M21-2 - Professor: Jennifer Plants PDF

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Professor: Jennifer Plants...

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Do we get gold? Let's make a rectangle somewhat like the Golden Rectangle. As before, start with a square; however, instead of cutting the base in half, cut it into thirds and draw the line from the upper right vertex of the square to the point on the base that is one-third of the way from the right bottom vertex. Now use this new line segment as the radius of the circle, and continue as we did in the construction of the Golden Rectangle. This produces a new, longer rectangle, as shown in the diagram. What is the ratio of the base to the height of this rectangle (that is, what is base/height for this new rectangle)? Let the sides of the square be 03 units long. Now remove the largest square possible from this new rectangle and notice that we are left with another rectangle. Are the proportions of the base/height of this smaller rectangle the same as the proportions of the big rectangle? The proportions are 1.721 Not the same. The right triangle has base 1, and height 3, so its hypotenuse has length √10. To tile or not to tile. Which of the following shapes can be used to tile the entire plane? Shape 01: Yes Shape 02: Yes Shape 03: No Shape 04: Yes Shape 05: Yes The first, second, fourth, and fifth shapes can be used to tile the plane. The star cannot. Flipping over symmetry. For each pattern below, describe a rigid symmetry corresponding to a flip. Which patterns have more than one flip symmetry? Pattern 01: (Fishscale) Symmetry around horizontal axis Pattern 02: (Hexagon) Symmetry around the axes repeating with intervals of 30 degrees Pattern 03: (Parquet) Symmetry around horizontal and vertical axes

The left-most and the right-most patterns have a flip symmetry obtained by flipping around a horizontal line drawn through the middle of each tile. The right-most pattern also has flip symmetry obtained by flipping through a vertical line. The center pattern has a symmetry around the axes repeating with intervals of 30 degrees. Come on baby, do the twist. Which patterns have a rigid symmetry corresponding to a rotation? First pattern (Fishscale) has no rotational symmetry Middle pattern (Hexagon) has rotational symmetry: through any multiple of 60 degrees Right-most (Parquet) pattern has only one rotational symmetry: through 180 degrees Do any of the patterns have more than one rotational symmetry? Yes Symmetric scaling. Each of the two patterns below has a symmetry of scale. For each pattern, determine how many small tiles are needed to create a super-tile. How many are required to build a super-super-tile? Pattern on the left. Super tile: 4 tiles Super-super tile: 9 tiles Pattern on the right. Super tile: 4 tiles Super-super tile: 9 tiles Expand again. Take your 4-unit equilateral triangle and surround it with 12 equilateral triangles to create a 16-unit super-triangle. Which way is it oriented? The 16-unit super-tile has the same orientation as the original (center) equilateral triangle. Unlike the 4-unit super-tile, the 16-unit super-tile has the same orientation as the original (center) equilateral triangle. Super total. Recall that the Pinwheel Triangle has sides of length 3, 6, and 3√5. The figure below shows a super triangle made of five Pinwheel Triangles. Find the lengths of the sides of the super triangle. 3√5, 6√5, 15 The Pinwheel Triangle has sides of length 3, 6, and 3√5. The super triangle has short leg equal to one hypotenuse of the Pinwheel Triangle, so it has length 3√5. The long leg of the super

triangle has length equal to two hypotenuses of the Pinwheel Triangle and so has length 6√5. The hypotenuse of the super triangle has length equal to two long legs plus a short leg of the Pinwheel Triangle and so has length 02(6) + 3 = 15. Describing distortion. What does it mean to say that two things are equivalent by distortion? Two objects are equivalent by distortion if we can stretch, shrink, bend, or twist one, without cutting or gluing, and deform it into the other. Your last sheet. You're in your bathroom reading the liner notes for a newly purchased CD. Then you discover that you've just run out of toilet paper. Is a toilet paper tube equivalent by distortion to a CD? Yes A toilet paper tube is equivalent by distortion to a CD, if we assume that the CD has thickness. Imagine the toilet paper tube is made of very squishy, malleable material. Just flatten it out until it's the diameter of a CD, then shrink the hole until it's the size of the hole in the CD. For the finishing touch, draw out the little tendrils from around the edges of the hole to match the "gripper teeth" that ring the edge of the center of the CD. Because the material of the original tube was only stretched or shrunk into a CD shape, the tube is equivalent by distortion to the CD. Rubber polygons. Find a large rubber band and stretch it with your fingers to make a triangle, then a square, and then a pentagon. Are these shapes equivalent by distortion? What other equivalent shapes can you make with the rubber band? Can you stretch it to make a rubber disk? Yes Circles and ellipses No The triangle, square, and pentagon are all equivalent by distortion because they can all be obtained by stretching or shrinking the same rubber band. All other polygons can also be obtained, as well as circles, ellipses, or shapes with curving boundaries. The rubber band cannot be stretched to make a rubber disk. Out, out red spot. Remove the red spot from the letters below. For each letter, how many pieces result? Are the original letters equivalent by distortion? Removing the red spot from the X leaves 4 piece(s), removing it from the Y leaves 3 piece(s),

from the Z leaves 2 piece(s). The original letters are not equivalent by distortion. That theta. Does there exist a pair of points on the theta curve whose removal breaks the curve into three pieces? If so, the existence of those two points would provide another proof that the circle is not equivalent by distortion to the theta curve. Yes Removing the two intersection points leaves the theta curve in three pieces, while removing any two points on the circle always leaves exactly two pieces. Suppose there were a set of deformations that turned the theta curve into a circle. Mark the two intersection points red and follow the deformation process. At any stage, the removal of the two red points should leave the distorted theta curve in three pieces, yet when we get to the final stage, the circle, we find that the object falls into just two pieces. This contradiction shows that our assumption is wrong. The theta curve is not equivalent to the circle. Your ABCs. Consider the following letters made of 01-dimensional line segments: AB C D E F G H I J K LM N O PQ R S TU V W X Y Z Which letters are equivalent to one another by distortion? Group equivalent letters together. (B), (P), (Q), (X) (A,R), (D,O), (H,K), (E,F,T,Y) (C,G,I,J,L,M,N,S,U,V,W,Z) Note that the category depends very much on the font. We have picked a simple font without serifs to simplify the categorization. There are nine groups (A,R), (B), (C,G,I,J,L,M,N,S,U,V,W,Z), (D,O), (E,F,T,Y), (H,K), (P), (Q), (X)....