Tile Design Project
The project is due by class time on Friday, Dec 12. Presentations will be held in class on Thursday, Dec 11 and Friday, Dec 12. We will schedule presentations on Dec 8.

You may work with a partner or small group on this project, but no more than three people per group. If you work with a group, you should turn in one project with the names of all involved.

Problem: A manufacturer of floor tiles has hired you as a consultant to design a new square floor tile. The company is prepared to provide for two colors in the tile but lacks the equipment necessary to have more than three simple regions for the colors. Through the grapevine, you have learned that the manufacturer likes symmetry and dislikes linear patterns. Following is an example of an acceptable tile design and corresponding floor pattern given to you by the manufacturer.



First, the manufacturer would like to know the relative amount of the two colors (percentage of tile devoted to each color) in the given example so that the cost estimates (which depend on how much of each color) can be made. Model the given tile with two functions in order to estimate the areas. You must justify your answer fully. Second, use graphs of other functions to design three other tile patterns which result in "pleasing" floor patterns. (For example, you might want to produce a design which yields a floor pattern of intersecting circles) Submit a written presentation of your designs and the corresponding floor patterns for review by the manufacturer. You should also indicate the relative amounts of each color for each of your patterns.

Grading Guidelines:The assignment will be graded in two parts.
• The first part is homework and will be worth 10 points. You must turn in a description of the given tile as well as three other tiles of your choosing. Each tile pattern will be graded out of 2 points for mathematical content (did you correctly compute the percentages of each color?). The remaining two points will be assigned based on timely completion and overall quality of writing. To receive full credit you must satisfy the condition that none of the tiles can be built from linear patterns.
• The second part of the assignment is a presentation and extra credit competition. The presentation will also be worth 10 points. There is a possibility of extra credit, should your project be voted class favorite. Judgements will be based on creativity (within the guidelines of the manufacturer) and presentation. You may use any tools you like in your presenatation, including using graphing utilities to draw your tiles, or even making model tiles for demonstration. You may use any (and as many!) functions you know to model your designs, just remember that you need to calculate the color percentages based on the designs you choose.
• As with the first project, please also turn in a short review of your group members (including yourself).

Adapted from the Tile Design project by John Ramsay (College of Wooster) from Problems for Student Investigation, MAA Notes Number 30, edited by Michael B. Jackson and John R. Ramsay.