Koch's Snowflake

in reality, all of this has been a total load of old bollocks
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Koch's Snowflake

Postby C » 03 Aug 2020, 13:59

kath's thread regarding her avatar and love of animated fractals reminded me of Koch's Snowflake that I have used in training a few times.

The really interesting concept regarding Koch's Snowflake is that it has an infinite perimeter but a finite area.

This animation shows that nicely where the first seven iterations are shown.

https://en.wikipedia.org/wiki/Koch_snow ... _curve.gif

A fractal can be generated ad infinitum so the perimeter will continue to increase for ever and ever.

However, as you can see the Snowflake is contained within the white square and therefore the area is enclosed within that square and will never 'burst out'. Therefore the area is finite (it doesn't increase forever).

In the example shown the square looks, on my screen, about 8cm by 8cm so the area will always be less than 64 sq cm

Similarly, the area of kath's avatar will remain in the square and will never spill out onto her posts!

Good ain't it?!




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Re: Koch's Snowflake

Postby C » 15 Aug 2020, 14:09

This is a nice fractal

Image






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Re: Koch's Snowflake

Postby Jimbo » 15 Aug 2020, 15:19

Image

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Re: Koch's Snowflake

Postby Positive Passion » 15 Aug 2020, 19:35

C wrote:kath's thread regarding her avatar and love of animated fractals reminded me of Koch's Snowflake that I have used in training a few times.

The really interesting concept regarding Koch's Snowflake is that it has an infinite perimeter but a finite area.

This animation shows that nicely where the first seven iterations are shown.

https://en.wikipedia.org/wiki/Koch_snow ... _curve.gif

A fractal can be generated ad infinitum so the perimeter will continue to increase for ever and ever.

However, as you can see the Snowflake is contained within the white square and therefore the area is enclosed within that square and will never 'burst out'. Therefore the area is finite (it doesn't increase forever).

In the example shown the square looks, on my screen, about 8cm by 8cm so the area will always be less than 64 sq cm

Similarly, the area of kath's avatar will remain in the square and will never spill out onto her posts!

Good ain't it?!




.


I am no mathematician, and I appreciate terms like "infinite" may have a technical meaning, but you say the area can't increase forever, so therefore it is finite. If the area can decrease forever, would that make infinite? Surely for something to be finite there has to be a lower limit as well as an upper limit?

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Re: Koch's Snowflake

Postby Fonz » 15 Aug 2020, 22:57

We’ve had this argument before.

The notion that, if you started at the centre of a circle, and jumped halfway to the edge, over a number of jumps (always jumping halfway from your new position towards the edge) you’d eventually get to the edge.

I don’t think you’d ever arrive there. The distance between you and the edge would get ever smaller, but you’d never get there.

Similarly, the square would never fill up.
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Re: Koch's Snowflake

Postby Fonz » 15 Aug 2020, 22:58

Btw

Halloween 81
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Re: Koch's Snowflake

Postby C » 15 Aug 2020, 23:19

Positive Passion wrote:Surely for something to be finite there has to be a lower limit as well as an upper limit?


Why?

The number of fingers on my right hand is finite

The number of grains of sand on a beach it is finite (although it would be quite difficult to count)

Something that is finite is countable.

It doesn't require limits.

The point about something which is infinite having a finite answer can best be seen with the fact that:

one-third equals 0.3333333... [0.3 recurring]. Nobody would dispute that

So two-thirds is twice one-third equals 2 times 0.3 recurring = 0.6666666... [0.6 recurring]Nobody would dispute that

So three thirds is three times 0.3 recurring equals 0.9999... [0.9 recurring]

But three thirds equals 1 which implies:

0.9999...... equals 1

True or false?

Well, logic tells us that if it was untrue then there must be something wrong with earlier steps in the process [which I think we agree there is not anything wrong]

Therefore, I suggest 0.999... equals 1.

It can be easily proven

Similarly Fonz's example 1/2 + 1/4 + 1/8 + 1/16 + .... sums to 1 [NOT a little bit short of 1 like 0.9 recurring NOT being a little be short of 1]

This is a lot more difficult to prove than my first example but ,if interested, it is shown here:

https://en.wikipedia.org/wiki/1/2_%2B_1 ... _1/16_%2B_






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Re: Koch's Snowflake

Postby Positive Passion » 16 Aug 2020, 06:21

C wrote:
Positive Passion wrote:Surely for something to be finite there has to be a lower limit as well as an upper limit?


Why?

The number of fingers on my right hand is finite

The number of grains of sand on a beach it is finite (although it would be quite difficult to count)

Something that is finite is countable.

It doesn't require limits.



Maybe I misunderstood, but you were the one who said if something can't increase forever, it is finite.

The point about something which is infinite having a finite answer can best be seen with the fact that:

one-third equals 0.3333333... [0.3 recurring]. Nobody would dispute that

So two-thirds is twice one-third equals 2 times 0.3 recurring = 0.6666666... [0.6 recurring]Nobody would dispute that

So three thirds is three times 0.3 recurring equals 0.9999... [0.9 recurring]

But three thirds equals 1 which implies:

0.9999...... equals 1

True or false?

Well, logic tells us that if it was untrue then there must be something wrong with earlier steps in the process [which I think we agree there is not anything wrong]

Therefore, I suggest 0.999... equals 1.

It can be easily proven

Similarly Fonz's example 1/2 + 1/4 + 1/8 + 1/16 + .... sums to 1 [NOT a little bit short of 1 like 0.9 recurring NOT being a little be short of 1]

This is a lot more difficult to prove than my first example but ,if interested, it is shown here:

https://en.wikipedia.org/wiki/1/2_%2B_1 ... _1/16_%2B_




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I don't think this answers my question, but as a side track, your comment about one third "equals" 0.333 recurring etc simply shows up a minor flaw in using base 10, in that base 10 terminology is not adequate to expressing a third. So one third does not "equal" 0.333 recurring, it is just that 0.333 recurring is the best approximation when one third is expressed using base 10 (which is the "something wrong" with the earlier steps in your process). It is not a "real" problem. 3 thirds equals 1. By contrast 3 times 333 thousandths does NOT equal 1, however much you try.
If you used base 3, base 9, base 12 etc the terminology problem you describe would not arise. Using base 3, a third would be 0.1, two thirds 0.2, and three thirds would be - ah! - 1. (though other problems in the same vein may arise with a different base).

My query really, I suppose, is "is the number of fractions infinite" -one half, one tenth, one hundred thousandth - and if so does that mean the area of the snowflake is infinite, because it could get infinitely smaller - an area a billionth cm x a billionth cm is smaller than an area a millionth cm x a millionth cm.

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Re: Koch's Snowflake

Postby C » 16 Aug 2020, 12:19

Positive Passion wrote:So one third does not "equal" 0.333 recurring, it is just that 0.333 recurring is the best approximation .


Untrue

I/3 = 0.3 recurring and it is exact and certainly not an approximation.

As for 0.9 recurring = 1 again, exact and not an approximation

Let x= 0.999999999999999999...

So 10x = 9.9999999999999999...

Subtract line 1 for line 2

We get
9x = 9

So x=1

Therefore 1 = 0.9 recurring







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Re: Koch's Snowflake

Postby Fonz » 16 Aug 2020, 19:40

I’m with PP.

The ‘recurring’ suffix is just a shorthand convenience for an infinite number of 3s, 6s, or 9s.
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Re: Koch's Snowflake

Postby Positive Passion » 16 Aug 2020, 19:50

I wrote a reply earlier on but lost it.

I love the philosphy of 0.999etc = 1.

But what I really want is an answer to my question about infinity.

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Re: Koch's Snowflake

Postby C » 17 Aug 2020, 08:03

Fonz wrote:I’m with PP.

The ‘recurring’ suffix is just a shorthand convenience for an infinite number of 3s, 6s, or 9s.


That is not true.

All sorts of numbers recur and a recurring digit can be anything 0 to 9.

Those that can be expressed as fractions {rational numbers} and those that can't {irrational numbers}

So 0.1717171717171717..... equals 17/99

or

0.123123123123123.... equals 123/999

or

0.15555555555555... equals 154/990

The above can be solved using the method I showed earlier

Finally, irrational numbers like π (pi) do not recur in a pattern:

The first few decimal places:

3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679

- that is why we use an approximation eg 22/7 or 3.14 [2dp]

Interestingly, the square root of any prime number is also irrational- that is the digits do not recur in a pattern so therefore can't be expressed as a fraction.

Any number that does recur in a pattern is rational - that includes all integers and terminating decimals

An integer: 6 equals 6.0000....

A terminating decimal: 0.5 equals 0.50000...







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Re: Koch's Snowflake

Postby C » 17 Aug 2020, 10:30

Positive Passion wrote:My query really, I suppose, is "is the number of fractions infinite" -one half, one tenth, one hundred thousandth - and if so does that mean the area of the snowflake is infinite, because it could get infinitely smaller - an area a billionth cm x a billionth cm is smaller than an area a millionth cm x a millionth cm.


The challenge with infinity is that it is not a number. If it were a number than that number would be finite.

The difficulty is that we never get to the end - as with the recurring 9s that equal 1. Children and many adults puzzle over that because they say it's only an approximation because the bit that is left off the end.

Mathematics when considering problems like 1/2+1/4+1/8 talk of 'tending to a limit'

For example what is 1 divided by 0?

Well, when I was at school we were told it is infinity (now we would say it was undefined)

But how can 1 divided by 0 = infinity?

Well consider this for 1/x where we start with x=2, 10 etc

1 divided by 2 = 0.5
1 divided by 10 = 0.1
1 divided by 100 = 0.01
1 divided by 1000 = 0.001
1 divided by 1000000 = 0.000001
1 divided by 10000000000 = 0.0000000001

We can see that as x gets larger 1/x gets smaller

So we can say that when x tends to infinity 1/x tends to 0

Now, it can be proven that all those little bits, that get smaller and smaller, that you mentioned do tend to a limit

So, if you had a frog in the centre of a pond and it jumped half the distance to the bank and then a quarter then an eighth etc it would eventually get to the other side. The proof I flagged up on wiki earlier and to which Fonz referred.

I try and explain it this way: I want to walk to the door - it is 6 feet away but I am going to do it in rapid steps first 3 feet, then 1.5 feet, then 0.75 feet etc. Do I ever get to the door? Well evidence suggests I do.....? ;)







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Re: Koch's Snowflake

Postby Fonz » 17 Aug 2020, 16:11

C wrote:I try and explain it this way: I want to walk to the door - it is 6 feet away but I am going to do it in rapid steps first 3 feet, then 1.5 feet, then 0.75 feet etc. Do I ever get to the door? Well evidence suggests I do.....? ;)


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Ok. Playing that game, let’s imagine that with every step you take you shrink to half your size.
When would you get to that notional doorway?
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Re: Koch's Snowflake

Postby C » 20 Aug 2020, 12:38

Fonz wrote:
C wrote:I try and explain it this way: I want to walk to the door - it is 6 feet away but I am going to do it in rapid steps first 3 feet, then 1.5 feet, then 0.75 feet etc. Do I ever get to the door? Well evidence suggests I do.....? ;)


.


Ok. Playing that game, let’s imagine that with every step you take you shrink to half your size.
When would you get to that notional doorway?


Does the doorway reduce in the form - 1/2 -1/4 -1/8 - ...?








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Re: Koch's Snowflake

Postby Fonz » 21 Aug 2020, 12:19

The size of the doorway is irrelevant to the discussion, isn’t it?

If you’re so sure that you would get to the doorway, you should be able to tell me when.
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Re: Koch's Snowflake

Postby C » 21 Aug 2020, 16:12

Fonz wrote:The size of the doorway is irrelevant to the discussion, isn’t it?


Correct

It was an attempt at humour





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Re: Koch's Snowflake

Postby Fonz » 21 Aug 2020, 20:04

:|
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Re: Koch's Snowflake

Postby Charlie O. » 21 Aug 2020, 20:10

:lol:
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Re: Koch's Snowflake

Postby C » 30 Aug 2020, 16:23

Charlie O. wrote: :lol:


;)







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