2/5/2013 Overview of Chapter 10 • We look at basic area and volume formulas. • We will discuss the ideas of dimensionality. • We will watch a video about Fractals. ▫ You will need to take notes and write a short paper about the Math for Liberal Arts video. MAT 110: Chapter 10 Notes Geometry & Fractals David J. Gisch Points, Lines, and Planes Dimension The dim ension of an object can be thought of as the number • A geometric point is imagined of independent directions in which you could move if you to have zero size. were on the object. We can also think about dimension by the number of • A geometric line is formed by coordinates required to locate a point. connecting two points along the shortest possible path. • Line segm ents are pieces of a line. • A geometric plane is a perfectly flat surface that has infinite length and width, but no thickness. 1
2/5/2013 Don’ t Limit Dimensions!! Multiple Dimensions • In Geometry we stick to 3D or less. It makes sense because that is all that we can visually see. • Dimensions do not have to be physical locations. • For example, I collect data on my students. ▫ Age (16-90) ▫ GPA (0-4.0) ▫ Grade in my Class (0-100) ▫ Height (inches) ▫ Male/Female (0=Male, 1=Female) • Each of these inputs can be thought of as a dimension. Dimensions and Measurement Role of Dimension Exam ples Dim ension Units Example 10.1: A box is increased so it is 3 times larger. Length How do the following change? Length Area Area Surface Area Volume Volume 2
2/5/2013 Role of Dimension Parts of a Circle Example 10.2: A pool has a floor with an area of 150 square Circles feet. It’s volume is 1200 cubic feet (or 8,977 gallons). What • All points on a circle are located happens if the owner wants a pool that is twice as big? at the same distance—the radius —from the circle’s Area of Floor center . • The diam eter of a circle is twice its radius. Volume Plane Geometry Perimeter and Area A polygon is any closed shape in the plane made from straight line segments. A regular polygon is a polygon in which all the sides have the same length and all interior angles are equal. Use � � 3.1415 or use the � button your calculator. 3
2/5/2013 Three-Dimensional Geometry Area and Perimeter Example 10.3: Find the area and perimeter of the following window. Volume Area and Perimeter Example 10.4: A local high school wants to install a new border around Example 10.3: Which can of soup holds more? the field of a track. They also wish to fertilize the grass to make it look good for an upcoming meet. Give them the need information (in feet) to buy brick bordering and fertilizer. 4
2/5/2013 Pythagorean Theorem Pythagorean Theorem Example 10.5: A piece of drywall needs to be cut for the side of the The Pythagorean theorem applies only to right triangles stairs shown below. What is the area of that piece? (those with one 90 angle). Also, if a piece of trim needs to be put along the hypotenuse, how long For a right triangle with side lengths a , b , and c , in which c is should it be? the longest side (or hypotenuse ), the Pythagorean theorem states that a 2 + b 2 = c 2 c a b Pitch, Grade, S lope Pitch, Grade, S lope Type of Calculation Exam ple Example 10.6: The board of tourism in a state brags that its bike trails Value are for the whole family with a maximum of 3%. What is the ����: ��� 9: 12 � 3: 4 corresponding pitch and slope? Pitch Ratio Put this information into practical terms. ���� 9 Grade Percent ��� % 12 � .75 � 75% ���� 12 � 3 9 Slope Fraction ��� 4 We use the stairs as examples in the table above. 5
2/5/2013 Fractals • Fractals are multiple (LOTS!!!!) of repetitions of a pattern. • Often the repetition creates smaller and smaller pieces for a more textured/real looking image. A little background for Tuesday’s Movie The S nowflake Curve The S nowflake Island Begin with a line segment L 0 of The snowflake island is a region (island) length 1. Then generate L 1 with the bounded by three snowflake curves. following steps: 1. Divide the line segment L 0 into three equal pieces. 2. Remove the middle piece. 3. Replace the middle piece with two segments of the same length arranged as two sides of an equilateral triangle. Repeat the steps for each line segment of the current figure to generate the next figure. 6
2/5/2013 S ierpinski Triangle S ierpinski Triangle The Sierpinski triangle is produced by starting with a solid black equilateral triangle and iterating with the following rule: For each black triangle in the current figure, connect the m idpoints of the sides and rem ove the resulting inner triangle. The Mandelbrot S et Two Views of Barnsley’s Fern Fractals are a huge part of computer generated imaging. The first fractal. Created by Benoit Mandelbrot. 7
2/5/2013 8 Fractal Images
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