The Golden Ratio tends to be oversold in its occurrences. Solving $f = 1 \frac$ yields the Golden Ratio as before. We divide by our growth factor to find the previous element given the current one. We've described the Golden Ratio with different phrasing: the scaling factor (f) must equal the original (1) plus the previous item (1/f). The key relationship is we move from one blob to the next, such that multiplication and addition have the same effect: The "growing blob" can represent the length of a line, a 2d shape, an angle - which can lead to interesting patterns: Growth symmetry: Addition and multiplication change the pattern identicallyĪha! That's a nice combo if I ever saw one.Multiplication symmetry: Each element is a scaled version of the previous.Addition symmetry: Each element is the sum of the previous two.Visually, I see a growing blob, like this:Īs we scale by Phi each time (1.618) we get: Just describing a ratio doesn't call out the symmetry we're able to achieve. ![]() I prefer the "growth factor" scenario, where we start with a single item (1.0) and evolve it, while keeping it linked to its ancestors with both arithmetic and multiplication. Many descriptions of the Golden Ratio describe splitting a whole into parts, each of which is in the golden ratio: There are slight differences as the decimals go on - computers have fixed precision. We only want a positive solution (the new part can't be negative), so we have We can find the "x" that makes this relationship true using the quadratic formula: That's the goal for the pattern, let's try to solve it.
0 Comments
Leave a Reply. |