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Paradoxes
Schrödinger’s Cat
Koch Snowflake
Coastline of Ireland
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ParadoxesParadoxes• Mathematics cannot be applied to everything, and
cannot explain everything. • A paradox is a set of statements which contain
contradictions. • The existence of paradoxes causes problems for
mathematical logicians which they cannot solve.
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• 20 people live on an island. • Some people cut their own hair, the rest use
the barber who only cuts the hair of people who do not cut their own hair.
• 10 people never cut their own hair. • Draw a Venn diagram of the people on the
island:
Russell’s ParadoxRussell’s Paradox
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Let A = people who cut their own hairLet B = people who do not cut their own hair
Russell’s ParadoxRussell’s Paradox
A B
In which circle is the barber?
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• A cat is placed inside a box, a device which operates randomly is placed inside the box which will release a deadly gas killing the cat.
• There is no way of knowing without opening the box if the cat is alive or dead.
• Logically the cat is either alive or dead, so we simply open the box to find out which!
Schrödinger’s CatSchrödinger’s Cat
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Schrödinger’s CatSchrödinger’s Cat
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Koch SnowflakeKoch Snowflake
Helge von Koch (1870 – 1924)
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Draw a simple equilateral triangle
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On each of the three sides place another equilateral triangle
exactly one third and in the middle of the side
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Repeat the process
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Derive a general formula for the perimeter of the nth curve in this sequence, Pn .
P0 = L
13
L––
13
L–– 13
L––
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13
L––
13
L–– 13
L––
43
L––P1 =
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43
L––P2 =2
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43
L––P1 =43
L––P2 =2
43
L––P3 =3
43
L––Pn =n
P0 = L
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43
L––P1 =
43
L––P2 =2
43
L––P3 =3
43
L––P4 =4
43
L––P5 =5
43
L––P6 =6
43
L––Pn =n
The perimeter of the curve is infinite.
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The area An of the nth curve is finite. This can be seen by
constructing the circumscribed circle about
the original triangle as shown.
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43
L––P1 =43
L––P2 =2
43
L––P3 =3
P0 = L
It is a surprising fact that the perimeter of the curve is infinite but the area is finite.
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Use a spreadsheet to compute the first 50 values for the perimeter. Set P0 = 1
1 1.33 11 23.68 21 420.45 31 7466.22 41 132583.11
2 1.78 12 31.57 22 560.60 32 9954.96 42 176777.48
3 2.37 13 42.09 23 747.47 3313273.2
8 43 235703.31
4 3.16 14 56.12 24 996.62 3417697.7
1 44 314271.07
5 4.21 15 74.83 251328.8
3 3523596.9
5 45 419028.10
6 5.62 16 99.77 261771.7
7 3631462.5
9 46 558704.13
7 7.49 17133.0
3 272362.3
6 3741950.1
2 47 744938.84
8 9.99 18177.3
8 283149.8
1 3855933.5
0 48 993251.79
9 13.32 19236.5
0 294199.7
5 3974578.0
0 491324335.7
2
10 17.76 20315.3
4 305599.6
7 4099437.3
3 501765780.9
6
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Coastline of IrelandCoastline of Ireland• How long is the coastline of Ireland?• If we use a 10 km stick to measure it, we might get
2,500 km. • As we measure it in more and more detail going in
and around every cove it will grow to 25,000 km.• If we look close enough we can make it 100,000 km
or any length we wish!• The coast line of Ireland is infinite, just like the
perimeter of the Von Koch Snowflake curve…
Yes No
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