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File:KIFS OpenCL 8K HQ 20200503 13m 8,3s.png

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Original file ‎(8,000 × 4,500 pixels, file size: 46.22 MB, MIME type: image/png)

Captions

Captions

A kaleidoscopic IFS focusing on triangle shapes.

Summary[edit]

Description
Deutsch: Kaleidoskopisches Iteriertes Funktionensystem mit Fokus auf Dreiecksformen (abgeleitet vom Sierpinski-Oktaeder). Jeweils eine Reihe (siehe Bild) folgt in kleinerer Ausführung der unteren darunter. Bis zur der Kante (oben im Bild) befinden sich unendlich viele immer kleiner werdende Reihen. Erstellt mit mandelbulber2. Renderzeit: 13 Minuten und 8,3 Sekunden. Die oberste Reihe im Bild hat theoretisch die Höhe des Wertes einer Infinitesimalzahl im Bild, wenn die größte Reihe eine Längeneinheit groß ist. Das Phänomen kann man mit folgender Analogie erläutern: Angenommen man befindet sich 1m weit weg vom Ziel. In der ersten Stunde bewegt sich die Person 0,5 Meter voran. In jeder weiteren Stunde bewegt sie sich jeweils 50 % weiter, als sie sich in der vorangegangenen Stunde bewegt hat (Stunde 1: 0,5 Meter; Stunde 2: 0,25 Meter; Stunde 3: 0,125 Meter usw.; die Werte addieren sich zu einer Gesamtdistanz). Die Person nähert sich immer weiter dem Ziel an, erreicht es jedoch nie. Siehe dazu auch das Paradoxon von Achilles und der Schildkröte.
Das Bild zeigt:
  • Selbstähnlichkeit
  • Dreiecke
  • Unendlich viele identische Reihen, die sich nur in der Position und Größe unterschieden
  • Asymptotische Annäherung der Höhe der Reihen an die Gesamthöhe des Fraktals
  • Eine symmetrische Ausrichtung einer symmetrischen Form
English: A kaleidoscopic IFS focusing on triangle shapes. Created with mandelbulber2. Rendering duration: 13m 8.3s. Every line in the image is smaller than the lower one. The edge on the top of the image is the limit. Underneath it, there are infinitely many rows. The highest line in this fractal is infinitesimally large in comparison of the line 1 with 1 lenght unit height. This phenomenom is analog to the following: If you travel per step only the half of the length you travelled in the step before (in the first step you travel one meter), you will never finish 2 meters, but you get with every step closer. If you add here the heights of every line (beginning with line 1), you will never get the actual height of all lines, but you get with adding more and more lines closer.
This image shows:
  • Self-similarity
  • Triangles
  • An infinite number of identical lines (despite of position and size)
  • Asymptotic approximation via the height of the lines to the height of this image crop of the fractal.
  • Symmetric perspective of a symmetric shape
Date
Source Own work
Author PantheraLeo1359531
Other versions File:Kantenannäherung 4500p60 20200526.webm: Intensive Begutachtung des gleichen Fraktals in anderer Farbe

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Reihe 1 hat 2284 Pixel (es wird jeweils nur die untere Kante gemessen)
Reihe 2 hat 1066 Pixel (≈46,7%)
Reihe 3 hat 0516 Pixel (≈48,4%)
Reihe 4 hat 0254 Pixel (≈49,2%)
Reihe 5 hat 0126 Pixel (≈49,6%)
Reihe 6 hat 0063 Pixel (50%)
Reihe 7 hat 0031 Pixel
Reihe 8 hat 0016 Pixel
Reihe 9 hat 0008 Pixel

Kleinere Reihen sind hier nicht mehr messbar. Mit jeder höheren Reihe wird die Reihenhöhe niedriger. Angenommen, Reihe 1 ist 1 LE hoch, dann ist Reihe 2 0,5 LE hoch. Welche Reihe welche Höhe hat, lässt sich mit folgender Formel ermitteln: 2(-(Reihennummer-1)). Mit einer Summenfolge kann man so bis zu einer Reihe x die bisherige Höhe ermitteln. In der oberen Bildhälfte sind mehr Reihen, aber dafür kleinere. Das heißt, dass das oberste Viertel des Bildes unendlich viele Reihen zeigt, während die untere Hälfte nur etwa eine Reihe zeigt.

Line 1 has 2284 pixels (only the lower edge is measured here, not the upper one)
Line 2 has 1066 pixels (≈46.7%)
Line 3 has 0516 pixels (≈48.4%)
Line 4 has 0254 pixels (≈49.2%)
Line 5 has 0126 pixels (≈49.6%)
Line 6 has 0063 pixels (50%)
Line 7 has 0031 pixels
Line 8 has 0016 pixels
Line 9 has 0008 pixels

Smaller lines are not measurable here anymore (image resolution too small). With every higher line, the line height gets smaller. When line 1 has a height of 1 lenght unit, then line 2 ist 0.5 length units high. Depending on the line number, you can calculate its height with the following formula: 2(-(Line number-1)). With a sum sequence, you can measure the height until line x. In the upper image half, there are more but smaller lines, in the lower one are less lines but bigger ones.

Licensing[edit]

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w:en:Creative Commons
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Date/TimeThumbnailDimensionsUserComment
current17:57, 3 May 2020Thumbnail for version as of 17:57, 3 May 20208,000 × 4,500 (46.22 MB)PantheraLeo1359531 (talk | contribs)32 Bit
17:56, 3 May 2020Thumbnail for version as of 17:56, 3 May 20208,000 × 4,500 (20.82 MB)PantheraLeo1359531 (talk | contribs)Tiefenkarte
17:54, 3 May 2020Thumbnail for version as of 17:54, 3 May 20208,000 × 4,500 (9.64 MB)PantheraLeo1359531 (talk | contribs)Diffuse-Kanal.
17:25, 3 May 2020Thumbnail for version as of 17:25, 3 May 20208,000 × 4,500 (158.59 MB)PantheraLeo1359531 (talk | contribs)Uploaded own work with UploadWizard

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