Question

Instructions: Complete the assignment on a separate sheet of paper. Be su project. Your write-up should be no longer than a page and a half To construct the fractal known as the dragon, we w need a long strip of paper and a few definitions. A valley fold (V) on a piece of paper is a crease where the paper has been folded up towards itself (see Figure la) while a mountain fold (M) on a piece of paper is a crease where the paper has been folded down towards itself (see Figure 1b) (a) A valley fold. (b) A mountain fold. Figure 1: Valley and mountain folds To create the nth step dragon fractal, follow the steps below: 1. Begin by holding a long strip of paper with the ends in each hand. 2. Create a valley fold in the middle of the paper by bringing your right hand up and over towards your left hand. 3. Crease the paper in the middle 4. Repeat steps 1, 2, and 3 with this new shorter strip of paper times 5. Unfold the paper so that each fold is at 90 and place the paper on its edge (see Figure 2) If we unfold the 1st step dragon fractal, we see that the paper contains only one fold, a valley fold, V. For the 2nd step dragon fractal, the unfolded paper contains 3 folds, VVM, in sequence from left to right. What sequence of folds appear on the paper (from left-to-right) on the 3rd step dragon fractal? Can you predict the result of the 4th step based on the 3rd step? Let vn count the https://en.vikipedia.org/wiki/Fractal Figure 2: The 2nd step dragon fractal. number of valley folds in the nth step of the dragon fractal. Let mn count the number of mountain folds in the nth step of the dragon fractal. Lastly, let tnnmn (the total number of folds on the nth step of the dragon fractal 1. Find a recurrence relation and initial conditions for vn, m, and tn when n 2 2 Solve each of the recurrence relation with initial conditions from part 1 3. Sketch the 4th step dragon fractal.

Answer 1

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