Koch's Snowflake
 C
 Robust
 Posts: 58640
 Joined: 22 Jul 2003, 19:06
Koch's Snowflake
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?!
.
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?!
.
kath wrote:(she squints, focusing all her concentration...)
inn dooooob ittt ableeee.
 C
 Robust
 Posts: 58640
 Joined: 22 Jul 2003, 19:06
Re: Koch's Snowflake
This is a nice fractal
.
.
kath wrote:(she squints, focusing all her concentration...)
inn dooooob ittt ableeee.

 Dribbling idiot airhead
 Posts: 18043
 Joined: 26 Dec 2009, 21:22
Re: Koch's Snowflake
Buckminster Fuller
kath wrote: *which is the real reason he can fucque off and rot for the rest of time.
Jimbo wrote: So Kath, put on your puka love beads ... Then go fuque yourself.

 Posts: 1346
 Joined: 05 Jul 2017, 23:05
Re: Koch's Snowflake
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?
 Fonz
 Posts: 3910
 Joined: 17 Feb 2014, 14:10
 Location: Nevermore
Re: Koch's Snowflake
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.
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.
Heyyyy!
"Fonz clearly has no fucks to give. I like the cut of his Cupicidal gib."
"Fonz clearly has no fucks to give. I like the cut of his Cupicidal gib."
 Fonz
 Posts: 3910
 Joined: 17 Feb 2014, 14:10
 Location: Nevermore
Re: Koch's Snowflake
Btw
Halloween 81
Halloween 81
Heyyyy!
"Fonz clearly has no fucks to give. I like the cut of his Cupicidal gib."
"Fonz clearly has no fucks to give. I like the cut of his Cupicidal gib."
 C
 Robust
 Posts: 58640
 Joined: 22 Jul 2003, 19:06
Re: Koch's Snowflake
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:
onethird equals 0.3333333... [0.3 recurring]. Nobody would dispute that
So twothirds is twice onethird 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_⋯
.
.
kath wrote:(she squints, focusing all her concentration...)
inn dooooob ittt ableeee.

 Posts: 1346
 Joined: 05 Jul 2017, 23:05
Re: Koch's Snowflake
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:
onethird equals 0.3333333... [0.3 recurring]. Nobody would dispute that
So twothirds is twice onethird 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_⋯
.
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.
 C
 Robust
 Posts: 58640
 Joined: 22 Jul 2003, 19:06
Re: Koch's Snowflake
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
.
kath wrote:(she squints, focusing all her concentration...)
inn dooooob ittt ableeee.
 Fonz
 Posts: 3910
 Joined: 17 Feb 2014, 14:10
 Location: Nevermore
Re: Koch's Snowflake
I’m with PP.
The ‘recurring’ suffix is just a shorthand convenience for an infinite number of 3s, 6s, or 9s.
The ‘recurring’ suffix is just a shorthand convenience for an infinite number of 3s, 6s, or 9s.
Heyyyy!
"Fonz clearly has no fucks to give. I like the cut of his Cupicidal gib."
"Fonz clearly has no fucks to give. I like the cut of his Cupicidal gib."

 Posts: 1346
 Joined: 05 Jul 2017, 23:05
Re: Koch's Snowflake
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.
I love the philosphy of 0.999etc = 1.
But what I really want is an answer to my question about infinity.
 C
 Robust
 Posts: 58640
 Joined: 22 Jul 2003, 19:06
Re: Koch's Snowflake
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...
.
kath wrote:(she squints, focusing all her concentration...)
inn dooooob ittt ableeee.
 C
 Robust
 Posts: 58640
 Joined: 22 Jul 2003, 19:06
Re: Koch's Snowflake
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.....?
.
kath wrote:(she squints, focusing all her concentration...)
inn dooooob ittt ableeee.
 Fonz
 Posts: 3910
 Joined: 17 Feb 2014, 14:10
 Location: Nevermore
Re: Koch's Snowflake
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?
Heyyyy!
"Fonz clearly has no fucks to give. I like the cut of his Cupicidal gib."
"Fonz clearly has no fucks to give. I like the cut of his Cupicidal gib."
 C
 Robust
 Posts: 58640
 Joined: 22 Jul 2003, 19:06
Re: Koch's Snowflake
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  ...?
.
kath wrote:(she squints, focusing all her concentration...)
inn dooooob ittt ableeee.
 Fonz
 Posts: 3910
 Joined: 17 Feb 2014, 14:10
 Location: Nevermore
Re: Koch's Snowflake
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.
If you’re so sure that you would get to the doorway, you should be able to tell me when.
Heyyyy!
"Fonz clearly has no fucks to give. I like the cut of his Cupicidal gib."
"Fonz clearly has no fucks to give. I like the cut of his Cupicidal gib."
 C
 Robust
 Posts: 58640
 Joined: 22 Jul 2003, 19:06
Re: Koch's Snowflake
Fonz wrote:The size of the doorway is irrelevant to the discussion, isn’t it?
Correct
It was an attempt at humour
.
kath wrote:(she squints, focusing all her concentration...)
inn dooooob ittt ableeee.
 Fonz
 Posts: 3910
 Joined: 17 Feb 2014, 14:10
 Location: Nevermore
Re: Koch's Snowflake
Heyyyy!
"Fonz clearly has no fucks to give. I like the cut of his Cupicidal gib."
"Fonz clearly has no fucks to give. I like the cut of his Cupicidal gib."
 Charlie O.
 Posts: 41611
 Joined: 21 Jul 2003, 19:53
 Location: InABaddaLaWadda, baybeh
 C
 Robust
 Posts: 58640
 Joined: 22 Jul 2003, 19:06
Re: Koch's Snowflake
Charlie O. wrote:
.
kath wrote:(she squints, focusing all her concentration...)
inn dooooob ittt ableeee.