Math Art

Computer Graphics Math Art

Grades 7-12 Colette Stemple

Coral Reef High School Miami, Florida

Math Art

Contents Page

Teacher Statement

1. Course Overview

Course description

Course Goals

2. Summary of State Standards

3. Technology Specifications

4. Lesson Plan Table

5. Detailed Lesson Plans

6. Evaluation Form

7. Lesson Ideas

8. Bibliography

9. Tutorials and Gallery of Projects

Adobe ® Photoshop™ proves that Math is Beautiful

Math Art

Computer Graphics Photoshop Mathematical Visualization

Colette Stemple Coral Reef High School

Miami, Florida

TEACHER STATEMENT The connection between Photoshop and Math became apparent to me when I launched the first version of Photoshop in the middle school art program. Students began experimenting with actions, paths, and filters and began applying "visual math” to their imaging projects. Because they were working on Art projects and could see the concepts, and watch changes as they altered curves and percentages or worked with graphs, they were able to apply math concepts with ease without really knowing that Math was what they were doing. Even though these exercises were not formal math projects math scores improved for each of these students as a result of their visual experimentations. This course will introduce a series of Math Art exercises that have been formalized by International Baccalaureate students at Coral Reef High School. The messages of these exercises is that reinforcing lessons from any discipline by creating hands on visuals with the Art teacher and approaching the process through play, imagination and creative exploration, is a wonderful extension of any subject or discipline.

Colette Stemple

Reviewing Math Art Project Digital Compass

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ART AS THE HUB If teachers from other disciplines will let the art teacher know what they are teaching, all concepts can be reinforced through visual exercises. This has been demonstrated for centuries by actual man made objects as evidenced in cave paintings, arts and crafts and architecture throughout history. Now we also have the option of using the computer to help achieve these goals. And, in the art classroom, the actual creation of an art work can often spontaneously spark the "genius "core in each child In many schools, students move from one room, one subject, and one teacher to another all day long, and never see connections from one subject to another. Most states have Math Rubrics and a Curriculum that teachers must follow. By working with the math teacher and applying these requirements to art exercises, the Art teacher can demolish the erroneous, long held, idea that Art is a frivolous elective and establish the art curriculum as the core of any discipline. The Art classroom is the place where all disciplines can and should all come together. As an Art teacher I have shown how this can happen by helping the Math teachers bring concepts to life and improving math comprehension by simply giving students some examples of how to be

aware of the mathematical principles inherent in the creation of visual images. With Adobe Photoshop, these principles are inherent in the process of creating art on the computer. Adobe Systems software used across the curriculum in all of the classrooms and computer labs helps administrators, teachers, parents, and the students to cooperatively get more out of, and put more into learning. Adobe software offers a cohesive visual language and form of expression that applies to every classroom situation. It "De-isolates", teachers and their subjects, it makes everyone's job easier and more fun.

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COURSE OVERVIEW Coarse Description: These lessons and exercises are not meant to be Math Workbook Lessons, though some could be used as such. Some of these examples illustrate actual math concepts literally and accurately, others are uninhibited play based in diverse mathematical themes or focusing on the “MATH TOOLS” inherent in Photoshop. The lessons were created for use by Art Teachers, to allow them to use art to reinforce mathematical concepts, rules and explorations. This course module was designed for artistic explorations in the realm of mathematical structures, laws, deviations and explorations. The course module I created is a nine week project for students from all academies, on each level of mathematical and artistic expertise. The only instructions they needed was to remind them of the incredibly diverse amount of information they have learned in Math classes and also Science classes and then challenge them take one or more principles or theories and create a simple work of visual art that reflects these principles and theories. This made it possible for students to take one discipline, Art and another discipline, Math, and put them together to see that they were inseparable. Of course, this leads too a great deal of lively discussion and collaboration among the students which also sparks learning. No emphasis was placed aesthetics. In some exercises it is easy to that this was not stressed. However, some students started to see it themselves as artists as their "Math ideas" became pictures and this sparked the wonderful result of making them want to keep working on to achieve an aesthetic that they considered acceptable. Course Goals: This workbook is intended to encourage all teachers to share their comprehensive tests, standards and lessons with the art teacher. Though it is not the Art teacher's job to teach Math, Science, Language Arts or Computer Literacy but that is exactly what happens when art class is combined with any other discipline.

The students were not obliged to get their creations "OK'd" by the Math teacher, although that could certainly be a wonderful way for the Math teacher to connect to the creative core of the person they are trying to teach. Exploring images made by students could indicate a "thinking" pattern in a student's mind that the teacher could tap into and exploit. Also, art works could start lively discussion about Math principles in the Math Room and not just in the Art Room. Math Art lessons should trigger the heart and imagination of each student and allow them to share those qualities in the Math classroom. If the Math teacher doesn't have the skills to create these opportunities for visual learning, then the Art should be called upon to help.

Digital Compass Project

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State Standards

Florida State Standards

Florida Department of Education Sunshine State Standards-Visual Arts 9-12 Skills and Techniques: Standard 1: The student understands and applies media, techniques and processes. VA.A.1.4 Cultural and Historical Connections Standard 1: The student understands the visual arts in relation to history and culture. VA.C.1.4 Aesthetic and Critical Analysis Standard 1: The student assesses, evaluates, and responds to the characteristics of works of art. VA.D.1.4 Applications to Life

Standard 1 The student makes connections between the

visual arts, other disciplines and the real world. VA.E.1.4

For more information about the Florida Sunshine State Competency Based Standards go to: www.firn.edu/doe/menu/sss.htm

Technology Specifications Computer Specifications: 20 PC computers, Pentium III Processors, 128.0 MB RAM 2 HP Deskjet 870 CXI color Printers 1 HP Deskjet 1600 CM Color Printer 2 HP Scanjet 6100 C Flatbed Scanner 2 Epson Stylus Photo 1270 Inkjet printers 1 Fuji FUJIX 3000 Dye Sublimation Printer 2 Canon CanoScan FS 2710 negative & transparency scanners 20 12” Wacom Tablets

Adobe Software List:

Adobe Photoshop 7.0 Adobe® GoLive ™ Adobe® Illustrator ™ Adobe® Atmosphere ™ Adobe® Acrobat™

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Lesson Plan Table 1. Review Math Concepts and Visualization

Review Photoshop Tools See Bibliography** Photoshop & Photoshop Elements Tutorials*

Start with the BLACK HOLE by O’Connor Photoshop or Elements*

2 Simple Algebraic Graphing Creating a simple x axis and y axis graph and elaborate

Graphing Equations from any Algebra book.

See DIGITAL COMPASS Exercise by Marraccini

3. Review Perspective (as in the Renaissance), Foreshortening and Atmospheric Perspective (as in Photo Realism)

1, 2, & 3 point perspective samples from Art texts

3D transform tool 9th Grade samples using & Sample PHOTOGRAPHS by Curtis Chan

4. 2D Design as defined in Abstract Art

Art History texts & URLs G. Yap & C. Chan samples.

5. Saturation and the Properties of Color and Light (as in Impressionism), Value Scale

Color Theory, Intensitiy, Saturation, Scales

Christina Jay and 9th grade Sample Color

6. Pattern and Symmetry (as in Islamic Art) Art History texts & URL's N. Salz Formal Symmetry samples

7. Digitization (as in Photo Imitation), Computer Graphics, Computer Engineering and Photo Realism

The Art of Chuck Close & Rasterization, Pixels etc.

Enlargements of any Rasterized Image Computer Engineering Photos by J. Jing Zeng

8.

Geometric Abstracts using all types of Angles Basic laws of motion in physics

The Art of Richard Mortensen. S. Delauney, M. Weber & Natural Elements

“Propriano”, Mortensen’s abstract painting; study of the measurement of angles. Sample by G. Obermeier (9 gr.)

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Detailed Lesson Plans

THE VISUAL ARTS LESSON PLAN LEVEL 1

All Lessons illustrated in this workbook were developed by the students in Visual Arts Level I. Each student was evaluated and graded according to the components shown below for this course. Materials: Photoshop or Photoshop Elements Media: Computer Imaging Visual Resources: Photoshop or Elements tool tutorial if necessary. Procedures: Show students how to open Photoshop and instruct with handout a basic diagram of the tools. Give a background with Art History examples of Formal and Radial Symmetry and discuss the concept of Compositional Centering

Show Art and Mathematical examples of both. Discuss OP ART, Convergence, Vortex, Perspective, Emphasis, Depth of Field, Concentricity, Space, Unity, Illusory Motion, & Space Vocabulary: All of the words in Procedures with emphasis on the correlation of Mathematics with the Elements of Visual Composition Sketchbook/ Journal or Homework: Do sketches in the Journal of the concepts introduced. List possible Design Applications List possible Mathematical Applications Show examples from the imagination, nature and existing Art and Mathematical forms or functions. Show how a simple concept can lead to advanced theories in Physics, Calculus, Geometry, and Algebra Show where Symmetry both Radial and Formal have dominated centuries of Design in different cultures and religious iconography.

Performance Assessment: Observation: Final Project: Group Critique of individualized re- interpretation of the Assignment Observation of student working Self - assessment by the student Journal Entries Sketches Base Assessment: Vocabulary test Design creation

See Evaluation Form following page.

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Florida Sunshine State Standards for Teacher and Student Evaluations.

Course Name___________________________________________________________________________________

Student Name__________________________________________________________________________________

Student

Evaluation

State Standards-Performance Assessment Teacher

evaluation

Grade

I. Skills and Techniques:

Standard 1: The student understands and applies media, techniques and processes.

VA.A.1.4

II. Cultural and Historical Connections

Standard 1: The student understands the visual arts in relation to history and culture.

VA.C.1.4

III. Aesthetic and Critical Analysis

Standard 1: The student assesses, evaluates, and responds to the characteristics of

works of art. VA.D.1.4

IV. Applications to Life

Standard 1 The student makes connections between the visual arts, other disciplines and

the real world. VA.E.1.4

V. Observation:

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MATH LESSON IDEAS 1. Analytic Geometry, Bi Lateral Symmetry, Rotational Symmetry, Curves

2. Graphs: Equations and Slopes of Lines, Tessellations or Tiling, Perspective, Histograms, Scale, Pattern _ Rhythm, Sierpinski Triangles, Fractals,

3. Tangents, 2D & 3D Geometric Shapes, Saturation, Spherical Coordinates,

4. Parametric Equations, Tangents, Drag: (e.g. Marine objects with least drag are torpedo shaped and are ¼ wide as they are long.)

5. Area & Perimeter, Diameter and Radii, Segments, Transformations, Rotations, Reflections, Positive vs. Negative, Bezier Curves, Mandalas,

6. Contours, Values, Shades, Series, Syncopation, Perimeter, Area, Volume Graphing, Histograms.

LESSONS WITH FOUNDATIONS IN ART AND CULTURES 1. Perspective (as in the Renaissance)

2. Value Scale

3. 2D & 3D Design, (as in Abstract Art)

4. Saturation, and the Properties of Color and Light, (as in Impressionism)

5. Pattern and Symmetry (as in Islamic Art)

6. Foreshortening and Atmospheric Perspective (as in Photo Realism)

7. Digitization (as in Photo Imitation)

8. Geometric Abstracts (as in Rayonism)

9. Mathematical Languages ( as in Abstraction, Symbolism, Architectural Construction, Cubism)

10. Proportion and Distortion and Illusion (as in Surrealism).

11. The SUMBA CULTURE of Indonesia 12. WESTERN APACHE CRAFTS and all Native American & African Crafts

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13. HELLENISTIC POTTERY of Greece 14. LANDSCAPE & STILL LIFE PAINTING from any period & place 15. MASTERS OF LIGHT DEFINITION from the Renaissance to the Present 16. CUBISM Its masters, prototypes, and imitators 17. OP ART 18. TROMPE L’OEIL especially in French & Dutch Art of the Renaissance 19. RAKED GARDENS OF JAPAN 20. ARCHITECTURE from any time and place 21. FUTURISM, DE STIJL, CONSTRUCTIVISM, SURREALISM, SUPREMATISM 22. NEOCLASSICISM especially Jacques Louis David & Jean Auguste Ingres 23. POST IMPRESSIONISM especially Paul Cezanne 24. EXPRESSIONISM especially Henri Matisse & Wassily Kandinsky

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BIBLIOGRAPHY 1. Art & Physics Parallel Visions in Space, Time & Light, by Leonard Shlain, Quill William Morrow New York, 1991

2. Exploring The Invisible (Art, Science And The Spiritual,) by Lynn Gamwell, Princeton University Press, 2002

3. Schoolworks Photoshop Elements , for Teachers, by Barbara and Nickolas Delikaris, Schoolworks, 2003

4. Discovering Geometry, An Inductive Approach, by Michael Serra, Key Curriculum Press

5. The Genesis of Form, From Chaos to Geometry, by Mark Verstockt, Muller, Blond & White

6. Symmetry in Chaos, A Search for Pattern in Mathematics, Art, & Nature, by Michael Field & Martin Golubitsky, Oxford Press

7. Visualization, The Second Computer Revolution, by Richard Mark Friedhoff and William Benzon, W.H. Freeman & Co.

8. Cellular Automata and Complexity, Collected Papers, by Stephen Wolfram, Westview Press, 1994

9. Algebra to Go, A Mathematics Handbook, by Great Source Education Group, Houghton Mifflin Co., 2000

10. Key to Geometry Skillbook Series # KJ100, by Lakeshore Basics

ON-LINE RESOURCES www.didkblick.com/zz720/64

Integrating Technology in Science & Math Instruction

www.bennett.edu/academic/nfs

Yahooligans! Arts & Entertainment: Art

www.yahooligans.com/Arts_and_Entertainment/Art

Welcome to Integer Jim’s Math Squad

www.mathsquad.com

Index Vance & Art’s Spell Bound Game and Math Puzzles and Games

http://web.pdx.edu/~manchest/ ADOBE EDUCATION SITE, wwwladobe.com/education

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Mathematics – k – 12 Internet Sites

http://archives.math.utk.edu/k12.html

PiRanch Math Camps, Unique Art, Algebra Camp, Geometry Camp etc.

www.piranch.com/schedule.htm

dvdream.com art instruction catalog

www.dvdream.com/dover_art_instruction_catalog.html

Kathy Schrock’s Guide for Educators – Mathematics Resources

school.discovery.com/schrockguide/math.html

Art Links

www.rcs.k12.va.us/csjh/artlinks.htm

2002 AMTNJ Symposium: Using Computers to Enhance Math Instruction

mathforum.com/~shelly/presentations/AMTNJ/10.2002.html

Art Instruction Books

www.rhodemap.com/store/books/artinstruction.htm

Cool Math Sites

www.cte.jhu.edu/theacademy/web/2000/heal/mathsites.htm

Fun Math

www.surprise.com/kids/math_skills/fun_math_program.cfm

Deliberately Distorting The Digital Mechanism Site

www.jodi.org,asdfg.jodi.org; 404.jodi.org, wrongbrowser.com & www.jodi.org, mirapaul@nytimes.com

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TUTORIALS Adobe Photoshop 7.0

M. O'Connor

Template Project

TEMPLATE TUTORIAL Photoshop 7.0

1. Pull down the File menu and select new. In the dialogue box set 50 Pixels/inch 5 in x 5 in 2. In the toolbox select the Geometric Lasso Tool Create a spiral in towards the center, and then follow it back out. 3. Double click the foreground color box in the Color Picker at the bottom of the toolbox and select the color Red as the foreground color 4. Pull down the Edit menu and select Fill In the dialogue box select foreground color The will fill selection with red. 5. Pull down the Filter menu and select Distort>Twirl In the dialogue box set the Amount to –150 to –100 or 100 to 150 depending on the direction of the twirl you want to create Select OK and the lines on the image will twirl into a spiral 6. In the Layer menu select New Layer A new layer will appear in the Layer window 7. In the Layer menu select Layer Properties In the dialogue box Add drop Shadow seta t 120 degrees Copy Layer Rotate New Layer 90 degrees

Copy Layer Rotate New Layer 90 degrees Copy Layer Rotate New Layer 90 degrees

TEMPLATE

M. O’Connor

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BLACK HOLE TUTORIAL Photoshop 7.0

File Menu: New Image 150 pixels/inch 5 inches x 5 inches Edit Menu: Fill Fill Layer with Black Image Menu: Mode Screen Filter Menu: Add Noise… Gaussian Monochrome Image menu: Adjustments: Adjust levels Drag Far Left (Black) Arrow Right. Filter Menu: Blur> Radial Blur Amount: 25

BLACK HOLE

M. O’Connor

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Tessellation Tutorial Photoshop 7.0

File: New Image 150 Pixels/inch 5 in x 5 in Edit: Fill >Background with black Toolbox: Rectangular Marquee In foreground, create a square that Is approximately 1/16th of the screen size Edit: Fill>Fill Square with color (red) Toolbox: Lasso tool: Create a selection including only parts from the right size of the square. Rectangular Marquee: Select half of square Edit Menu: Copy Selection Paste selection on a new layer Move selection until it is flush with opposing side. Layer Menu: Merge New layer with square layer Repeat process for top and bottom. Do not cross outside of the square when making selection.

Layer Menu: Copy Layer. Fill Selection with black. Edit Menu: Fill, Fill background with white. Layer menu: Align black and red selections. Merge Layers. Edit Menu: Copy Layer with black and red. Paste Continue copying until image is full. Fill Background with black

TESSELATION M. O’Connor

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Cage and Curl Tutorial

Photoshop 7.0

1. Pull down File Menu and create a New Image file In the dialogue box, use the following parameters

150 pixels/inch 5 inches x 5 inches 2. Select the Paint Brush from the Toolbox and the Airbrush tool from the menu bar Choose Color: Black Set the brush size to Size: 35 (Soft) 3. On the blank image Create Lines

Create several Vertical Lines and Several Horizontal Lines. Do not make them completely straight.

4. Pull down the Filter menu and select Distort>Pinch Set the Pinch to 100% 5. Pull down the Filter menu and select Distort>Twirl

Set the Twirl to 72% Select OK and the original pinched lines will twirl

6. In the Layer menu Create a New Layer. Be sure this layer is selected in the Layer Window

7. In the toolbox select the Rectangular Selection Tool Make Vertical selections that extend from the top to the bottom of the image area 8. Pull down the Edit Menu and select Fill:

Fill each selection with 100% Black. Each rectangle should look like a line

Repeat Process with Horizontal Lines These lines will form boxes that indicate a cage The boxes should look tilted because of the swirl behind them.

CAGE CURL

M. O’Connor

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TUTORIAL PHOTOSHOP 7.0 UNRAVELLING

File Menu: New>New Image 150 Pixels/inch 5 Inches x 5 inches Toolbox: Paintbrush tool Color: Black Size: 35 (Soft) Create Lines

Create several Vertical Lines and Several Horizontal Lines. Do not make them completely straight.

Filter Menu: Polar Coordinates:>Rectangular to Polar

UNRAVELLING M. O’Connor

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TUTORIALS PHOTOSHOP 7.0

MATHEMATICAL DISTORTION R. Fackler

Science Major In this exercise, I took a photo of my brother, added mathematical principles of geometry and algebraic graphing and turned the picture into a work of art. I believe you can make any subject fun by applying basic math principles to photographic images. This project allowed me to use my creativity while using the math concepts I have learned over the last four years. Rendered Brother- applied to a photograph based on the geometric principles Head embellished by using a sphere shape on a 35-degree angle field of view: 1. Pull Down the filter menu> Render>3-D transform. 2. Choose the sphere shape from the toolbox 3. Draw around the head 4. Tilt the sphere at a 35 –degree angle. This made the head almost look 3-D while the rest of the picture was flat. I thought about using other shapes to accent his body shapes, however it drew away from the essence of the geometric figures.

POLAR COORDINATES R. Fackler

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MATHEMATICAL DISTORTION R. Fackler

Polar Coordinate Brother- Modifications to a photographic image using mathematical concepts of algebraic graphing. Distortion One:

1. Start with the equation x = y 2. Set the image on a vertical slope at the points

(0.0) on the x/y axis.

This pulled the picture into the center of the graph distorting the picture into an oval shape. Distortion Two:

1. Select the Filter Menu>Distort> polar coordinate. 2. Choose the option rectangular to polar.

Mathematical Distortion

R. Fackler

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Math Concepts

R. Fackler Science Major

Polar Coordinates-Graphs: Centering equidistant to the midpoint: Pulled image area into 0-0 on x-y axis Created an Oval of the original shape

Illustrating Geometric Shapes: Filter/Render/3D Images Selected Cube Tilted Axis by 15 degrees

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DIGITAL COMPASS P. Marriccinni

Step 1. Open a new file that is 1200 by 1600 pixels and make a graph that ranges from at least 6 to –6 on the x and y axis using the line tool. Make a mark the x-axis and y-axis every 100 pixels. Make sure you use the rulers to make sure the space between each number is exact. Step 2. Draw the line y=x using the line tool. Start at point (-5, -5) which is 1300 pixels on the vertical line and at 100 pixels on the horizontal ruler. Then draw the line to point (5,5) which is 300 pixels on vertical ruler and 1100 pixels on the horizontal ruler. Draw the line y=-x, which starts at (-5,5) which, is 300 pixels on vertical and 100 pixels on the horizontal ruler. Then end at point (5, -5) which is 1300 pixels on the vertical and 100 pixels on the horizontal ruler..

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Step 3. Draw the line y=0 and the line x=0. They are on your axis but only draw them from 6 to –6.

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Step 4. Select the pen tool and make 3 anchor points on the line y=-x. One at where it started, one on the origin (0,0) which is 800 pixels on the vertical and 600 pixels on the horizontal ruler.

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Step 5. Hold down shift and click on the anchor point at the origin. Then move the mouse to make a curve. Then let go of the shift key and the mouse.

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(Step 5. Hold down shift and click on the anchor point at the origin. Then move the mouse to make a curve. Then let go of the shift key and the mouse.) Step 6. Repeat Steps 5 except make the curve the opposite way

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Step 7. Repeat step 6 for new curve (Step 6 Go to the anchor point on the end of the line that does not have the curve hold down shift and click on it. It will automatically make the curve for you.)

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Step 8. Repeat for line y=x

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Step 9. Repeat for line y=0

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Tutorials

Step 10. Finish line y=0

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Tutorials

Step 11. Make curves on line x=0

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Step 12. Make a new layer with 30% opacity. Then write in the equations for some of the lines used next to them to label them.

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Step 13. Make new layer 100% opacity and label the units on the graph.

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Final Step. (Optional) Color in graph and add filter.

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STEP BY STEP THROUGH THE HISTORY PALLETTE

History: - paint bucket - ellipse - ellipse - ellipse - ellipse - pattern fill 1 - liquefy - liquefy - line - line - line - line - line - line - line - line - pattern fill 1 - lens flare

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History: - pen - pen - pattern fill - ellipse - ellipse - pattern fill - ellipse - pattern fill - liquefy - liquefy - 3D transform

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TUTORIALS Adobe Photoshop 7.0

The figure above was drawn with the polygonal and the elliptical selection tools. The Foreground and Background colors were set to Black and White and the Gradient Tool was used to fill in the shapes. The letters were created by hand using the Pencil Tool, working with black in the white areas and white in the black areas for visibility. The figure above illustrates a surprising fact about triangles and circles. Given any triangle, within the A, B, C, points there is a circle that contains all of the following nine points:

1. The midpoints K, L, and M of the sides of the triangle ABC. 2. The points X, Y, and Z, where AX, BY, and CZ are the altitudes of triangle A, B, C. 3. The points R, S, and T, which are the midpoints of the segments AH, BH, and CH that join the vertices of

triangle ABC to the point H where the lines containing the altitudes intersect.

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GALLERY OF STUDENT PROJECTS

J. Williams To create this image I used the ellipse tool, the rectangle tool, the custom shape tool,

the rounded rectangle tool and the polygon tool. I duplicated the shapes and used the Emboss Filter tool in the Stylize section of the Filters menu.

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This image was inspired by the lesson in geometric relationships on the classifications of quadrilaterals.

It was created with the polygonal lasso, the gradient tool, the rectangular selection tool and the paint brush tool.

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This painting was done with the Adobe Photoshop.

Tools used were the Paintbrush, Pencil, Polygonal lasso tool, Paint Bucket and Color Selection Tools. The Lasso was used to draw the shapes and the Paint Bucket and Brush Tools were used to fill in the colors. The paint Brush and the Pencil were used to draw in the lines and details. This lesson was designed to teach Mathematical Concepts using Art History for inspiration, in this case Richard Mortensen’s work, and Adobe Photoshop for execution.

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To make the picture above:

1. Create a background with the gradient tool 2. Use the Filter Menu to create and color shapes:

- Create shapes with the 3D Transform filter and the Cutout filter - Create effects with the Glass filter and the Lighting Effects filters.

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Shapes One Shapes Two

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STELLAR NAVIGATION G. Yap