ملخص

About this image

I made this animation Cuddlyable3 15:36, 13 March 2007 (UTC) The Koch curve, in its fully (infinitely) iterated form is not a line that anyone has ever seen! That is to say, it is not like any of the lines that we are familiar with in Euclidean geometry because it somehow spans a finite distance while being infinitely long. The cross-section of an ordinary 2-D line is a point, but the cross-section of the Koch curve would have to be a probability distribution. That makes displaying the Koch curve challenging and we may need to rethink what we mean by aliasing error. The quantising imposed to make this animation is:

• Resolution: 200 x 100 pixels
• Colours: Just 2 i.e. monochrome
• Time: 10 frames that recycle at 10 frames/second

Each of the above is a potential source of aliasing error, but I think the dominant cause of comments about this is the restricted colour scale. Any pixel that the Koch "distribution" touches, no matter how slightly, is painted black. It is that simple.

Other details of this animation:

• the line actually drawn is first described numerically by 4097 points i.e. a it is a highly developed but not infinite Koch curve. This model exceeds the display resolution so much that I am sure that further Koch iteration would make no difference.
• the points are joined in order by the Bresenham line-drawing algorithm. At this scale I think I would have got exactly the same result by merely plotting the points because there is no line span long enough for the B. algorithm to paint intermediate points, but now you know what I told the computer to do.
• the animation was assembled using The Gimp software into a .gif file of only 4 331 bytes.

Finally, I judiciously panned the view of the sequence so the last frame smoothly recycles to the first frame i.e. the center of the curve seems to remain still. Although that gives a nice subjective effect, it may encourage a misinterpretation that forms are being emitted from the center. What you see is really just a zoom and pan view of a strange but static "self-similar" object.Cuddlyable3 19:14, 7 May 2007 (UTC)

^I got rid of the deprecated tag, and I'll add this tag. Plus, nominated for featured.Temperal^xy 19:12, 6 May 2007 (UTC)

See my Discussion page for some variations of this animation. Cuddlyable3 19:07, 2 June 2007 (UTC)

ترخيص

أنا، مالِك حقوق تأليف ونشر هذا العمل، أجعله في النِّطاق العامِّ، يسري هذا في أرجاء العالم كلِّه. في بعض البلدان، قد يكون هذا التَّرخيص غيرَ مُمكنٍ قانونيَّاً، في هذه الحالة: أمنح الجميع حق استخدام هذا العمل لأي غرض دون أي شرط ما لم يفرض القانون شروطًا إضافية.