Basic 2D design, a toolbox for artists, Part 1 by Visual Unfolding

Basic 2D design

a toolbox for artists

Mårten Strömquist

All copyrights reserved, Mårten Strömquist, BUS, Sweden 2004

â&#x20AC;?Things live longer than men and the form of things, still longer.â&#x20AC;? R. Broby-Johansen

Part Two

INDEX

Point, line and surface The Point ................................... 45 The Open Line.......................... 53 Spirals ........................................ 63 Round, Triangular and Square............................ 68 Working with Circles ............. 71 Working with Squares........... 89 Combining Circle and Square.................................. 103 Working with Triangles....... 113 The Polygons.......................... 122 About regular surfaces........ 119 Five sides: The Pentagon.... 128 Six sides: The Hexagon....... 133 Eight sides: The Octagon.... 141 Nine sides: The Nonagon ... 146 Ten sides: The Decagon ...... 148 Twelve sides: The Dodecagon ................. 150 Proportions ............................ 153 Surface filling........................ 164 Word List ................................ 170 Literature ................................ 171

Foreword .................. 5 Image Index............. 6

Part One Basic Concepts The Dimensions.... 14 On Perception....... 16 Grids........................ 20 Point location....... 27 Reflection .............. 28 Symmetries ........... 32 Negative and Positive Form .... 34 Add, subtract and intersect ............. 39 Visual Movement and Rotation ..... 41

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FOREWORD Ethnic art Every human culture has its own treasure of visual expression for decorative or esthetic use or for ritual or pedagogical or communicative reasons or to establish or maintain cultural identity. On every kind of surface humans have stamped their visual messages: on their own skin, on the walls of caves, on clay, wood, bone, leather, fabric, and later on paper and today on the screens of computers. Some of these designs have become classical and timeless, others forgotten. When it comes to the designs on fabrics, woven, printed or stitched, an endless row of anonymous women have contributed to this outstanding heritage of visual designs. Today, when it is so easy for almost anyone with a simple drawing program to make visual creations it is maybe even more important to connect to the knowledge and visual experiences of this great tradition.

The starting point of this study is an interest in visual qualities of patterns, symbols and abstract, decorative imagery in folk art. These visual traditions, sometimes called ethnic art, have a history as long as that of humanity itself. In the present work examples of designs from a number of cultures, from all corners of the world, from all ages are represented. However, the stress is not on the presentation of these designs, their development or history but on certain visual qualities showing different principles of a more general interest for those working with 2D design.

A toolbox This work can be seen as an attempt to create a toolbox for developing a relation ship to the world of forms. Here is shown a number of principles of generating and modifying, of combining and recombining points, lines and surfaces from a visual point of view. The aim of the book is fundamentally practical. The theories and principles that are examined serve the purpose of enriching the creative possibilities when working with form in a visual context.

The structure As I have structured this work it begins with introductory sections in which I go through some basic concepts, helpful for a better understanding of the sections to follow. These basic concepts are the concepts of dimensions and of perception, of grids and point location, of symmetry and reflection, of the relationship of negative and positive form and of visual movement and rotation. The sections that follow are developments from the categories of point, line, round, triangular and square forms and polygons.

The limits When looking into this vast field of human expression one soon realizes that a study must limit itself in order to be manageable. Here I have chosen to let a number of traditional, two dimensional, “flat”, designs from different cultures and ages, serve as starting points. When choosing these designs I have also limited myself to those that can be called abstract or nonrepresentative in contrast to those designs that originate from a graphical abstraction of real things.

The illustrations At last something about the illustrations, most of which have been made digitally in a vector program, in a very strict way. These drawings are by no means supposed to be artistic but are here for pedagogical reasons in the hope that they can serve as a starting point for far more personal expressions.

The tradition Man has been occupied with visual symbols, signs, images and patterns as long as we have been able to trace human activity.

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Mårten Strömquist April, 2004, Malmö, Sweden

Image Index What follows is an index with the most significant images shown with a page number referring to the relevant page in the right corner. When looking up a design in this index it is advisable to look at the previous and following pages to gain a more comprehensive understanding of the design shown. At the end of the book there is an alphabetical index with significant words listed in alphabetical order. In many cases a word refers to many different pages. Generally the first page mentioned is the one with the most general information.

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127

150

128

118

121

115

151

100

120

152

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144

138

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135

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110

161

132

121

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147

130

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166

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100

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Part One

Basic Concepts

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The Dimensions This chart shows the minimum requirements or the minimum of information needed to achieve the following categories: point, line, surface or body. It also shows the relations between length, width and depth, that is to say, the relations of one, two and three dimensions. Each new step on this staircase contains the qualities from the steps before. You can also go down the stairs and see how the body, in this case a tetrahedron can be defined by its surfaces, its sides, by lines or by the spatial relation of the four points.

3D 2D 1D 0D

Four points

Three points

Body

Two points

Surface One point

body volume room space

Line

Point A location, a place, a starting Length, distance, measure, border and outline. point, a point of reference.

Surface, field, area, plane, something ”flat”.

Radiuses, diameters, sides, diagonals, heights, are all examples of lines defined by two points. A line can be understood as a transportation between points. The so called ”vectors” are lines defined by a number of points.

This includes anything that can be surrounded by an outline. The regular developments are circles, ellipses, and all polygons and combinations of them.

The basic condition for form. Occupies no space and is best represented by a thin cross. There are many types of points: points of gravity, centre points, touching points, points of tension, breaking points etc.

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The ”real world” as we experience it. A world of volumes and spaces. This world, so to say, ”contains” the other dimensions. Surfaces, lines and points are all found in the 3D world.

This can also be a ”negative” surface, an interspace.

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Our perception tells us that we are living in a three dimensional world: a world of objects or bodies and spaces where we perceive three different axes, a vertical which separates up from down, a horizontal which separates right from left and a depth axis that separates behind from in front. In nature we hardly perceive two dimensional worlds, maybe we can think of a shadow falling on a completely flat surface as something with only two dimensions, but apart from exceptions like this, when we perceive something two dimensional it is an artifact, something man made. Our perception works in another way when looking at something two dimensional compared to looking into the real world of objects and spaces. There seems to be a ”desire” for a sort of visual understanding as simple and direct as when we look at the real world. We easily see depths in flat surfaces, we try to find consistent, recognizable shapes although they hardly are there and so on. The primary two dimensional shapes of the circle, the square and the triangle seem to exercise a strong influence in this process.

The primary shapes seem to take precedence over points and lines.

In relation to the chart of the staircase of the dimensions on the previous page it seems as if our perception easily finds lines where there are points, it finds surfaces where there are lines, it finds depth and space where there are surfaces. It is as if our perception always wants to climb to the next step. So, when we think of the world of images, all these drawings, signs, symbols, patterns etc. in a way represent something more connected to the human consciousness than to the world. One could say that an image is, by the power of its two dimensionality, a mental reflection, a genuinely human activity both when created and perceived. Although the triangular shape covers less than half of its area, it dominates the perception.

Our vision spontaneously connects points to lines

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We tend to see the shape at the top right as a triangle under another. In other words, we tend to see depth and space where there is only surface.

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On perception

In visual communication it is our perception that sets the limits. In the last century research in the psychology of perception has created a number of more or less well-known images which give us clues about how human perception works. In this work I am not going further with these images, but it is important to recognize the fact that it is how we perceive that is crucial when working with visual design. The use of geometry in this book is just a tool for structuring and analyzing the world of forms, and by no means a way to judge if a design is working visually or not. The eye is always the final judge, not the compass or the ruler.

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These illustrations showing unexpected perception prove the simple fact that the very foundation of how we sense or perceive is limited, relative and subjective. When working with visual communication this fact is a starting point. You have to look, feel, sense and experience. In this world there are no objective truths, no mathematical or geometrical final solutions. What which counts is the impressions, sensations and feelings within the person who is looking. Therefore it is within this field that an artist truly can develop quality when working with the visual.

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In the image above a triangle is seen although it does not really exist. The primary forms of the triangle, the square and the circle seem somehow as something the eye “wants” to see.

Image by Poggendorff

In these two sets the squares have the same distance between themselves. To create the impression that they have a similar distance you would have to narrow the distance between them. So again, it ís the eye not the ruler that is needed.

In the image above it is not a band of three lines cut from vision by the rectangles as you can see in the image below it. This again proves that the eye wants an understandable image. Our vision prefers a certain simplicity and sees what it wants to see.

Image by Muller - Lyer

Here you probably assume that the grey shape behind the white circle also is a circle.

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This is the classic example of how two lines of equal length are seen as different lengths. – 17 –

Looking at the image below leads us first to assume that it is constructed from overlapping triangles. When we see that it is created from just one single line we still go on looking for the triangles. In this case, as in some of the other images on the previous pages, it seems as if our visual perception so to speak “prefers” simple geometrical shapes like circles, triangles and squares. Later in this book I have devoted a section to these shapes, called “Round, Triangular and Square” see page 68.

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Shown here are some possible juxapositions of primary shapes. All have the same distance (4 m.m.) but visually you do not get that impression.

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There is, so to speak, more or less “visual energy” in these juxtapositions. An altogether different visual “event” is created depending upon the shapes that are involved.

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Grids A grid might be percieved as an underlying structure as if you were working on transparent paper using certain points, lines, curves and surfaces from the grid beneath. When working with geometrical or regular designs, patterns and even with single images, the use of grids is one of the most helpful tools. You can invent any number of grids but two of them are more important than others. Since the circle is the shape that is defined by the least possible information, i.e. only two point locations, the close packing of circles of equal sizes seems to create grids with great visual potential. Circles can be packed regularly in two ways. One gives a grid that enables development of shapes with 90 or 45 degree angles, and the other, the dense packing, gives shapes with 60 or 30 degree angles. This has been done in theory also in 3D by packing spheres and thus generating an endless number of regular and semi-regular bodies. This work was started by Buckminster Fuller and Keith Critchlow. (See page 171, literature) Apart from these basic grids any of the regular polygons or combinations of them has been used to create potent grids.

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Two ways of packing circles This simple difference of stacking circles gives two different kinds of grids. The first way gives the structure for the 90 and 45 degree angles, for square and octagonal developments. The second gives the structure for 90, 60 and 30 degree angles, for triangular and hexagonal developments.

Circle packing at 90 degrees The grid for square developments based on 45 and 90 degrees

Circle packing at 60 degrees (dense packing) The grid for triangular and hexagonal developments

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Some basic linear possibilities with 90 degree circle packing.

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Some basic linear possibilities with 60 degrees dense circle packing.

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Another variation of this linear 60 degree grid. Variations of this particular development are found in many cultures. The one below can for instance be found woven on the jackets of certain samurai clans in Japan from the 15th century.

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A development of the previous grid, found in both the Persian, the Moorish and the Japanese traditions. It has been used as it is, with lines, or further developed with differently coloured surfaces.

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Convex and concave If we define geometrical forms, regular or irregular polygons, as a structure of point locations, these points are normally connected with straight lines. As mentioned later (see page 164) the properties of the forms, i.e. its symmetries, its ability to cover a surface etc. will remain even if the straight line is replaced by another kind of line. In this simple illustration we choose the square as an example of a form, and a segment of a circle as an example of a line with which you replace the straight line. As you will understand, this is only a very simple development of a principle that can be developed endlessly.

minimum

maximum

Some variations. Of course the same thing can be done with any shape and any wave line, zigzag line or segment of an ellipse. Notice the changes of the interspaces.

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Point location The concept of grids is closely related to the concept of point location. Using primary grids means connecting points with lines. These points may be centre points or contact/touching points. What the grid does is that it fixes these points in a spatial relationship. To clarify this we can take the following example.

These point locations or fixed spatial relations of points we find in the 60 degree grid (the dense packing). See page 23. Connected with straight lines these point locations give a number of possibilities. Here are six examples.

These point locations are related to the regular hexagon (See page 133). One of the properties of the hexagon is that it is one of the “surface filling” units. (see page 164) When copied these units can cover a surface without leaving any interspaces. The interesting thing is that if you connect these points with any other line, curved or irregular, this property will remain.

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Reflection

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Vertical and horizontal reflection Development by reflection with the letter “R” as an example. As we have seen earlier, reflection is fundamental for all kinds of symmetries. A reflected form is like the image in a mirror. Shown here is a 90 degree (vertical) reflection axis that makes left to right and right to left. A 180 degree (horizontal) reflection axis makes top to bottom and bottom to top. This can also be understood as if a transparent sheet of paper is turned, like in a book. In fig. 1 to the left the first type of reflection is shown. In fig. 2 the horizontal reflection.

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Another way of creating a pattern is to take any design copy it and reflect it. In the example below the grey image is a copy and a reflection of the black. In the fourfold pattern these two images are copied and reflected around a 180 degree axis. Obviously many patterns in nature that appear on for example snakeskins, butterfly wings, furs etc. can be understood this way.

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With any irregular design a regular pattern can be created by copying and reflecting the original unit. Shown here is the patterns with two, four and eight copied and reflected original units. As you can see this development can be repeated endlessly.

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Symmetries The principles of reflection as shown on page 29 are the foundation for understanding the concepts of symmetry. Symmetry, as reflection, can be understood from folding a piece paper. Cutting with a pair of scissors in a piece of paper that is folded one or several times creates different kinds of symmetries.

There are three different kinds of symmetries. The single symmetry, where one half of the shape is reflected once. The double symmetry where one quarter of the shape is reflected vertically and horizontally. And poly symmetry where more than four reflection axes are used.

Single symmetry

Double symmetry

Poly symmetry

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Even if the diffrence between single and double symmetries seems to be simple, it in fact results in innumerable possiblities. In the examples below an asymmetric unit (shown here to the right) is used four times to create either a single or double symmetric shape. There are of course many more combinations possible than those shown here.

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Negative and positive space When working with shapes laid out on a surface, “a background”, one sooner or later realizes that this “background” also is made up from forms. One can as well work and change and develop these negative forms, these interspaces. It will effect the positive forms to the same degree as if we where working directly upon the positive forms, “the objects”.

The relationship between negative and positive form is the fundamental polarity upon which all visual impact is made possible. There is a parallel to the audio world where the relation between silence and sound makes audio impact possible. The phenomena of rhythm build upon a similar relationship of polarity of sound and silence. The pauses and the breaks, the things that “are not”, is the key to rhythm.

The totality of the visual impact is the sum of all positive and all negative forms, where, visually, everything is of equal importance.

As we can see with the “Yin Yang” symbol (see page 74) this awareness is the basis of most classical designs. In this case we look at the Greek “Meander” pattern named after a winding river, nowadays situated in Turkey. In this pattern the positive form is a reflection of the negative. This concept can be developed endlessly.

One could say that the visual impact is depend upon the relationship of all positive and all negative forms. In other words: that which is, is depends upon that which is not.

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In the drawings below there is a shift of the negative and the positive forms from the right to the left side.

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An example of the relation between negative and positive space can be seen when working with letters. In this example the negative space is always the same.

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The use of black paper and a sharp knife can give some insights in the relationships of space and interspace or negative and positive form. In the simple example above you can see how cutting out a face from a square black paper leaves all the pieces necessary to create an identical negative copy. This is a good example of the fact that, whenever you create a positive shape, you automatically create its negative equivalent. Between negative and positive shape there is not only a relationship but an absolute dependency.

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Below is, let us say, a still life: “Pear on table”. The conventional way of looking is object centered, which in this case means that the outline around the pear is the outer border of the pear. But if you experience this outline as the inner border of the surrounding space the visual result is the same. To work creatively with form you need to develop the visual receptivity that allows seeing both these aspects simultaneously.

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When working with 2D design the relations of the positive and negative surfaces must be regarded as parameters, things that can be gradually changed. A surface, negative or positive can shrink or grow and the balance that is chosen might be the one that gives the strongest visual impact when looking at the image. Shown here are some simple developments with gradual changes which end in an inversion. It is striking how the visual impact changes although the shapes themselves are basically the same.

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The relation between the negative and the positive form can be related to symmetries, to reflection or to some kind of visual interchange. Another area for this relation is the change of density. In the example to the right the same rhomb has been used for all four images.

The development shows a change in density. The negative space becomes denser and the six pointed star appears. The change of density can also be understood as a change of proportions. (See page 153.) This is one of the main parameters when working with form. In this example you can see how there is a radical change in visual impact although only this parameter is changed.

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Add, subtract, intersect One result of treating the negative space as of equal visual importance to the positive space is the possibility of using the logics of addition, subtraction and intersection to create or modify shapes. Here are some examples showing the basic principle. If you look at the columns from top to bottom they work like this:

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1. The combination of the outlines of two shapes 2. The two shapes added to each other 3. The left shape subtracted by the right 4. The shapes without the intersection 5. Only the intersection

The principle of adding, subtracting and using the intersection can be useful in creating single images. Shown here are some simple developments using three circles. Three circles can be combined in the following three regular positions. Each combination gives a number of possibilities for new shapes.

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A series of irregular and semi regular shapes are generated from the intersection of three circles. If the centre points and touching points are also connected with lines, new sets of forms are made possible. From this simple starting point many archetypal symbols, the drop, the star, the flower, the moon etc. are geometrically derived.

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Rotation and movement

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Before going into further developments of the primary shapes we can look into some principles of visual movement. Forms can give the impression of rotating movement. It is worth noting that the visual impact differs according to what turn it takes. A clockwise rotation gives a different impression from a counter clockwise. Both in Taoist and Buddhist texts it is supposed that the counter clockwise movement, going to the left, is passive and receptive whereas the clockwise is expressive and active. The two last images on this page show how the illusion of depth can help creating movement. Here overlapping or mesh work is used to create the illusion.

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Visual acceleration Visual movement can also be developed as acceleration or to a dynamic or rhythmic movement. Shown below are some examples. These developments are based on the change of parameters like direction, placement, density and rotation.

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The image at the bottom of this page shows a dynamic development based on simple geometry. A series of regular squares is put in a row and a diagonal line is drawn from one corner to the opposite corner of the following squares.

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Part Two

Point, line and surface

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The Point The point is that graphic unit which can be described with the least possible information. It is the foundation of all other developments of form. Sometimes the point in this sense has been called “the morphic point” from the Greek word “morph”, which means form.

So a point in this sense, a way to define a location, does not itself represent any dimension but can be used to define any form in all dimensions. When we look at the two dimensional world of forms the point serves as the starting point for two different, theoretically interesting, developments. The first is the creation of a circle, which can be developed as a row of equal intersecting circles, and the second is when the point serves as a centre of radiation. Here we are going to look into these two different developments.

We let the first point serve as a centre and move the second point all the way around. This way we have created a circle.

These things may at a first glance seem too simple! But nevertheless these simple beginnings are if rightly understood the basis of innumerable intricate design developments.

Now we just change the points. Point “B” serves as the central point and point “A” moves around. Notice that no more information about location or distance is added! So, here is the point. Next to nothing but the one little thing which makes everything possible…

We move a distance from this first point “A” and stop at point “B”. This length is the only information we have added.

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When these two equal and intersecting circles have been created, we just connect the points where lines cross with the original centre points. You can see the number of new shapes this generates on the next page.

Here is a regular triangle with the three heights inscribed. This is a detail from a Latin translation of a manuscript written by the Arab mathematician al Khwarizmis in the ninth century A.D. He based it on Euclid's Elementa. Al - Khwarizmis worked in what was called “the Academy of Baghdad”. In this academy many new algebraic and geometric relations were discovered. Descartes studied these discoveries in the seventeenth century and integrated them into Western science.

This is a rhomb which appears frequently in this book see page 99.

This rhomb’s smaller diagonal is the same as the long diagonal in the previous rhomb.

It is remarkable what amount of surface shapes this intersection of two equal circles generates. Here we only pick out the most common. Agains it is worth noticing that they all originate from a minimum of information, i.e. two point locations.

The regular triangle appears in two sizes. This unit is the basis of grids and other developments which you can read more about on page 20.

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Rectangles are also to be found within this first grid.

Another basic development of form is the principle of radiation. This can be seen in nature as the basis of several growth patterns and it also plays a role in forming crystals etc. From a central point lines, so to speak, beam out in different directions. From the point of view of economy of information this is interesting since all lines have one point in common. For the regular polygons you can find the numbers of the central angles on page 126. If you put the center of a compass at this central point and make a circle crossing the radiating lines you will have point locations for every possible polygon, regular or irregular.

The center of an agave plant from Arizona, photo MS

Shown here is a simple way of finding the point locations of a square, a pentagon and finally of an eighteen sided polygon where these points are connected in a regular way.

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This simple primary shape, the “almond” or “eye” shape, generated by two equal circles whose peripheries touch each other’s centre points, can be developed in many ways. Shown here is crossings of the shapes. This can be seen as a repetition of the shape in a horizontal and a vertical position. At the bottom is shown how the rhomb achieved by connecting the points in the first design develops a regular octagon when crossed.

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The Bull or the Ox As shown in the section “The Dimensions”, the line, the surface and the solid body or the space are categories that can be defined by connecting points with lines. A simple application of connecting points is found in books for children where points, sometimes numbered, when connected result in an image. This way of creating forms has an ancient application in the way almost every culture have seen images in the sky by connecting stars with lines. To the right are shown four of the most common images that are still in use although their roots date far back in the cultures of Greece and India. Orion It is worth noting that, at the time these images where conceived the stars were supposed to be fixed on a flat surface, as if painted on a ceiling. Later we have come to know that enormous distances in depth separate the stars. This means that the flat image in these cases is a representation of a thre edimensional reality. This shows an interesting difference between 2D and 3D where two dimensional images can be understood as projections on a flat surface of a 3D reality. Again we can see how the world of images, the two dimensional world, is an artifact, something manmade. The image is the result, both when created and conceived, of human mental activity. One of the few cases where 2D exists in nature is when three dimensional objects cast a shadow on a flat surface.

Leo

Virgo

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So, let us say we have four points and want to see what image is created when we connect them with lines. As shown here we easily get eleven completely different images. And still more are possible. Already at this level we are faced with the complexity of possibilities that simple forms generate. Connecting a few points with lines generates an unexpected number of images. Here is one of the clues to why “less is more” when working with graphic design.

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Painting: Uta Uta Tjangala, without title, from Louisiana Revue

The Aratjara, the art of the Aboriginal people, is perhaps the oldest tradition still alive. Some of its designs date back tens of thousands years. As with the sand paintings of the American Indians these designs are supposed to be connected to the subconscious or other worlds. Without delving into this it is worth mentioning that images, 2D designs, have always been something different from the “real world” which is three dimensional. These, both when created and perceived, are always dependant of human consciousness. In the context of this book the design above is another example of the relation of points and lines, where lines can be understood as tranportations between points.

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This is a tradition still alive in India. It is practised by village women by pouring rice flour on the ground outside the door. This tradition is called “Rangoli” or “Rangavalli” or in South India “Kolam” and is mentioned in the literary epic Ramayana which dates back to maybe eight centuries BC.

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Here is another example of the relation of points and lines and you can see the use of both the primary grids explained on page 21.

The open line

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When working with open lines we come into we discover the phenomenon that lines tend to create surfaces, as points tend to create lines. Here we see how an open line visually successively creates a more and more complex play of surfaces. The final design was found on a piece of African pottery.

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If you make a line like this where one shape is repeatedly reflected…

…and then reflect this line…

…you will have the principle for many interesting developments. The pattern below appears in the rich Mexican tradition and is constructed by the creation of surfaces from the lines above. Again it is striking how, from a very simple beginning, the positive and negative shapes are woven together in a visually complex way.

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Here is another development of a single, unclosed line. The unit that is reflected and repeated is shown here to the right. The first part of the line is reflected 90 degrees and the next part is reflected 180 degrees. (See page 29.) All open line developments can be used as outlines for surfaces, for example as shown below.

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Shown here are two developments of a single open line found for instance in the traditions of Mexico and Guatemala. It is common both in weaving and decorative painting. There is a clear relationship to the classic meander line, but instead of using 90 degree angles there is in these cases a use of 45 degree angles.

This is another variation from the same tradition. Of course still more variations on this theme are possible.

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This is a line found on Persian carpets. The grid for the unit that is repeated and reflected is the 60 x 120 degree rhomb (see page 100). Looms set up in a more complex way enable angles other than the 90 and the 45 degree angles.

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Yet another development of a single open line is the wave. Wave lines appear in nature and have been used in many different design developments. Wave lines of a certain regularity can be described mathematically and are called sinus waves. Here are two examples of simple developments, the first using half circles and the next quarter circles.

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This page shows some linear developments made possible by the use of a wave line made by quarter circles.

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This is still another development of open lines with a strong connection to the classical Greek meander design. (See page 34.) However this pattern is copied from carvings in bone from paleolithic times (8000 BC).

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Another development of the open line can be understood from branching. The first example below shows a development when each line is divided in two. In nature this is called dichonomy branching. The next example shows a branching pattern where every new unit is a downscaling of the first. The third example is a variation of the same principle. An interesting thing is that these growth patterns relate to the formula of the “Golden Section”, which says, “the smaller relates the the bigger as the bigger

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to the whole”. (See page 157.) The same goes for the modern discovery of fractals. Below is shown “Koch’s Curve”, a fractal development that can develop from left to right endlessly. Fractals are built by a mathematical formula which so to speak “gives a recipe” for how a line (graph) should evolve. Fractals were mathematically developed by Professor Mandelbrot. Below is a photo from a fractal developed by Jurgen, Peitgen and Saupe. Recently fractals found a practical application as antennas in cell phones.

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Spirals

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Spirals The spiral is a common development of a line that you can also find in nature. You only have to look at the water running out of a bathtub to realize that. The spiral can be found in patterns, designs and symbols in most cultures. Also the labyrinth, a design often laid out with stones on the ground, found in so many different old traditions, often builds upon a spiraling form. The simplest spiral is the one that builds evenly.

Another kind of accelerating spiral is the Fibonacci spiral or the spiral of the golden section. Based on quarter circles with radiuses increasing according to the numerical series of Fibonacci. (See more about this on page 162.)

This even development is used here in this Danish bronze shield, a design from 1300 – 1000 BC.

A spiral with an accelerated movement can be described as a mathematical graph, i.e. a logaritmic spiral. Such a spiral is based on the formula: R= 2(L

) 360

and is shown above right. Of course any accelerated spiral can also be drawn entirely freehand.

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The spiraling movement appears in growth patterns in nature. Below is a photo from a sprouting fern. As you can see this is a logarithmic spiral development.

The spiral above is constituted by two lines that do not meet at the centre. If you go on and move all the halves on the right still one step further down, this kind of spiral is created.

Another way of working with spirals graphically can be done from the starting point of concentric circles. In this example the smallest has half the radius of the next. The next circles are increased by the difference of the first two.

If you take away every second line from the image above, a spiral formed by only one line is created.

Now, if all these circles are cut into two halves vertically and all the circle segments of the right half are moved downwards one step, this kind of spiral is created.

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This spiral design, carved out in red marble about 1500 BC. for Atreus treasurehouse, Mykene, Greece, is supposed to build upon designs that originated from

the earlier Minoan culture on Creete. From there it spread slowly north, to appear in Scandinavian bronze designs about a thousand years later.

Yet another kind of spiral development is found in the geometrical fact that the diagonal is slightly longer than the side of a rectangle. In this example we use a rectangle with the proportions 2:1, and fit another rectangle with the length of the first diagonal as the length of the new one’s sides, along the diagonal of the first. This way a complex grid for a spiral development is created.

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While working with spirals it is worth noting that the visual impression changes when the spiral changes clockwise or counter clockwise. Shown here are four possibilities, turned or reflected.

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Round, square, triangular When discussing form one always comes back to these basic principles: the rounded shapes, the square shapes and the triangular shapes. These three seem to be a kind of “ABC” when one tries to analyze images, symbols, signs, patterns etc. Each one of them has a significant character of their own as you can see in the column to the left. To the right you can see that they acquire another visual significance when packed in patterns.

The triangle represents the minimal area that can be enclosed by one line. Squares and triangles belong to the family of surface covering units. This is a concept of a form that, when repeated, can cover a surface completely. See more about this on page 164. These three different principles of form also give three different developments of lines as shown on the following page.

From one point of view the circle represents a maximum, a maximal area enclosed by one line. The regular square represents some kind of middle way, that can be understood as the crossing of two pairs of parallel lines.

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Here are some freehand developments of lines emanating from the circle, the square and the triangle. You can see how the significant character of the original form is also retained in these developments. The first row is an open line without crossings. The second row’s open lines allow crossing. Notice the close step from lines to surfaces.

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Without going too far in interpretation most people agree that the circular developments create an impression of something soft, organic, friendly. The square of something regular, balanced, constructive and the triangle of something moving, energetic, sharp. So, these are the three extremes from which endless variations and combinations can be drawn.

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Here are three examples of closed lines built upon the three different primary shapes. These are designed to show that the negative form and the positive form have the same character. There is a visual interchange between them so that the positive changes place with the negative and vice versa.

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After this introduction of the different characters of the primary shapes we will go on to a closer look at the possibilities of each of them.

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Working with circles

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From one to six

Shown in the vertical column to the left is how inscribed circles may constitute regular polygons. In the horizontal rows are shown how the triangle gives the geometry for the hexagon, the square for the octagon, the pentagon for the decagon and the hexagon for the dodecagon.

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This may be a more than 3,000 year old symbol, created by an anonymous person in China, one of the most timeless visual designs ever created by man. It is still used as some kind of free logotype. We are not going to look into its esoteric significance as a symbol of the unification of the two opposite forces that constitute our universe, but only look into its visual qualities. Anyhow it is worth noting that what the symbol literally means: two in one, it also “says” geometrically with its two circles housed in one. Here is the main example of a design playing with the contrast of positive and negative form, where one form is a reflection of the other and of visual movement. Below is shown how the grid works and how the curved line comes from jumping from one periphery to the other at the touching point. At the top right you can see how the two smaller equal circles also give the proportions of the 1:2 rectangle or the “double square” widely used i China and Japan. (See page 156.)

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The same principle can be used to inscribe any number of equal circles. Shown below is a symbol from the Japanese Shinto religion, where three circles are used. When more than two circles are to be inscribed like this you can use a regular polygon (see page 122). The diameters of the touching circles should be of the same length as one side of the polygon in use.

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In the grid for this classic design two circles are inscribed in two larger circles. This leads to a comparison with the grid for the Yin Yang symbol, built up in a similar way, where the negative space is a turned reflection of the negative.

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This design is usually associated with the Ionian culture in Greece but appears in different variations also in many other times and places. It is common in floor mosaics in Roman baths and in the Aescylous temples from the Greek Roman period.

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This is another classic design found on pottery from the Greek islands from 1500 BC and onwards. This particular pattern is copied from a bronze vase to be seen in the National Museum in Athens. As you can see on the grid, it is a development of the so called “dogtail” design just going on including more circles. More turns in the spiral come from adding pairs of the smaller circles.

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A freehand version of the double spiraling waves. This time on a clay pot from Pan Shan, China, about 2000 BC.

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You can use grids made out of circles in a number of ways. This first example is found in many traditons.

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Here is still another variation of a similar grid. On the right is a double version. See also “Wavelines”, page 59.

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If we make three circles where the bigger ones increase by the measure of the radius of the smallest, we will have a design as shown at top right. These units can be arranged in a row as shown below. This design serves as a grid for the mesh work and the knot. Designs like these appear in many cultures. The Nordic Viking, the Celtic, the Indian Rangolla tradition have all been working with grids like these.

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In this design, based on using two quarter segments of two equal circles, the same principle is used as that used in constructing the Yin Yang symbol is used. At the point where the circles touch you move from the outline of one circle to the other.

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This grid is contained in a rectangle with the proportions 1:2. You can see more about this under “Proportions” on page 153. Here is still another example of a classical shape which creates an identical but reflected negative shape.

Again, if we take the wave created by two quarter circles many designs are possible with this unit. As a leafshape it creates an identical mirrored negative form as seen in the pattern below. Put in a circular grid (see more about this under “Decagon” on page 149) it creates a lotus mandala common in the Indian culture where the lotus petal is used in designs in both the Hindu and the Buddhistic traditions. A special form of these images are the so called Yantras, images used as objects of meditation.

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Here are two lotus mandalas from the Yantric tradition, one building upon the triangle and the other upon the square.

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This is a variation of the grid wth of dense packing. Here we start with a row of circles where the periphery of each circle touches the centre of the following circle. On the left is shown the final design. And below is shown the development of its grid. The row of circles is packed as densely as possible. The area where they meet has to be cleaned up. The top and bottom parts of circles have to be removed. A rectangular section is taken out and intersects with a copy of the first grid to create the vivid pattern shown below. As you understand this is just one possible development of a very potent grid.

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To the left is shown the grid of a design for a European cathedral window from about 1500 AD with another application of the dense packing of double circles shown on the previous page. During the Renaissance the use of circular grids and the use of half and quarter circles became a predominant feature in interior and exterior design.

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This development of covering a surface with segments of circles is one of the oldest patterns found. It is sometimes called “fish scale” pattern. At right there is a freehand version carved in bone from the Stone Age found in many places. Below follows two different geometrical developments where the last has a visual similarity to the “axe shape” found in the Minoan culture in Crete about 1500 BC. The photo at the bottom shows a detail of a snake sculpture in the Acropolis Museum in Athens from about 450 BC.

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In a way the circle can be seen as a special case of the ellipse where the proportions of the vertical and the horizontal axis are 1:1. Above you can see how an ellipse can be created with simple tools. The needles mark the nodes, two points inside the ellipse with also a visual interest. The longer axis X is sometimes called the primary axis and the shorter Y the secondary.

The two dots placed off centre on the x - axis are the nodes of the ellipse. They are placed at those points on the x - axis where the distance is the same, vertical and horizontal, to the circumference of the ellipse. This gives some geometrical options as shown in the drawing to the left.

The proportions x:y can be changed to any other proportions when working with ellipses. As examples we can use some proportions from page 155.

5:8

1:2 The “Superellipse” above was created by the Danish poet and mathematician Piet Hein. This shape is supposed to be the perfect balance of circle and square. 2.5 2.5 It’s formula is X + Y =1 Basic 2D design

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Working with squares

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A traditional Persian pattern where the lines coming from the corners and from the middle of each side provides opportunities creating a regular new shape. This new crosslike shape gives space for a starlike negative form. This is a striking example of a widely used traditional pattern that with great simplicity plays with positive and negative form.

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Another example of using the inherent lines in a regular square to build a unit that in a pattern creates two distinct different shapes with an unexpected interplay between the negative and the positive surfaces.

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Another example of using the potential lines in a square to make a new regular shape. Again, when a pattern is built with this unit, a distinct negative shape appears.

When working with patterns like this the scale is of utmost importance for the visual impact. The relation between the size of the whole surface and the size of the units that build up the pattern is fundamental for the impression that is created. (See more about this in the section “Proportions", page 153.)

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Another way of working with squares is shown here. This grid is fixed to 5 x 7 regular squares. By filling some of them an interplay between negative and positive surfaces is created. This design is found, as below, in a single row or in a covering pattern in the Chinese tradition. Here the form that follows is a reflection and the following row is also a reflection of the first. With this principle as a starting point it is easy to construct similar patterns or images. You can work with 5 x 5 squares or any other number. It is worthwhile noticing that there is a strong resemblance between these Chinese patterns and all the different “Meander” patterns from Greece. One of the differences is that the visual movement in the meander patterns in the Chinese versions is often completely “stable”.

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A pattern built up by units of a five times five square grid, where each new unit is rotated counter clockwise a quarter of a circle. This creates a pattern where the negative space is identical with the positive in an intricate way.

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Again another variation of the square grid. In this example the starting point is a grid of six times six lines. The development starts by “avoiding” the central square from which a rotating movement is created. To build the row as in the pattern to the right, these 6 x 6 squares are simply stacked on top of each other. On the following page you can see a variation where the centre is closed.

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This little sculpture is from a tantric tradition in Tibet. The design on its chest is developed from a square grid with seven times seven squares. The white forms looking like a “T” are five squares broad and three squares high. In this case, as in many other in this book, it becomes obvious that the artistic quality, the visual strength, gets more or less lost when analyzed in a strictly geometrical drawing. But as mentioned in the foreword, these drawings are not there for artistic reasons but to help to understand how the images are structured.

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This is a design based on a 9 x 9 square grid involving a rotation (you can read more about this on page 41). All these grids made up from regular squares refer to the basic 90 degree grid shown on page 21.

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Still another development made possible by a grid of squares. The “swastika”, sometimes called the sun cross, is found in many early cultures. Shown here are, to the right, a traditional design from the Indian “Rangavalli” tradition which dates back several centuries BC. And at the bottom a photo from the Acropolis Museum in Athens showing a design which dates back to the 5th century BC. The fact that the Nazi party in Germany in the middle of the last century used this swastika as its symbol associates it with negative feelings. Sometimes it is argued that the movement of the Nazi symbol is turned clockwise. You can read more about these things in Carl G Liungman’s book Thought signs (see literature list).

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The rhomb is used in several traditional patterns. Here are some examples from the rich treasure of visual design in the African countries Mali and Senegal. It is a striking fact that peoples living in or near a desert often develop strong and simple geometrical designs. The patterns of Rajastan in India are another example.

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So far we have been looking at regular squares. But in the family of squares we can also include the rhomb. In this case a rhomb mentioned earlier (see page XX): the one where two corners have 60 degree angles and two 120. In the examples at the bottom of the page this unit is put around a regular square. The intersection of two equal circles, where each circle’s periphery crosses the centre of the other circles, gives four points, that when joined, create a rhomb. This rhomb is sometimes called “diamond shape”. It can also be seen as a reflected regular triangle. This unit has a capacity to form a number of complex designs and patterns mostly belonging to the family of triangular or hexagonal grids (see page, 21, 134).

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