Math Art

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					       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                              Adobe ® Photoshop™ proves that Math is Beautiful
    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
                                                         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
                                                                                Colette Stemple
and approaching the process through play, imagination
                                                                    Reviewing Math Art Project Digital Compass
and creative exploration, is a wonderful extension of any
subject or discipline.
                                                          Math Art

                     ART AS THE HUB
If teachers from other disciplines will let the art teacher     aware of the mathematical principles inherent in the creation
know what they are teaching, all concepts can be                of visual images. With Adobe Photoshop, these principles
reinforced through visual exercises. This has been              are inherent in the process of creating art on the computer.
demonstrated for centuries by actual man made objects as
evidenced in cave paintings, arts and crafts and                Adobe Systems software used across the curriculum in all of
architecture throughout history. Now we also have the           the classrooms and computer labs helps administrators,
option of using the computer to help achieve these goals.       teachers, parents, and the students to cooperatively get
And, in the art classroom, the actual creation of an art work   more out of, and put more into learning. Adobe software
can often spontaneously spark the "genius "core in each         offers a cohesive visual language and form of expression
child                                                           that applies to every classroom situation. It "De-isolates",
                                                                teachers and their subjects, it makes everyone's job easier
In many schools, students move from one room, one               and more fun.
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
                                                                    Math Art

                      COURSE OVERVIEW                                     The students were not obliged to get their creations "OK'd" by the Math
Coarse Description:                                                       teacher, although that could certainly be a wonderful way for the Math
These lessons and exercises are not meant to be Math Workbook             teacher to connect to the creative core of the person they are trying to
Lessons, though some could be used as such. Some of these                 teach. Exploring images made by students could indicate a "thinking"
examples illustrate actual math concepts literally and accurately,        pattern in a student's mind that the teacher could tap into and exploit.
others are uninhibited play based in diverse mathematical themes or       Also, art works could start lively discussion about Math principles in the
focusing on the “MATH TOOLS” inherent in Photoshop. The lessons           Math Room and not just in the Art Room. Math Art lessons should
were created for use by Art Teachers, to allow them to use art to         trigger the heart and imagination of each student and allow them to
reinforce mathematical concepts, rules and explorations. This course      share those qualities in the Math classroom. If the Math teacher doesn't
module was designed for artistic explorations in the realm of             have the skills to create these opportunities for visual learning, then the
mathematical structures, laws, deviations and explorations.               Art should be called upon to help.

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                             Digital Compass Project
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.
                                                              Math Art

                                                     State Standards

                  Florida State Standards                                        Technology Specifications

Florida Department of Education Sunshine State                    Computer Specifications:
Standards-Visual Arts 9-12                                        20 PC computers, Pentium III Processors, 128.0 MB RAM
                                                                  2 HP Deskjet 870 CXI color Printers
Skills and Techniques:                                            1 HP Deskjet 1600 CM Color Printer
Standard 1: The student understands and applies media,            2 HP Scanjet 6100 C Flatbed Scanner
techniques and processes. VA.A.1.4                                2 Epson Stylus Photo 1270 Inkjet printers
                                                                  1 Fuji FUJIX 3000 Dye Sublimation Printer
                                                                  2 Canon CanoScan FS 2710 negative & transparency scanners
Cultural and Historical Connections                               20 12” Wacom Tablets
Standard 1: The student understands the visual arts in
relation to history and culture. VA.C.1.4                         Adobe Software List:
Aesthetic and Critical Analysis                                   Adobe Photoshop 7.0
Standard 1: The student assesses, evaluates, and                  Adobe® GoLive ™
responds to the characteristics of works of art. VA.D.1.4         Adobe® Illustrator ™
Applications to Life                                              Adobe® Atmosphere ™
                                                                  Adobe® Acrobat™
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:
                                                                   Math Art

                                                          Lesson Plan Table
1.   Review Math Concepts and Visualization                                          See Bibliography**            Start with the BLACK HOLE
     Review Photoshop Tools                                                          Photoshop & Photoshop         by O’Connor
                                                                                     Elements Tutorials*           Photoshop or Elements*
2    Simple Algebraic Graphing                                                       Graphing Equations from any   See DIGITAL COMPASS
     Creating a simple x axis and y axis graph and elaborate                         Algebra book.                 Exercise by Marraccini
3.   Review Perspective (as in the Renaissance), Foreshortening and Atmospheric      1, 2, & 3 point perspective   3D transform tool 9th
     Perspective (as in Photo Realism)                                               samples                       Grade samples using &
                                                                                     from Art texts                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   Color Theory, Intensitiy,     Christina Jay and 9th grade
     Scale                                                                           Saturation, Scales            Sample Color
6.   Pattern and Symmetry (as in Islamic Art)                                        Art History texts & URL's     N. Salz Formal Symmetry
7.   Digitization (as in Photo Imitation), Computer Graphics,                        The Art of Chuck Close &      Enlargements of any
     Computer Engineering and Photo Realism                                          Rasterization, Pixels etc.    Rasterized Image
                                                                                                                   Computer Engineering
                                                                                                                   Photos by J. Jing Zeng
8.   Geometric Abstracts using all types of Angles                                   The Art of Richard            “Propriano”, Mortensen’s
     Basic laws of motion in physics                                                 Mortensen. S. Delauney, M.    abstract painting; study of the
                                                                                     Weber                         measurement of angles. Sample
                                                                                     & Natural Elements            by G. Obermeier (9 gr.)
                                                                Math Art

                                                      Detailed Lesson Plans

 THE VISUAL ARTS LESSON PLAN                   Show Art and Mathematical examples of both.
            LEVEL 1                                  Discuss OP ART, Convergence,              Performance Assessment:
                                                     Vortex, Perspective,
                                                     Emphasis, Depth of Field,                 Observation:
All Lessons illustrated in this workbook             Concentricity, Space, Unity,
were developed by the students in                    Illusory Motion, & Space                  Final Project:
Visual Arts Level I.                                                                                  Group Critique of individualized re-
                                                                                                      interpretation of the
                                               Vocabulary:                                            Assignment
Each student was evaluated and                 All of the words in Procedures with emphasis           Observation of student working
graded according to the components             on the correlation of Mathematics with the             Self - assessment by the student
shown below for this course.                   Elements of Visual Composition                         Journal Entries
Photoshop or Photoshop Elements                Sketchbook/ Journal or Homework:                Base Assessment:
                                                       Do sketches in the Journal of the                Vocabulary test
Media:                                                 concepts introduced.                             Design creation
Computer Imaging                                        List possible Design Applications
                                                        List possible Mathematical
Visual Resources: Photoshop or Elements
tool tutorial if necessary.                                                                     See Evaluation Form following page.
                                               Show examples from the imagination, nature
                                               and existing Art and Mathematical forms or
Show students how to open Photoshop and        Show how a simple concept can lead to
instruct with handout a basic diagram of the   advanced theories in Physics, Calculus,
tools.                                         Geometry, and Algebra

Give a background with Art History examples    Show where Symmetry both Radial and
of Formal and Radial Symmetry and discuss      Formal have dominated centuries of Design in
the concept of Compositional Centering         different cultures and religious iconography.
                                                                  Math Art

                    Florida Sunshine State Standards for Teacher and Student Evaluations.

Course Name___________________________________________________________________________________
Student Name__________________________________________________________________________________

Student                                           State Standards-Performance Assessment                           Teacher      Grade
Evaluation                                                                                                         evaluation

             I.          Skills and Techniques:

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

             II.         Cultural and Historical Connections

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

             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:
                                                                    Math Art

                                                             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
                                                               Math Art


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


20. ARCHITECTURE from any time and place


22. NEOCLASSICISM especially Jacques Louis David & Jean Auguste Ingres

23. POST IMPRESSIONISM especially Paul Cezanne

24. EXPRESSIONISM especially Henri Matisse & Wassily Kandinsky
                                                            Math Art

     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
Integrating Technology in Science & Math Instruction
Yahooligans! Arts & Entertainment: Art
Welcome to Integer Jim’s Math Squad
Index Vance & Art’s Spell Bound Game and Math Puzzles and Games
                                                              Math Art

Mathematics – k – 12 Internet Sites
PiRanch Math Camps, Unique Art, Algebra Camp, Geometry Camp etc. art instruction catalog
Kathy Schrock’s Guide for Educators – Mathematics Resources
Art Links
2002 AMTNJ Symposium: Using Computers to Enhance Math Instruction
Art Instruction Books
Cool Math Sites
Fun Math
Deliberately Distorting The Digital Mechanism Site,;, &,
                                                                           Math Art

                                                                      Adobe Photoshop 7.0

                                                                          M. O'Connor
                                                                        Template Project

                        TEMPLATE TUTORIAL                                       Copy Layer
                           Photoshop 7.0                                        Rotate New Layer 90 degrees
1. Pull down the File menu and select new.                                      Copy Layer
         In the dialogue box set                                                Rotate New Layer 90 degrees
         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
                                                                                                              M. O’Connor
       Rotate New Layer 90 degrees
                                            Math Art

                      BLACK HOLE TUTORIAL              BLACK HOLE
                          Photoshop 7.0
File Menu:                                             M. O’Connor
New Image
        150 pixels/inch
        5 inches x 5 inches

Edit Menu:
   Fill Layer with Black

Image Menu:

Filter Menu:
Add Noise…

Image menu:
      Adjust levels
       Drag Far Left (Black) Arrow Right.

Filter Menu:
         Blur> Radial Blur
         Amount: 25
                                                                             Math Art

                          Tessellation Tutorial                                  Layer Menu:
                            Photoshop 7.0                                        Copy Layer. Fill Selection with black.
New Image                                                                        Edit Menu:
       150 Pixels/inch                                                           Fill, Fill background with white.
       5 in x 5 in
                                                                                 Layer menu:
Edit:                                                                            Align black and red selections. Merge Layers.
Fill >Background with black
                                                                                 Edit Menu:
Toolbox:                                                                         Copy Layer with black and red. Paste
Rectangular Marquee                                                              Continue copying until image is full.
In foreground, create a square that Is approximately 1/16th of the screen size   Fill Background with black

Edit:                                                                                                           TESSELATION
Fill>Fill Square with color (red)                                                                                M. O’Connor

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.
                                                                            Math Art

                       Cage and Curl Tutorial                                    7. In the toolbox select the Rectangular Selection Tool
                          Photoshop 7.0                                          Make Vertical selections that extend from the top to the bottom of the image
1. Pull down File Menu and create a New Image file
         In the dialogue box, use the following parameters                       8. Pull down the Edit Menu and select Fill:
         150 pixels/inch                                                                  Fill each selection with 100% Black.
         5 inches x 5 inches                                                              Each rectangle should look like a line
                                                                                          Repeat Process with Horizontal Lines
2. Select the Paint Brush from the Toolbox and the Airbrush tool from the                 These lines will form boxes that indicate a cage
menu bar
          Choose Color: Black                                                    The boxes should look tilted because of the swirl behind them.
          Set the brush size to Size: 35 (Soft)
                                                                                                                 CAGE CURL
3. On the blank image Create Lines
         Create several Vertical Lines and Several Horizontal Lines.                                             M. O’Connor
         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
                                                       Math Art

              TUTORIAL PHOTOSHOP 7.0                              UNRAVELLING
                   UNRAVELLING                                     M. O’Connor

File Menu:
New>New Image
       150 Pixels/inch
       5 Inches x 5 inches

Paintbrush tool
      Color: Black
      Size: 35 (Soft)

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

Filter Menu:
Polar Coordinates:>Rectangular to Polar
                                                        Math Art

                    PHOTOSHOP 7.0
                           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
         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
                                                           Math Art

                         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
                               Math Art

Math Concepts   Polar Coordinates-Graphs:                Illustrating Geometric Shapes:
 R. Fackler     Centering equidistant to the midpoint:   Filter/Render/3D Images
Science Major   Pulled image area into 0-0 on x-y axis   Selected Cube
                Created an Oval of the original shape    Tilted Axis by 15 degrees
 Math Art

  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

                   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..
Math Art

           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.
Math Art

           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.
Math Art

            (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
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           Step 7. Repeat step 6 for new
           (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 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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            Step 10. Finish line y=0
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            Step 11. Make curves on line
Math Art

           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.
Math Art

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

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


                                            - paint bucket
                                            - ellipse
                                            - ellipse
                                            - ellipse
                                            - ellipse
                                            - pattern fill 1
                                            - liquefy
                                            - liquefy
                                            - line
                                            - line
                                            - line
                                            - line
                                            - line
                                            - line
                                            - line
                                            - line
                                            - pattern fill 1
                                            - lens flare
Math Art

           - pen
           - pen
           - pattern fill
           - ellipse
           - ellipse
           - pattern fill
           - ellipse
           - pattern fill
           - liquefy
           - liquefy
           - 3D
Math Art
                                                                 Math Art

                                                         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.
                                             Math Art

                         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.
                           Math Art

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.
                                                    Math Art

                          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.
                   Math Art

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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      Math Art

      G. Yap

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