Problem 1MS: A search for self. What does self-similarity mean? Problem 2MS: Desperately seeking similarity. Which of these objects display self-similarity at different scales:... Problem 3MS: Too many triangles? At stage 0, the Sierpinski triangle consists of a single, filled-in triangle.... Problem 4MS: Counting Koch. Look at the early stages of the Koch curve on page 470. The top figure is stage 1; it... Problem 5MS: Argyle art. Take four line segments and place them to form the X configuration as shown in the... Problem 6MS: Natures way. Find several examples of objects in nature that, like broccoli, display self-similarity... Problem 7MS: Whos the fairest? Can you position three mirrors in such a way that in theory you could see... Problem 8MS: Billiards and mirrors. On an idealized, square billiard table, you want to hit a shot that traces a... Problem 9MS: MTV. Youve become a rock star and consequently feel that there can never be too many of you. So you... Problem 10MS: Photo op. Suppose you arrange two mirrors facing each other at a slight angle, as shown. Place a... Problem 11MS: How many mes? Arrange mirrors and camera as in Mindscape 10, but put yourself in the picture right... Problem 12MS: Quacker, Quacker, Quacker. Suppose you had the job of shooting the photograph for the next Quacked... Problem 13MS: Sierpinski hexed (S). Take an equilateral triangle and remove a hexagon from the center as shown at... Problem 14MS: The Kinks (ExH). Kochs kinky curve is created by starting with a straight segment and replacing it... Problem 15MS: Four times (H). Draw a picture on a square piece of paper. Now make a collage by reducing your... Problem 16MS: Burger heaven (S). Sketch a picture of a hamburger, and make three reduced copies, two reduced to... Problem 17MS: Ice cream cones. Draw a picture of an ice cream cone, and follow the same instructions as in... Problem 18MS: Sierpinski boundary. Take the boundary of a triangle. Make three reduced copies, each reduced to 1/2... Problem 19MS: Catching Zs (H). Take a Z. Put in nine smaller Zs, as shown, to create the second stage. If the... Problem 20MS: Replacement pinwheel. Take a 1,2,2 right triangle. Divide it into five identical similar right... Problem 21MS: Koch Stool (ExH). Start with a line segment, mark it into three equal pieces, and replace the middle... Problem 22MS: Koch collage stool. Given the Koch Stool described in Mindscape 21, explain how you can create the... Problem 23MS: Sierpinski shooting. Suppose that you were playing the Chaos Game to create the Sierpinski Triangle... Problem 24MS: Sierpinski target practice (H). What sequence of numbers will land you in the sub-sub-sub-triangle... Problem 25MS: Cantor Set. Start with the interval [0, 1]. Build a 1-dimensional collage by making two copies of... Problem 26MS: Cantor luck (H). Start with the point 0. Flip a coin. If it comes up heads, move 2/3 of the way... Problem 27MS: Cantor Square (S). Take a square. Make four copies, each reduced to 1/3 the length and width of the... Problem 28MS: Cantor Square shrunk. Take the Cantor Square. Describe why applying the collage-making instructions... Problem 29MS: Cantor Squared. Draw the four corners of a square, label them 1, 2, 3, 4, and start at one corner.... Problem 30MS: Hexed again. Suppose you start with the three vertices of an equilateral triangle labeled 1, 2, 3.... Problem 31MS Problem 32MS: Antoines necklace. Antoines necklace is a delicate necklace indeed. The first stage is a solid... Problem 33MS: Menger jacks. For the game of jacks, lets imagine constructing a six-pointed metal jack whose ends... Problem 34MS: A tighter weave. Look back at the Tight Weave story in Chapter 1 (page 9). There you constructed a... Problem 35MS: A looser weave. Lets modify the carpet-designing process yet again. Instead of dividing the original... Problem 36MS: From where? Look at this fractal. What collage instructions would produce it? Problem 37MS: Treed. Describe collage-making instructions that would result in a fractal collage that resembles a... Problem 38MS: Flaky (H). Describe collage making instructions that would result in a fractal collage that... Problem 39MS: How big a hole? In the Tighter weave (34) and Looser weave (35) Mindscapes, how much of the total... Problem 40MS: 4D fractal. Describe a fractal in 4-dimensional space that is analogous to the Menger Sponge. How... Problem 41MS: Personal perspectives. Write a short essay describing the most interesting or surprising discovery... Problem 42MS: With a group of folks. In a small group, discuss the infinite collage-making process. After your... Problem 45MS: Graphing Koch. Plot the following points in the xy-plane: A=(0,0),B=(6,0),C=(9,4),D=(12,0),E=(18,0)... Problem 46MS: Koch adds life. Look at the points given in the previous Mindscape. Find an equation for the line... Problem 47MS: White out. Each stage of the Sierpinski triangle is a figure composed of white and green triangles... Problem 48MS: Measuring the length of your Koch. The Koch stool is presented in Mindscape 21. If the starting line... Problem 49MS: Counting Koch segments (H). Suppose you start with a line segment of length 1 and begin to construct... format_list_bulleted