Lesson 1:
A look at the Mandala as an art form

Lesson 1
Ritual Geometry


Lesson 2
Group Elements

Color Theory

Lesson 3
Groups and Groups Acting on Sets

Block printing

Lesson 4
Klimt and the Computer

Color and symmetry in modern art I

Lesson 5

Islamic art

Lesson 6
Penrose and Rice

Color and symmetry in modern art II

Lesson 7
Escher 1

Escher 2

Lesson 8
Hundertwasser & Griffeath

Pattern and Modern Painting

Brian P. Hoke: Cellular Automata and Art

Student's Work


Opening questions on examples of student mandalas

More Student Mandalas

What do you see in these Mandalas?
What kind of marks, lines, shapes?
What kind of media was used to create them?
How does the media affect the design?
Which designs are symmetrical?
How can we define symmetry?

Symmetry often functions as a paradigm of the balance and harmony found in the natural world and great works of art. But mathematicians define symmetry in a more specific way. They describe the repetition of forms as movement in space, and distill symmetry and pattern into a set of laws that operate on an abstract level.

From Hargittai: "SYMMETRY A Unifying Concept", p.47
Shelter Publications, Inc., Bolinas, California, 1994

A 2-dimensional motif can repeat in only four different ways. These four types of repetition or symmetry operations are:

  • 1) Rotation - the motif turns

    From István and Magdolna Hargittai: "SYMMETRY   A Unifying Concept"
    Shelter publications, Inc.

    Folk Art, p.49
    Decorations displaying exclusively rotational symmetry often occur in folk arts. Old Native American pottery has decorations with a wealth of rotational-only symmetry.

  • 2) Reflection - the motif reflects, as in a mirror

    From Hargittai: "SYMMETRY A Unifying Concept",p.2

    From Hargittai: "SYMMETRY A Unifying Concept", p.73

  • 3) Glide reflection - the motif translates and reflects,

    From Hargittai: "SYMMETRY A Unifying Concept",p.135

  • 4) Translation - one motif moves up or down, left or right or diagonally while keeping the same orientation,

    From Hargittai: "SYMMETRY A Unifying Concept",p.183

How does symmetry relate to pattern?

Artists use patterns more intuitively and expressively. Today we will be looking at important elements of design: line, shape and value. We'll also observe that these elements of design can form patterns, and the effects these patterns produce on the viewer.

Design Elements

Lines can exhibit many different traits. Lines can be straight, curved, zigzag (angular), or broken.

This drawing by Steinberg shows a regular repetition of straight lines. What sort of experience or mood do these repeating line patterns convey?

Saul Steinberg: "THE DISCOVERY OF AMERICA", p. 9
Alfred A. Knopf Inc., New York, 1992

The curvilinear lines in this drawing by Matisse incorporate many qualities: sensuality elegance, economy, sensuality, and elegance.

From Bert Dodson: "Key to Drawing", p.47
Published by North Light, 1985

Van Gogh uses circular, energetic, emotionally charged lines.

From Bert Dodson: "Key to Drawing", p.49
Vincent van Gogh, Cypresses, Saint-Remy. 1889

The smooth undulating lines of this Japanese woodcut create an atmosphere that is natural, harmonious and decorative. The density of line builds depth into the compositon

Andrew W. Tuer: "Japanese stencil Designs", p. 40

Dover Publications, Inc., New York, 1967

In this sketch of sleeping figures Rembrandt builds up loose marks portraying the presence of light and shadow with a "free and supple hand."

From Bert Dodson: "Key to Drawing", p.45
Rembrandt van Rijn, Saskia Asleep.

Using a pen or pencil, draw lines with different qualities (angular, curved, etc) in your notebook.

Shapes are visually perceived areas of value, color or line and any combination of angular and curvilinear configurations. There are endless types of shapes: soft; hard; round, etc. These traits allow artists to express different feelings, ideas, personalities, i.e. shy, angry, loving. Geometric (angular) shapes are machine like, dynamic, rigid. Biomorphic (curved) shapes suggest the possibility of life, emotions, nature. Shapes can also help to create balance and consistency in a composition. (Footnote1: Ocvirk, Otto G.:"Art Fundamentals") 2

  • Angular shapes:

    N. Simakoff: "Islamic Designs in color", p. 43
    Dover Publications, Inc., New York, 1993

    From Norbert Lynton: "The Story of Modern Art",
    Cornell University Press, Ithaca, New York, 1980

    Figure 146. Chales Sheeler: Church Street El.

  • Soft open shapes:

    Figure 54. Robert Delaunay: Window on the City

    Figure 71. Vassily Kandinsky: Improvisation 13.

  • Biomorpic shapes:

    From Norbert Lynton: "The Story of Modern Art",
    Figure 197. Jean Arp: Amphora

  • Shapes combine to create patterns:

    John Elderfield: "The cut-outs of Hehri Matisse"

    Portrait of Fritza Riedler, 1906, p.56

    Gustav Klimt

    Portrait of Fritza Riedler, 1906, p.55

How does Klimt use shape?

Would you describe the shape combinations of Klimt and Matisse as patterns?

Negative and Positive Shapes
The dynamic of positive and negative space is the primary organization of space. Areas that represent the artist's initial choice of shapes are called positive areas. They can be recognizable or not. The empty spaces are called negative spaces, but are just as important. (Footnote2: Ocvirk, Otto G.:"Art Fundamentals")

John Elderfield: "The cut-outs of Hehri Matisse"

Portrait of Fritza Riedler, 1906, p.51

  • Are Matisse's shapes geometric or biomorphic?

  • How do you respond to this composition?

Value is the name given areas that are white to black or dark to light. The value of an area is seen by the amount of light reflected from it. Value has expressive power. (Footnote3: Ocvirk, Otto G.:"Art Fundamentals")

The World of M. C. Escher,
Figure 109. Three Spheres II. p. 94

Light hitting a solid creates shadow and a sense of volume. Variations in light and dark create planar space. Value gradations evoke gradual movements across a plane or around a sphere. Even,flowing tones create a gentle curve, while abrupt changes in value produce sharp or abrupt angles in space.

Observing an unfocused slide of a black and white photograph to draw and explore the values of its composition: With your pen, draw a small rectangle (about 4"x6") in your notebook. Depict in your own way the pattern of dark and light values you see within the rectangle.

After the exercise, the slide is focused so that image is clear.
What was it like for you to make an abstract composition?
All compositions have an abstract foundation of lines, values, and shapes.


For Homework students design two Mandalas
A Mandala is sansckrit for circle - slide
Mathematically a circle is defined to be the set of all points in a plane equidistant from a given point called the center of the circle. Geometriclally the circle is divided into units called degrees. A circle has an infinite amount of symmetries. How are the designs in the mandala format different from a rectilinear composition?

A line web of ideas expands outward from a center -- computer graphics.

Mandalas are found everywhere and have powerful symbolism across all cultures

All circles have a center. This is principle of the center. The center is everywhere.
The center represents expansion and integration of opposites, micro and macrocosms.
The center also represents the soul, god or the individual.
Circles symbolize unity/time/universal cycles/life creation/eternity/god/sun.
Circles/mandalas contain all symmetries.
All of Nature's harmonious principles are exhibited through it's harmonious relationships.

From Jose and Miriam Arguelles: "Mandala"
Shambala: Berkeley and London, 1972

Mandala of Kalachakra, p.14

Two mandala designs due in one week.

Parts a and b to be started and completed in class1 if possible, and turned in with homework at critique.

a and b: pencil, ruler, brushes, black and white paint, white paper. For c and d add, black construction paper, a compass and black marker

Part a: Gray Scale

The purpose of this exercise is to develop skill in seeing and using the gradations of value that range between black and white. This will provide background for color studies, since many colors combine hue with. black and/or white. Neutral areas of light and dark are sufficient within themselves to create powerful visual statements, as evidenced by many black and white photographs, prints, drawings, paintings and fabrics.

On a small sheet of drawing paper, draw a vertical or horizontal bar of 9 one inch squares with a pencil and ruler. Observe the range of grays in your photograph. Try to find at least 9 shades of gray, mixing paint to match. Fill in the squares with thee gradations in a light to dark arrangement. Mix your paint with water to about the consistency of slightly melted ice cream.

Part b: Irregular geometric design.
Students work from black and white photographs.

The purpose of Part b is to translate recognizable images into abstract shapes and define them in relationship to each other with a subtle range of values.

  • 1. On a piece of white paper outline a 12" square or rectangle.

  • 2. Sketch an abstract design of irregular geometric shapes based on observation of your photograph.

  • 3. You may follow the relative composition of the photograph.

  • 4. Paint the design using your knowledge of grays from the previous exercise. Be as sensitive to the negatinve shapes as you are to the positive shapes. Observe how scale (size and value can create spatial relationships).

Part c: Extract a shape for mandala

In Part c you will extract a particular shape from your painting to use as the main repeating motif in your mandala. A good method is to cut a small triangular window in the center of a piece of paper and scan your design for a shape that you like. Once you have isolated a motif, trace or draw it. Make several templates by placing the shape on black construction paper, tracing around it and cutting it out.

  • 1) Draw a circle with a radius of 5 1/2 inches on white paper and lay out your shapes. Your mandala can grow larger than this dimension. Play with the shapes, moving them around to see what kind of negative areas they make in different locations and directions. If it's not right, modify the shape until you like it by cutting or redesigning it and making new templates. Use two different sizes of the final shape and arrange your mandala.

  • 2) When you have decided on the design, trace around each shape with a pencil. Notice the kind of momentum the circle creates and how you can work with it or counter it. Include the center as an important part of the design even if it is blank. Maintain symmetry in the center. This mandala must have at least 3 fold symmetry.

  • 3) Decide which areas you want to be negative or positive. Start outlining and filling in with black marker.

  • 4) Use negative areas for freehand embellishment, creating repetitions and rhythms with points, lines or other small shapes. Be conscious of the density and intervals of the repetition in relation to the rest of your mandala design. Repetition is regular repeat ing . . . . . .
    A rhythm is repetition organized in intervals .o.. .o .o.. .o

Part d:
Create a second mandala by reinterpreting your first black and white design with at least four different values. These grays can be mixed in a gradient ranges of high key (lighter) or low key (darker) grays. Alter the new design as much as you want to make it successful. You can keep the two designs visually related or depart widely from the first mandala. Suggestion: explore using different values to create overlapping shapes.

From Wucius Wong: "Principles of Color Design",
p.32, Figure 87, 88

M. C. Escher:"ESCHER on ESCHER Exploring the Infinite"
Circle limit IV, p. 42

Jose and Miriam Arguelles: "MANDALA",

From Ocvirk, Otto G. : "Art Fundamentals"
Design Elements


Gouache is a water based opaque paint mixed With a gum base. When storing mixed colors, add a drop of water to the top

Jars for water
Small containers with lids for storing mixed colors or a palette
Containers for water
Paper towels

Open the tube of paint and get rid of any clear liquid on the top. This is an oil and it will streak.

Mix your colors on a pallet, or if you are mixing a larger amount use a container with a lid. Use a cheap brush for mixing. For your first assignment d palette is fine. Always add some water to your paints, but use water sparingly. The consistency of the paint should be like thick melted ice cream.

Dark colors require a lot of white to get to a light color, so start mixing from lighter to dark. Begin by adding a small amount of black paint on you palette to a larger amount of White paint.

For each darker gray continue adding small amounts of black into the Whites moving toward the a near black range at the end. Note that's some areas will appear to look the same even though more black was added to the mixture. This is because at a certain point the amount added to the mixture must increase in order to effect a visual change.

Use a small brush to paint the edge of shapes. Paint away from the edge uncorking from one side of the shape to the other.


Pencil/ eraser

tracing paper
measuring tools
graph paper
black marker
block cutting tools
white paper
ink roller
safety kut block
Speedball printing inks
mat board scraps for mixing ink
x-acto blade or mat knife

You may get printing ink on your clothing, so bring or wear something you can print in.

Instead of working out initial ideas on the White paper in your kit, you could purchase a pad of newsprint. You can also use your graph paper for this but the newsprint is better for block printing tests.

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© Copyright 1996, Pippa Drew and Dorothy Wallace, Dartmouth College