Math Is Fun Forum

Kurre

Member

Registered: 2006-07-18

Posts: 280

Perfect Squares forming some kind of wave

i was programming a program that can generate all numbers, then delete all primes/squares/palindromes etc, or keep them, and also painting a picture with dots at the numbers positions (and no dot where a nmber has been deleted).
when playing around with it i noticed a very cool thing
i rendered all numbers from 0 to 10000. then it looks like this:
http://www.geocities.com/tibiihost/all_numbers.bmp
every green dot is a number, and it is 10 numbers (10 pixels) wide. so for example the first line are numbers 0 to 9, 2nd are 10 to 19, the last line is 990 to 999 and the last dot is 1000 etc.

then i kept all perfect squares, removed all other numbers:
http://www.geocities.com/tibiihost/all_Squares.bmp
look at the pattern! it forms some kind of wave. btu that isnt everything
i removed all odd numbers, kept all even squares, and the dots remaining formed another wave shape, but reversed
http://www.geocities.com/tibiihost/all_Even_Squares.bmp
cool eh??

Patrick

Real Member

Registered: 2006-02-24

Posts: 1,005

Re: Perfect Squares forming some kind of wave

neat indeed

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krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Perfect Squares forming some kind of wave

Interesting. Here's a picture 10x10000:

Last edited by krassi_holmz (2006-08-02 07:34:54)

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IPBLE:  Increasing Performance By Lowering Expectations.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Perfect Squares forming some kind of wave

You mean 10x1000, no?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Perfect Squares forming some kind of wave

Here's better:(It's flipped)

Last edited by krassi_holmz (2006-08-02 08:42:13)

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IPBLE:  Increasing Performance By Lowering Expectations.

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Perfect Squares forming some kind of wave

You mean 10x1000, no?

Yes

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IPBLE:  Increasing Performance By Lowering Expectations.

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Perfect Squares forming some kind of wave

Connecting the points:

Last edited by krassi_holmz (2006-08-02 08:47:06)

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IPBLE:  Increasing Performance By Lowering Expectations.

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Perfect Squares forming some kind of wave

I have played for a while with mathematica, and I founded interesting patterns.
The code:

ToCoords[x_][c_] := {Floor[c/x], c - x Floor[c/x]};
SqueezedDotPlot[x_, list_, ops___] := Show[Graphics[Point /@ (ToCoords[x] /@ list)], ops];
SqueezedLinePlot[x_, list_, ops___] := Show[Graphics[Line[ToCoords[x] /@ list]], ops];

Here I will actually use only the first plotting function.

Picture 1: this is the actual square function.

SqueezedDotPlot[10, Range[100]^2]

Picture 2: If you use irrational numbers, there can be interestiong results.

SqueezedDotPlot[4, Range[1000]^Sqrt[2]]

Picture 3: But for most of the numbers, you will get the ordinary noise.

SqueezedDotPlot[Sqrt[10], Range[10000]^3]

Picture 4: Some non-trivial structure

SqueezedDotPlot[Sqrt[2], Range[10000]^2]

Picture 5: The previous zoomed

SqueezedDotPlot[Sqrt[2], Range[1000]^2]

Picture 6: Noise again, but different from the ordinary.

SqueezedDotPlot[3, Range[10000]^1.5]

Picture 7: And what if the exponent is smaller than 1?

SqueezedDotPlot[2, Range[10000]^0.9]

Picture 8: interesting...

SqueezedDotPlot[1, Range[10000]^0.99]

Picture 9: What a wave!

SqueezedDotPlot[2, Range[10000]^0.999]

Picture 10: this is beautiful!!!

SqueezedDotPlot[0.1, Range[10000]^0.5]

Last edited by krassi_holmz (2006-08-02 09:55:32)

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IPBLE:  Increasing Performance By Lowering Expectations.

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Perfect Squares forming some kind of wave

I'll upload the pictures soon...

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IPBLE:  Increasing Performance By Lowering Expectations.

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: Perfect Squares forming some kind of wave

http://www.geocities.com/tibiihost/Squarelines.bmp
seems like the odd numbers are like the line including all numbers but much more angular with less dots
edit :

I'll upload the pictures soon...

nice, cool, im waiting

Last edited by Kurre (2006-08-02 09:57:54)

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Perfect Squares forming some kind of wave

Pictures 1-5:

Last edited by krassi_holmz (2006-08-02 09:58:38)

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IPBLE:  Increasing Performance By Lowering Expectations.

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Perfect Squares forming some kind of wave

Pictures 6-10:

Last edited by krassi_holmz (2006-08-02 10:05:37)

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IPBLE:  Increasing Performance By Lowering Expectations.