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# Geometry In Art

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This is the Geometry Project Handout I gave to students

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### Geometry In Art

1. 1. Geometry in Art, Architecture, and Nature <br />Due Date: March 16 Presentations: March 16 -18<br />Introduction<br /> Our world is a collection of geometric figures in many different shapes and sizes.  Artists, architects, and naturalists have long been inspired by these geometric relationships and they discovered early in their lives that our world is rich in abundance of these geometric properties.  Fortunately, their work has been (and can be) captured in photographs and images.  It is your job to search the world around you for examples of this phenomenon.  You may take your own photographs using a digital camera or search the web for appropriate images.  Your photographs or images must highlight ten specific geometric properties and you will present your findings in a PowerPoint presentation.  Our wonderful world is full of these images and your imagination and creativity will be your greatest tools! <br />Task  <br />Your task is to explore the world around you and find occurrences of specific geometric relationships.  Think of your mission as having to teach visitors from Mars about Earth's geometric properties.  Your mission will be divided into three parts.  The first part will require you to take ten photographs (using either a digital camera or images found on the web)  that illustrate specific geometric properties.  The second part will be to provide an analysis of the photographs or images and explain and discuss the geometric relationships.  The third part will be to create a PowerPoint presentation to present your findings. <br />Process<br />Part One - The Photographs and Images<br />  You will take specific photographs or find images on the Web of the following geometric relationships: <br />Find an example of two parallel lines being cut by a transversal, not at right angles, which show that the alternate interior (corresponding, alternate exterior, etc.) angles are congruent.  <br />Find two (or more) congruent triangles which show that their corresponding parts are also congruent.  <br />Find a triangle inscribed in a circle which shows that the center of the circle is also the circumcenter of the triangle. <br />Find a regular polygon with more than 4 sides; find the measure of one interior angle and the sum of all interior angles. <br />Find a (non-square) rectangle, and (non-square) rhombus and a square either in the same picture or separate pictures that shows the relationship between all three polygons. <br />Find two similar triangles which demonstrate the idea of proportionality or scale factor. <br />Find a circle and its diameter, radius and tangent which show the properties at the point of tangency. <br />Find two non-congruent polygons whose areas are equal. <br />Find a cylinder and a cone that display the relationship between their volumes. <br />Your Choice – may be any geometric relationship you find intriguing, interesting or perhaps puzzling.  Explain why you chose this picture or image and its geometric significance.                              <br />Part Two - The Analyses <br />Choose three out of the ten relationships above and provide some mathematical analysis to further enhance your conclusions.  You may do your measurements by hand using a ruler or protractor.  Think of the analysis as a way to verify the mathematical relationship. <br /> For example, if you choose to analyze relationship 6 (Similar Triangles), you may measure corresponding sides of the triangles to verify their proportionality.  Or you can determine the scale factor and prove that taking a side of the smaller triangle and multiplying it by that number will yield the corresponding side of the larger triangle.  The analysis section involves taking actual measurements and verifying the mathematics behind the visual representations.                                 <br />Part Three - The Presentation<br />Make a title slide and choose an aesthetic slide template. <br />You will need one example of each of the above geometric relationships (10 photographs or images). <br />Each photograph or image will be labeled (location and date) and inserted into an MS PowerPoint slide. <br />On the same slide, add the relevant geometric information about the picture.  For example, if you are writing about scenario #1 (parallel lines cut by a transversal), you may want to include which conjecture or definition allows you to conclude that the corresponding angles are congruent.  You can add information about other angle relationships that are seen in the picture as well or what may happen to those angles if the lines were not parallel.  Add about 4 bullet points of information for each picture. <br />The three mathematical analyses slides should each be on a separate slide and should come after the 10 pictures. <br />Write a summary slide at the end (see Conclusion below).  <br />  <br />Resources                  <br />Use the following resources to help you focus your photographs or images so that they target a specific geometric relationship:<br />The Math Forum @ Drexel, K-12 Geometry, http://mathforum.org/geometry/k12.geometry.html <br />Math.com, The World of Math Online, http://www.math.com/homeworkhelp/Geometry.html <br />The Geometry Junkyard, http://www.ics.uci.edu/~eppstein/junkyard/ <br />Home School Math, Online math quizzes, and interactive tutorials, http://www.homeschoolmath.net/math_high_school.php#geometry <br />Welcome to Zona   Land , The Geometry Section, http://id.mind.net/~zona/mmts/geometrySection/geometrySection.html <br />Geometry Online, Cynthia Lanius, http://math.rice.edu/~lanius/Geom/index.html <br />Geometry in Action, Architecture, http://www.ics.uci.edu/~eppstein/gina/arch.html <br />Congruent Figures and Triangles, http://www.ies.co.jp/math/java/geo/congruent.html <br />Math-kitecture, Student Gallery, http://www.math-kitecture.com/gallery.htm#ss <br />Conjectures in Geometry, http://www.geom.uiuc.edu/~dwiggins/mainpage.html <br />Evaluation         <br />You will be evaluated on the following three parts of the Webquest: the photographs and images, the analyses, and the presentation.  You must have ten photographs or images that adequately show the specific geometric relationships.    The analyses should be convincing and use the mathematical properties (conjectures) that you learned in this course.  The presentation should be in PowerPoint and include the elements outlined above in the Process section.  <br /> <br />Beginning4Developing6Accomplished8Exemplary10Score Photographs and Geometric Relationships  Photographs of at least two geometric relationships.  Photographs of at least four geometric relationships.  Photographs of at least eight geometric relationships.  Photographs of ten geometric relationships.    Photographs Exhibits a Unified Theme  Photographs do not adhere to a single unifying theme.  Photographs exhibited in each of the three unifying themes (Art, Architecture, Nature)  Photographs exhibited in two of the three unifying themes.  Photographs exhibited all belong to one of the three unifying themes.    Analyses - Quantity   Analyses of at least two geometric relationships.  Analyses of at least four geometric relationships.  Analyses of at least eight geometric relationships.  Analyses of ten geometric relationships.    Analyses - Quality  Analyses fails to demonstrate the geometric relationships.  Analyses attempts to demonstrate the geometric relationships, with several errors.  Analyses somewhat demonstrates the geometric relationships, with some errors.  Analyses clearly demonstrates the geometric relationships with no errors.    Presentation  Title slide, aesthetic template, summary included  All of previous plus: 10 photos per slide included  All of previous plus: labels for each photo included  All of previous plus: geometric information for each photo included   <br /> <br />Conclusion <br />The last slide in your presentation will provide a summary of your Webquest.  The summary should include some overall thoughts on the project:  What was your method of finding pictures and images?  Did you enlist any help?  What was the most difficult picture or image to find?  What was the easiest?  Did you enjoy doing this project? <br /> <br />