icmp_seq=4 ttl=252 time=18.682 ms64 bytes from 218.102.23.36: icmp_seq=5 ttl=252 time=18.263 ms64 bytes from 218.102.23.36: icmp_seq=6 ttl=252 time=18.311 ms64 bytes from 218.102.23.36: icmp_seq=7 ttl=252 time=18.213 ms64 bytes from 218.102.23.36: icmp_seq=8 ttl=252 time=18.636 ms64 bytes from 218.102.23.36: icmp_seq=9 ttl=252 time=18.768 ms64 bytes from 218.102.23.36: icmp_seq=10 ttl=252 time=17.535 ms

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)

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/~jevb

ratt/1_to_1/index_ng.html

Creative Applications

• Babel• By Simon Biggs

• http://hosted.simonbiggs.easynet.co.uk/babel/babel.htm