Introduction
There are a multitude of connections between mathematics, art, and architecture. Geometry is one area of mathematics that is particularly well-suited to exploring some of these connections. In this Webquest you will learn about the Golden Ratio, Polygons and Tiling, the Golden Triangle, and Penrose Tiles. To complete your Webquest, you will explore various websites and answer questions based on information from the websites. You will complete a worksheet about the Golden Ratio, and engage in two interactive/creative projects about Penrose Tiles and modern day connections between geometry and art, respectively.
The Process
Visit the following websites to learn about various aspects of the relationship between geometry and art as they relate to the following:
Geometry in Art and Architecture: http://www.dartmouth.edu/~matc/math5.geometry/unit1/INTRO.html
Exploring the Golden Ratio: http://jwilson.coe.uga.edu/EMAT6680Fa10/Jones/6690/Golden%20Ratio/JonesGoldenRatio.html
The Incredible Tesselations Page: http://www.incredibleart.org/lessons/middle/tessell.htm
Penrose Tiling Applet: http://www.cgl.uwaterloo.ca/~csk/software/penrose/
Polyhedra and Non- Polyhedra: http://www.mathsisfun.com/geometry/solid-geometry.html
Next, with the aid of these websites, work through the following activities which are designed to help you learn more about the relationship between geometry and art with respect to:
· The Golden Ratio
· Polygons and Tiling
· The Golden Triangle
· Penrose Tiles
These tasks will require the following supplies:
· Pencil and paper
· Ruler (must measure in., cm, and mm)
· Java-enabled computer with internet access.
This project is worth 100 points overall:
· Ten questions worth four points each
· One worksheet worth twenty points
· Two projects worth twenty points each
Question/Assessment (20 points)
Answer the following questions. Each question is worth two points.
Golden Ratio Worksheet (20 points)
Find a photograph of a person’s face in a magazine or advertisement. The photograph must be large enough for you to measure with a ruler. Work carefully to take the following measurements, then fill in the values below. Be sure to clearly mark your measurements on the photograph.
a. Top of the head to chin: ____cm.
b. Top of head to pupil: ____cm.
c. Pupil to tip of nose: ____cm.
d. Pupil to lip: ____cm.
e. Width of nose: ____cm.
f. Outside distance between eyes: ____cm.
g. Width of head: ____cm.
h. Hairline to pupil: ____cm.
i. Tip of nose to chin: ____cm.
j. Lips to chin: ____cm.
k. Length of lips: ____cm.
l. Tip of nose to lips: ____cm.
Next, calculate the following ratios:
a/g = ____cm
b/d = ____cm
i/j = ____cm
i/c = ____cm
e/l = ____cm
f/h = ____cm
k/e = ____cm
Now answer the following summary questions:
1. Did any of your ratios come close to the Golden Ratio?
2. Would you expect all faces to contain the Golden Ratio? Why or why not?
3. Why do you think the Golden Ratio occurs so often in nature?
Scoring Rubric for Golden Ratio Worksheet
Measurements
Target- All measurements are fully and accurately documented (5-6 pts)
Acceptable- Most measurements are fully and accurately documented (3-4 pts)
Needs Improvement- There are several errors in measurement and/or measurements are not fully documented (1-2 pts)
Calculations
Target- All calculations are fully and accurately documented (7-8 pts)
Acceptable- Most calculations are fully and accurately documented (4-6 pts.)
Needs Improvement- There are several errors in calculation and/or calculations are not fully documented (1-3 pts)
Summary Questions
Target- All summary questions are answered thoroughly and with detail (5-6 pts)
Acceptable- All summary questions are answered with some detail (3-4 pts)
Needs Improvement- Summary questions may or not be answered, and/or detail may be insufficient (1-2 pts)
Project 1: Penrose Tiles (20 points)
Visit the following website to access the Penrose Tiling Applet- http://www.cgl.uwaterloo.ca/~csk/software/penrose/ . Java must be enabled on your computer in order to access the applet. Use the applet to explore various Penrose patterns. Select a pattern that you like and create a sketch of it. You may add color to your sketch, but this is optional and no additional points will be given for color.
Now answer the following summary questions:
1. How is Penrose Tiling different from other types of geometric patterns? (Hint: Include the Golden Triangle in you discussion)
2. Describe how the Golden Ratio is connected to Penrose tiling
Scoring Rubric for Project 1:
Sketch
Target- The sketch pattern is consistent with Penrose Tiling and is done neatly. (9-10 pts)
Acceptable- The sketch pattern is somewhat consistent with Penrose tiling and is done neatly. (5-8 pts)
Needs Improvement- The sketch pattern is inconsistent with Penrose tiling and/or is not neatly done (1-4 pts)
Questions
Target- Student’s answers are detailed reflect understanding of the content. (9-10 pts)
Acceptable- Student’s answers are somewhat detailed and/or reflect some understanding of the content. (5-8 pts)
Needs Improvement- Student’s answers are incomplete and/or do not reflect understanding of the content (1-4 pts)
.
Project 2: Modern Connections (20 points)
Use the internet to search for artists who use their creativity to explore mathematical topics in geometry. Choose one artist whose work interests you and write a two paragraph essay which includes the following:
· The artist’s name
· The artist’s preferred medium
· Geometry topic(s) explored in the artist’s work.
· Do you think the artist’s chosen medium is a good one for the area of geometry being explored in his or her work? Why or why not?
· Can you think of another creative way to represent the area of geometry explored by this artist through art or architecture?
Scoring Rubric for Project 2:
Information Management
Target- All requested biographical information is included along with a complete description of geometry topic(s) explored (6-8 pts)
Acceptable- Most requested biographical information is included along with a basic description of geometry topic(s) explored (4-6 pts)
Needs Improvement- Little or no requested biographical information is included and/or there is a minimal description of geometry topic(s) explored (1-3 pts)
Analysis
Target- Student provides a detailed, thoughtful analysis of the artist’s work as it relates to geometry and discusses alternative methods for representing concepts explored by the artist. (6-8 pts)
Acceptable- Student provides a basic analysis of the artist’s work as it relates to geometry and/or does not discuss alternative methods for representing concepts explored by the artist (4-6 pts)
Needs Improvement- Student provides little to no analysis of the artist’s work as it relates to geometry and/or does not discuss alternative methods for representing concepts explored by the artist.
(1-3 pts)
Conventions
Target- Contains little or no errors in spelling, punctuation, and grammar (4 pts)
Acceptable- Contains some errors in spelling, punctuation, and grammar (3 pts)
Needs Improvement- Contains many errors in spelling, punctuation, and grammar (1-2 pts)
References
Calter, P. (1998). Geometry in art and architecture unit one. Retrieved from http://www.dartmouth.edu/~matc/math5.geometry/unit1/INTRO.html
Jones, A. (n.d.). Around the world: Finding the golden ratio. Retrieved from http://jwilson.coe.uga.edu/EMAT6680Fa10/Jones/6690/Golden Ratio/JonesGoldenRatio.html
Kaplan, C. (n.d.). The craig web experience: Penrose tiling applet. Retrieved from http://www.cgl.uwaterloo.ca/~csk/software/penrose/
n.a. (2013). Solid geometry. Retrieved from http://www.mathsisfun.com/geometry/solid-geometry.html
n.a. (n.d.). The incredible tesselations page. Retrieved from http://www.incredibleart.org/lessons/middle/tessell.htm
There are a multitude of connections between mathematics, art, and architecture. Geometry is one area of mathematics that is particularly well-suited to exploring some of these connections. In this Webquest you will learn about the Golden Ratio, Polygons and Tiling, the Golden Triangle, and Penrose Tiles. To complete your Webquest, you will explore various websites and answer questions based on information from the websites. You will complete a worksheet about the Golden Ratio, and engage in two interactive/creative projects about Penrose Tiles and modern day connections between geometry and art, respectively.
The Process
Visit the following websites to learn about various aspects of the relationship between geometry and art as they relate to the following:
Geometry in Art and Architecture: http://www.dartmouth.edu/~matc/math5.geometry/unit1/INTRO.html
Exploring the Golden Ratio: http://jwilson.coe.uga.edu/EMAT6680Fa10/Jones/6690/Golden%20Ratio/JonesGoldenRatio.html
The Incredible Tesselations Page: http://www.incredibleart.org/lessons/middle/tessell.htm
Penrose Tiling Applet: http://www.cgl.uwaterloo.ca/~csk/software/penrose/
Polyhedra and Non- Polyhedra: http://www.mathsisfun.com/geometry/solid-geometry.html
Next, with the aid of these websites, work through the following activities which are designed to help you learn more about the relationship between geometry and art with respect to:
· The Golden Ratio
· Polygons and Tiling
· The Golden Triangle
· Penrose Tiles
These tasks will require the following supplies:
· Pencil and paper
· Ruler (must measure in., cm, and mm)
· Java-enabled computer with internet access.
This project is worth 100 points overall:
· Ten questions worth four points each
· One worksheet worth twenty points
· Two projects worth twenty points each
Question/Assessment (20 points)
Answer the following questions. Each question is worth two points.
- Describe a ratio. What is the Golden Ratio and why is it important?
- Give two examples of the Golden Ratio in art and/or architecture.
- Explain the difference between regular and semi-regular polygons.
- What are tessellations and how do artists use them to create optical effects?
- What are Golden Triangles and how are they related to Penrose Tiling?
- A polyhedron is a solid bounded by plane polygons. The polygons are called_______; they intersect in _______, and the points where three or more edges intersect are called _______.
- Give an example of a polyhedron in architecture.
- What are the Platonic solids?
- Name four solids that are not polyhedra and explain why they are not.
- Search for a modern day artist whose art has mathematical connections. Give the artist’s name, primary medium, and explain how his or her art relates to mathematics.
Golden Ratio Worksheet (20 points)
Find a photograph of a person’s face in a magazine or advertisement. The photograph must be large enough for you to measure with a ruler. Work carefully to take the following measurements, then fill in the values below. Be sure to clearly mark your measurements on the photograph.
a. Top of the head to chin: ____cm.
b. Top of head to pupil: ____cm.
c. Pupil to tip of nose: ____cm.
d. Pupil to lip: ____cm.
e. Width of nose: ____cm.
f. Outside distance between eyes: ____cm.
g. Width of head: ____cm.
h. Hairline to pupil: ____cm.
i. Tip of nose to chin: ____cm.
j. Lips to chin: ____cm.
k. Length of lips: ____cm.
l. Tip of nose to lips: ____cm.
Next, calculate the following ratios:
a/g = ____cm
b/d = ____cm
i/j = ____cm
i/c = ____cm
e/l = ____cm
f/h = ____cm
k/e = ____cm
Now answer the following summary questions:
1. Did any of your ratios come close to the Golden Ratio?
2. Would you expect all faces to contain the Golden Ratio? Why or why not?
3. Why do you think the Golden Ratio occurs so often in nature?
Scoring Rubric for Golden Ratio Worksheet
Measurements
Target- All measurements are fully and accurately documented (5-6 pts)
Acceptable- Most measurements are fully and accurately documented (3-4 pts)
Needs Improvement- There are several errors in measurement and/or measurements are not fully documented (1-2 pts)
Calculations
Target- All calculations are fully and accurately documented (7-8 pts)
Acceptable- Most calculations are fully and accurately documented (4-6 pts.)
Needs Improvement- There are several errors in calculation and/or calculations are not fully documented (1-3 pts)
Summary Questions
Target- All summary questions are answered thoroughly and with detail (5-6 pts)
Acceptable- All summary questions are answered with some detail (3-4 pts)
Needs Improvement- Summary questions may or not be answered, and/or detail may be insufficient (1-2 pts)
Project 1: Penrose Tiles (20 points)
Visit the following website to access the Penrose Tiling Applet- http://www.cgl.uwaterloo.ca/~csk/software/penrose/ . Java must be enabled on your computer in order to access the applet. Use the applet to explore various Penrose patterns. Select a pattern that you like and create a sketch of it. You may add color to your sketch, but this is optional and no additional points will be given for color.
Now answer the following summary questions:
1. How is Penrose Tiling different from other types of geometric patterns? (Hint: Include the Golden Triangle in you discussion)
2. Describe how the Golden Ratio is connected to Penrose tiling
Scoring Rubric for Project 1:
Sketch
Target- The sketch pattern is consistent with Penrose Tiling and is done neatly. (9-10 pts)
Acceptable- The sketch pattern is somewhat consistent with Penrose tiling and is done neatly. (5-8 pts)
Needs Improvement- The sketch pattern is inconsistent with Penrose tiling and/or is not neatly done (1-4 pts)
Questions
Target- Student’s answers are detailed reflect understanding of the content. (9-10 pts)
Acceptable- Student’s answers are somewhat detailed and/or reflect some understanding of the content. (5-8 pts)
Needs Improvement- Student’s answers are incomplete and/or do not reflect understanding of the content (1-4 pts)
.
Project 2: Modern Connections (20 points)
Use the internet to search for artists who use their creativity to explore mathematical topics in geometry. Choose one artist whose work interests you and write a two paragraph essay which includes the following:
· The artist’s name
· The artist’s preferred medium
· Geometry topic(s) explored in the artist’s work.
· Do you think the artist’s chosen medium is a good one for the area of geometry being explored in his or her work? Why or why not?
· Can you think of another creative way to represent the area of geometry explored by this artist through art or architecture?
Scoring Rubric for Project 2:
Information Management
Target- All requested biographical information is included along with a complete description of geometry topic(s) explored (6-8 pts)
Acceptable- Most requested biographical information is included along with a basic description of geometry topic(s) explored (4-6 pts)
Needs Improvement- Little or no requested biographical information is included and/or there is a minimal description of geometry topic(s) explored (1-3 pts)
Analysis
Target- Student provides a detailed, thoughtful analysis of the artist’s work as it relates to geometry and discusses alternative methods for representing concepts explored by the artist. (6-8 pts)
Acceptable- Student provides a basic analysis of the artist’s work as it relates to geometry and/or does not discuss alternative methods for representing concepts explored by the artist (4-6 pts)
Needs Improvement- Student provides little to no analysis of the artist’s work as it relates to geometry and/or does not discuss alternative methods for representing concepts explored by the artist.
(1-3 pts)
Conventions
Target- Contains little or no errors in spelling, punctuation, and grammar (4 pts)
Acceptable- Contains some errors in spelling, punctuation, and grammar (3 pts)
Needs Improvement- Contains many errors in spelling, punctuation, and grammar (1-2 pts)
References
Calter, P. (1998). Geometry in art and architecture unit one. Retrieved from http://www.dartmouth.edu/~matc/math5.geometry/unit1/INTRO.html
Jones, A. (n.d.). Around the world: Finding the golden ratio. Retrieved from http://jwilson.coe.uga.edu/EMAT6680Fa10/Jones/6690/Golden Ratio/JonesGoldenRatio.html
Kaplan, C. (n.d.). The craig web experience: Penrose tiling applet. Retrieved from http://www.cgl.uwaterloo.ca/~csk/software/penrose/
n.a. (2013). Solid geometry. Retrieved from http://www.mathsisfun.com/geometry/solid-geometry.html
n.a. (n.d.). The incredible tesselations page. Retrieved from http://www.incredibleart.org/lessons/middle/tessell.htm