Something to start with....... What is the same and what is different ......
Masterclasses Autumn 2013
Masterclasses Autumn 2013
Topology..... ....is the study of shapes. Specifically, it is the study of the properties that do not change when the shapes are twisted or stretched. Size and proportion have no meaning in topology. A small oval is the same as an enormous circle. A sphere the size of the sun is the same as a dumbell you hold in your hand. To topologists, what matters is the number of holes and twists. Masterclasses Autumn 2013
Möbius strip Take one strip Twist the strip once Tape the two ends together - you have a Möbius strip
Masterclasses Autumn 2013
Step 1 Take your Möbius strip Cut it along the middle of the strip Make a prediction – what will you get? Now cut your strip … was your prediction correct? One strip – one half-turn – cut in two → 1 strip, length doubled http://www.youtube.com/watch?v=6dEnz4tSKNk&feature=related
Masterclasses Autumn 2013
Step 2 - investigating What happens if you cut one third from the edge – one quarter from the edge – one fifth… Make a prediction – what will you get? Now cut your strips Work as a team, share the cutting
… were your predictions correct? How have you organised your results? http://www.youtube.com/watch?v=6dEnz4tSKNk&feature=related
Masterclasses Autumn 2013
Twister What happens if you do the same with 2 twists? Make a prediction – what will you get? What happens if you try 3 twists?
Double cross What happens if you do start with a cross? Make a prediction – what will you get?
Two loops with no twists… …one twist in one loop… …one twist in each loop…
http://www.youtube.com/watch?v=JNtKcK27x1s from 5 mins in
Masterclasses Autumn 2013
Some results Halftwists 1 1
1 1
Placing the cuts ÷2 ÷3
1 1
2 2
÷4 ÷5
2 2 2
1 2 3
÷2 ÷3 ÷4
Cuts
How many bands 1 1 and 1 2 2 and 1 2 3 4
Masterclasses Autumn 2013
Band length 2 2 1 2 2 1 1 1 1
Moebius Strips
Masterclasses Autumn 2013
Moebius Strips
Conveyor belts – why? Masterclasses Autumn 2013
Escher
Masterclasses Autumn 2013
Mobius strip discovered by… Johann Benedict Listing Discovered 1858
August Ferdinand Mobius Discovered 1858 … but later Masterclasses Autumn 2013
Möbius facts German Astronomer born 1790, died 1868 Descended from Martin Luther (church reformer who began Reformation)
Studied at Leipzig including for Gauss A crater on the moon is named after him, and an asteroid… … and many mathematical concepts/constructs Masterclasses Autumn 2013
Further exploration
Torus Paradromic ring Klein bottle Masterclasses Autumn 2013
Masterclasses Autumn 2013
A Topological Rope Trick
Masterclasses Autumn 2013
What is the least number of colours needed to colour in the map of Australia? States sharing a border cannot be the same colour.
Masterclasses Autumn 2013
Masterclasses Autumn 2013
The ‘maps’ could be simplified….
Masterclasses Autumn 2013
More Map Colouring....... TASK 1
TASK 2
Invent a country with states where the minimal number of colours needed is four.
Invent a country with states where the minimal number of colours needed is five.
Masterclasses Autumn 2013
The Four Colour Theorem 1852: Augustus De Morgan (1806-1871) 1878: Arthur Cayley (1821-1895) 1879: Alfred Kempe (1849-1922) 1880: P.G. Tait (1831-1901) – Knot Theory 1976 at the University of Illinois, Kenneth Appel and Wolfgang Haken eventually completed a solution to the Four Colour Theorem using a computer to test all map configurations. Masterclasses Autumn 2013
Some uses of Topology • It has made huge contributions to biology where it helps to describe and understand how proteins, DNA and other molecules fold and twist. • Cosmologists need topology to determine the shape of the Universe. • Topology is vital in understanding the structure of graphs in network science. Masterclasses Autumn 2013
The London Tube Map Designed by HARRY BECK in 1933 Masterclasses Autumn 2013
