Paradoxes Schrödinger’s Cat Koch Snowflake Coastline of Ireland

Paradoxes 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.

Russell’s Paradox 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 Paradox Let A = people who cut their own hair Let B = people who do not cut their own hair A B In which circle is the barber?

Schrödinger’s Cat 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! But, as we enlarge a fractal object we keep seeing ever smaller pieces. For example, this series of pictures could show first the inside of the intestine, then the crypts between the cells, then the microvilli on each cell. The smaller pieces are copies of the larger pieces. They are not exact smaller copies, but they are smaller replicas that are kind-of-like the larger pieces.

Schrödinger’s Cat But, as we enlarge a fractal object we keep seeing ever smaller pieces. For example, this series of pictures could show first the inside of the intestine, then the crypts between the cells, then the microvilli on each cell. The smaller pieces are copies of the larger pieces. They are not exact smaller copies, but they are smaller replicas that are kind-of-like the larger pieces.

Koch Snowflake Helge von Koch (1870 – 1924)

Draw a simple equilateral triangle

On each of the three sides place another equilateral triangle exactly one third and in the middle of the side

Repeat the process

1 3 L –– P0 = L 1 3 L –– Derive a general formula for the perimeter of the nth curve in this sequence, Pn . 1 3 L ––

1 3 L –– 1 3 L –– 4 3 L –– P1 = 1 3 L ––

4 3 L –– P2 = 2 ö æ ø è

ö æ ø è ö æ ø è ö æ ø è 4 3 L –– P1 = 4 3 L –– P2 = P0 = L 4 3 L –– Pn = n ö æ ø è

The perimeter of the curve is infinite. 4 3 L –– P6 = 6 ö æ ø è 4 3 L –– P5 = 5 ö æ ø è 4 3 L –– P4 = ö æ ø è 4 3 L –– P3 = ö æ ø è 4 3 L –– Pn = n ö æ ø è 4 3 L –– P2 = 2 ö æ ø è The perimeter of the curve is infinite. 4 3 L –– P1 =

The area An of the nth curve is finite The area An of the nth curve is finite. This can be seen by constructing the circumscribed circle about the original triangle as shown.

ö æ ø è ö æ ø è 4 3 L –– P1 = 4 3 L –– P2 = P0 = L It is a surprising fact that the perimeter of the curve is infinite but the area is finite. 4 3 L –– P3 = ö æ ø è

Use a spreadsheet to compute the first 50 values for the perimeter 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 33 13273.28 43 235703.31 4 3.16 14 56.12 24 996.62 34 17697.71 44 314271.07 5 4.21 15 74.83 25 1328.83 35 23596.95 45 419028.10 6 5.62 16 99.77 26 1771.77 36 31462.59 46 558704.13 7 7.49 17 133.03 27 2362.36 37 41950.12 47 744938.84 8 9.99 18 177.38 28 3149.81 38 55933.50 48 993251.79 9 13.32 19 236.50 29 4199.75 39 74578.00 49 1324335.72 10 17.76 20 315.34 30 5599.67 40 99437.33 50 1765780.96

Coastline 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…