flat-unit-iteratively-cumulated-icosahedron-pre-assembly : 無料・フリー素材/写真
flat-unit-iteratively-cumulated-icosahedron-pre-assembly / Ardonik
| ライセンス | クリエイティブ・コモンズ 表示-継承 2.1 |
|---|---|
| 説明 | To create a Koch Curve, start with a line segment, subdivide it into three equal segments, and replace the center segment with a triangular peak. The original line segment goes from having three equal parts to having four, increasing the perimeter by 33%. Applying the algorithm an infinite number of times yields the true curve, which has infinite length but encloses a finite area. If we start with an equilateral triangle and apply this algorithm to all three sides, we get the well-known Koch Snowflake fractal. What is the third-dimensional analog of this "snowflake" figure? Perhaps instead of starting with a line segment, we might start with a triangle, subdividing it into four equal triangles and replacing the central triangle with a tetrahedral peak. Thus the original triangle goes from having four equal parts to having six, increasing the surface area by 33%. And, of course, applying the algorithm an infinite number of times would create a curve with an infinite surface area though it would still enclose a finite volume. If we start with a regular tetrahedron and apply this algorithm to all four faces, we get...something. But what? Naturally, I'm not the first person to ask this question. As far as I can tell, the consensus is that this shape, or something like it, is the answer. All of the peaks end up touching each other and leaving no gaps, so the 3D analog of the Koch Snowflake ends up being a plain old, ordinary cube (albeit one with infinite surface area). You wouldn't even be able to tell that it was a fractal from a distance. How very disappointing! In order to keep things interesting, I modified the algorithm. I would generate the initial peak in the first iteration, yes, but instead of applying the algorithm to all six of the new faces, I would "lock" the three outer triangles (the large triangles in the photo) and only subdivide the three triangles belonging to the central peak. This would give the nascent polyhedron some much-needed "breathing room" and keep the peaks from touching one another in that boring, interlocking way. And while this process could be continued indefinitely, I decided to only fold a representation of this shape at its second iteration for the sake of my own sanity and paper supply. One more thing: instead of starting with a run-of-the-mill tetrahedron, I decided to start with an icosahedron just for the heck of it. The result would obviously no longer be self-similar throughout and would thus be unworthy of the name "fractal". Oh, well! |
| 撮影日 | 2011-05-08 23:24:10 |
| 撮影者 | Ardonik |
| タグ | |
| 撮影地 | |
| カメラ | Canon PowerShot A495 , Canon |
| 露出 | 0.017 sec (1/60) |
| 開放F値 | f/3.0 |
| 焦点距離 | 15136.92941 dpi |
