Скачать или смотреть Fill in the Void with Golden Ratio ~ 4K - 1 Million Frames

𝗧𝗵𝗲 𝗦𝗲𝘁𝘂𝗽 :

The setup features two rotating arms: the outer arm rotates precisely "Golden Ratio" times faster than the inner arm throughout the entire animation.
This visually represents the intricate interplay between irrationality and infinity . As theta approaches infinity , the distance between adjacent lines approaches zero , but it never reaches that point.
In an ideal scenario, the simulation can run for days, weeks, years, or even decades, yet there will always be an infinitely thin gap between two adjacent lines, disregarding computation limits and floating-point precision.

𝗥𝗲𝗻𝗱𝗲𝗿𝗲𝗿 :

The simulation was created with programming using Python, utilizing the Matplotlib library for rendering. It consists of 50+ short clips seamlessly stitched together, with a total rendering time exceeding around 150 hours on my old laptop.

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𝙆𝙚𝙮 𝙈𝙤𝙢𝙚𝙣𝙩𝙨 :

00:00 equation & credits
10:41 1st near miss
17:20 2nd near miss
28:20 3rd near miss
46:10 4th near miss
01:14:45 5th near miss
01:32:30 6th near miss
01:43:40 7th near miss
02:01:10 8th near miss
02:30:00 9th near miss
02:48:42 10th near miss
03:00:30 1st zoom in 15x
04:23:20 11th near miss
04:24:50 zoom out
04:25:31 fast forward 15x. The curve becomes zig-zag line because of the lack of sample points.
04:37:50 more sample points are added manually to the curve before the final zoom to correct(smoothen) the zig-zag line.
04:38:00 last zoom in 10x

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   / @fascinating.fractals  

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𝗧𝗵𝗶𝘀 𝘀𝗲𝗰𝘁𝗶𝗼𝗻 𝗶𝘀 𝗿𝗲𝘃𝗶𝘀𝗲𝗱 𝗯𝘆 𝗖𝗵𝗮𝘁𝗚𝗣𝗧 , 𝗴𝗶𝘃𝗲𝘀 𝗺𝗼𝗿𝗲 𝗱𝗲𝘁𝗮𝗶𝗹𝘀 𝗮𝗯𝗼𝘂𝘁 𝘁𝗵𝗲 𝗚𝗼𝗹𝗱𝗲𝗻 𝗥𝗮𝘁𝗶𝗼 𝗳𝗿𝗼𝗺 𝗱𝗶𝗳𝗳𝗲𝗿𝗲𝗻𝘁 𝗽𝗲𝗿𝘀𝗽𝗲𝗰𝘁𝗶𝘃𝗲𝘀 : ⬇️

What makes the Golden Ratio unique from a mathematical perspective?

○ 𝙄𝙧𝙧𝙖𝙩𝙞𝙤𝙣𝙖𝙡𝙞𝙩𝙮 : The golden ratio is an irrational number, meaning it cannot be expressed as a fraction of two integers. Its decimal representation goes on forever without repeating, making it a non-terminating, non-repeating decimal.
○ 𝘼𝙡𝙜𝙚𝙗𝙧𝙖𝙞𝙘 𝙋𝙧𝙤𝙥𝙚𝙧𝙩𝙮 : The golden ratio has a unique algebraic property. It is the positive solution to the quadratic equation ϕ = 1 + 1/ϕ , which contributes to its distinctive value and mathematical significance.
○ 𝙎𝙚𝙡𝙛-𝙎𝙞𝙢𝙞𝙡𝙖𝙧𝙞𝙩𝙮 : The golden ratio exhibits a self-similar property. When you divide a line into two parts such that the ratio of the whole line to the longer segment is the same as the ratio of the longer segment to the shorter segment, you get the golden ratio.
○ 𝙂𝙚𝙤𝙢𝙚𝙩𝙧𝙞𝙘 𝘾𝙤𝙣𝙨𝙩𝙧𝙪𝙘𝙩𝙞𝙤𝙣 : The golden ratio can be geometrically constructed using a compass and straightedge by creating a specific relationship between the side and diagonal of a regular pentagon.
○ 𝙁𝙞𝙗𝙤𝙣𝙖𝙘𝙘𝙞 𝘾𝙤𝙣𝙣𝙚𝙘𝙩𝙞𝙤𝙣 : The golden ratio is intimately connected to the Fibonacci sequence, with the ratio of consecutive Fibonacci numbers converging to the golden ratio as the sequence progresses.

What makes the Golden Ratio unique from a nature perspective?

○ 𝙋𝙝𝙮𝙡𝙡𝙤𝙩𝙖𝙭𝙞𝙨 : The arrangement of leaves, petals, and seeds on plants often follows the Golden Ratio. This phenomenon is known as phyllotaxis, and the spiral patterns formed by these arrangements can be described using Fibonacci numbers and the Golden Ratio.
○ 𝙎𝙝𝙚𝙡𝙡 𝙎𝙥𝙞𝙧𝙖𝙡𝙨 : Certain mollusk shells, such as nautilus shells, exhibit spiral patterns that approximate the Golden Ratio. The growth of these shells follows a logarithmic spiral, and the ratio of successive radii corresponds to the Golden Ratio.
○ 𝙎𝙪𝙣𝙛𝙡𝙤𝙬𝙚𝙧𝙨 : The seed patterns in the center of a sunflower often follow a spiral arrangement based on the Fibonacci sequence, which is closely related to the Golden Ratio. The spiral arrangement allows each seed to have optimal access to sunlight.
○ 𝙋𝙞𝙣𝙚𝙘𝙤𝙣𝙚𝙨 𝙖𝙣𝙙 𝙋𝙞𝙣𝙚𝙖𝙥𝙥𝙡𝙚𝙨 : The spirals on pinecones and pineapples also demonstrate patterns consistent with the Fibonacci sequence and the Golden Ratio. The arrangement allows for efficient packing of seeds and optimal exposure to sunlight.
○ 𝙃𝙪𝙢𝙖𝙣 𝘽𝙤𝙙𝙮 𝙋𝙧𝙤𝙥𝙤𝙧𝙩𝙞𝙤𝙣𝙨 : Some studies suggest that the proportions of certain features in the human body, such as the length of the bones in a finger, exhibit relationships that approximate the Golden Ratio. However, this is a more debated and less universally accepted aspect of the Golden Ratio in nature.