basic 2d design, a toolbox for artists, part 1

a toolbox for artists

Basic 2D design

by

Mårten Strömquist

All copyrights reserved, Mårten Strömquist, BUS, Sweden 2004

”Things live longer than men and the form ofthings, still longer.”

R. Broby-Johansen

INDEX

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

Part OneBasic Concepts

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

Part TwoPoint, line and surface

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

– 3 –Basic 2D design © All copyrights reserved

Ethnic artThe starting point of this study is aninterest in visual qualities of patterns,symbols and abstract, decorative imageryin folk art. These visual traditions, some-times called ethnic art, have a historyas long as that of humanity itself. In the present work examples ofdesigns from a number of cultures, fromall corners of the world, from all agesare represented. However, the stress is not on thepresentation of these designs, theirdevelopment or history but on certainvisual qualities showing differentprinciples of a more general interestfor those working with 2D design.

A toolboxThis work can be seen as an attemptto create a toolbox for developing a rela-tion ship to the world of forms. Here isshown a number of principles of gene-rating and modifying, of combining andrecombining points, lines and surfacesfrom a visual point of view. The aimof the book is fundamentally practical.The theories and principles that areexamined serve the purpose of enrich-ing the creative possibilities when work-ing with form in a visual context.

The limitsWhen looking into this vast field ofhuman expression one soon realizes thata study must limit itself in order to bemanageable. Here I have chosen to leta number of traditional, two dimen-sional, “flat”, designs from differentcultures and ages, serve as starting points.When choosing these designs I have alsolimited myself to those that can be calledabstract or nonrepresentative in contrastto those designs that originate from agraphical abstraction of real things.

The traditionMan has been occupied with visualsymbols, signs, images and patterns as longas we have been able to trace human activity.

Every human culture has its own trea-sure of visual expression for decorativeor esthetic use or for ritual or pedagogicalor communicative reasons or to establishor maintain cultural identity. On every kind of surface humans havestamped their visual messages: on theirown skin, on the walls of caves, on clay,wood, bone, leather, fabric, and later onpaper and today on the screens of com-puters. Some of these designs have be-come classical and timeless, others for-gotten. When it comes to the designs onfabrics, woven, printed or stitched, anendless row of anonymous women havecontributed to this outstanding heritageof visual designs. Today, when it is soeasy for almost anyone with a simpledrawing program to make visual creationsit is maybe even more important toconnect to the knowledge and visualexperiences of this great tradition.

The structureAs I have structured this work it beginswith introductory sections in which I gothrough some basic concepts, helpful fora better understanding of the sections tofollow. These basic concepts are the con-cepts of dimensions and of perception, ofgrids and point location, of symmetry andreflection, of the relationship of negativeand positive form and of visual movementand rotation. The sections that follow are develop-ments from the categories of point, line,round, triangular and square forms andpolygons.

The illustrationsAt last something about the illustrations,most of which have been made digitallyin a vector program, in a very strict way.These drawings are by no means sup-posed to be artistic but are here forpedagogical reasons in the hope thatthey can serve as a starting point forfar more personal expressions.

Mårten Strömquist April, 2004, Malmö, Sweden

FOREWORD


What follows is an index with the most significant imagesshown with a page number referring to the relevant page inthe right corner. When looking up a design in this index it isadvisable to look at the previous and following pages to gaina more comprehensive understanding of the design shown.

At the end of the book there is an alphabetical index withsignificant words listed in alphabetical order. In manycases a word refers to many different pages. Generally thefirst page mentioned is the one with the most generalinformation.

Image Index


63 127 150

128 118 96

121 94 115

151 40 100

120 152 75


95 71 147

46 110 40

144 138 116

107 143 74

110 9159


40 40 93

100 85 79

90 135 104

76 85 101

83 40 38


161 61 110

161 132 121

67 52 110

75 40 147

130 60 58


106 166 106

81 81 106

56 83 100

77 62 18

42 42 42


167 168 164

164164 115

115 115 16

40 135 161

117 108112


Part One

Basic Concepts


The Dimensions

This chart shows the minimum requirements or theminimum of information needed to achieve the followingcategories: point, line, surface or body. It also showsthe relations between length, width and depth, that isto say, the relations of one, two and three dimensions.Each new step on this staircase contains the qualitiesfrom the steps before.

You can also go down the stairs and see howthe body, in this case a tetrahedron can be defined byits surfaces, its sides, by lines or by the spatial relationof the four points.

0D1D

2D3D

One point

Two points

Three points

Four points

Length, distance, measure,border and outline.

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

This includes anythingthat can be surroundedby an outline. The regu-lar developments arecircles, ellipses, and allpolygons and combina-tions of them.

bodyvolumeroomspace

A location, a place, a startingpoint, a point of reference.

The basic condition forform. Occupies no spaceand is best representedby a thin cross.

There are many types ofpoints: points of gravity,centre points, touchingpoints, points of tension,breaking points etc.

Point

Line

Radiuses, diameters, sides,diagonals, heights, areall examples of linesdefined by two points.A line can be understoodas a transportationbetween points. The socalled ”vectors” are linesdefined by a number ofpoints.

Surface

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

Body

The ”real world” as weexperience it. A worldof volumes and spaces.

This world, so to say,”contains” the otherdimensions. Surfaces,lines and points are allfound in the 3D world.


Our perception tells us that we are living in a threedimensional world: a world of objects or bodies andspaces where we perceive three different axes, avertical which separates up from down, a horizontalwhich separates right from left and a depth axis thatseparates behind from in front.

In nature we hardly perceive two dimensional worlds,maybe we can think of a shadow falling on acompletely flat surface as something with only twodimensions, but apart from exceptions like this, whenwe perceive something two dimensional it is an arti-fact, something man made.

Our perception works in another way when looking atsomething two dimensional compared to looking intothe real world of objects and spaces. There seems tobe a ”desire” for a sort of visual understanding assimple and direct as when we look at the real world.We easily see depths in flat surfaces, we try to findconsistent, recognizable shapes although they hardlyare there and so on. The primary two dimensionalshapes of the circle, the square and the triangleseem to exercise a strong influence in this process.

In relation to the chart of the staircase of thedimensions on the previous page it seems as if ourperception easily finds lines where there are points,it finds surfaces where there are lines, it finds depthand space where there are surfaces. It is as if our per-ception always wants to climb to the next step.

So, when we think of the world of images, all thesedrawings, signs, symbols, patterns etc. in a wayrepresent something more connected to the humanconsciousness than to the world. One could say thatan image is, by the power of its two dimensionality,a mental reflection, a genuinely human activity bothwhen created and perceived.

Our vision spontaneously connects points to lines

The primary shapes seem to take precedence overpoints and lines.

Although the triangular shape covers less than halfof its area, it dominates the perception.

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


On perception

In visual communication it is our perception that setsthe limits. In the last century research in the psycho-logy of perception has created a number of more orless well-known images which give us clues abouthow 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 howwe perceive that is crucial when working with visualdesign. The use of geometry in this book is just a toolfor structuring and analyzing the world of forms, andby no means a way to judge if a design is workingvisually or not. The eye is always the final judge, notthe compass or the ruler.

These illustrations showing unexpected perceptionprove the simple fact that the very foundation ofhow we sense or perceive is limited, relative andsubjective.

When working with visual communication this factis a starting point. You have to look, feel, sense andexperience. In this world there are no objectivetruths, no mathematical or geometrical final solu-tions. What which counts is the impressions, sensa-tions and feelings within the person who is looking.Therefore it is within this field that an artist trulycan develop quality when working with the visual.


Image by Muller - Lyer

Image by Poggendorff

In the image above a triangle is seen although it doesnot really exist. The primary forms of the triangle,the square and the circle seem somehow as some-thing the eye “wants” to see.

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

Here you probably assume that the grey shape be-hind the white circle also is a circle.

In the image above it is not a band of three linescut from vision by the rectangles as you can see inthe image below it. This again proves that the eyewants an understandable image. Our vision prefersa certain simplicity and sees what it wants to see.

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


Looking at the image below leads us first to assumethat it is constructed from overlapping triangles.When we see that it is created from just one singleline we still go on looking for the triangles.

In this case, as in some of the other images on theprevious pages, it seems as if our visual perceptionso to speak “prefers” simple geometrical shapes likecircles, triangles and squares.

Later in this book I have devoted a section to theseshapes, called “Round, Triangular and Square”see page 68.


Shown here are some possible juxapositions of primaryshapes. All have the same distance (4 m.m.) butvisually you do not get that impression.

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


Grids

A grid might be percieved as an underlyingstructure as if you were working on transparentpaper using certain points, lines, curves and surfacesfrom the grid beneath.

When working with geometrical or regular designs,patterns and even with single images, the use of gridsis one of the most helpful tools.

You can invent any number of grids but two of them aremore important than others.

Since the circle is the shape that is defined by the leastpossible information, i.e. only two point locations, theclose packing of circles of equal sizes seems to creategrids with great visual potential.

Circles can be packed regularly in two ways. One givesa grid that enables development of shapes with 90 or45 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 packingspheres and thus generating an endless number ofregular and semi-regular bodies. This work was startedby Buckminster Fuller and Keith Critchlow. (See page171, literature)

Apart from these basic grids any of the regular polygonsor combinations of them has been used to create potentgrids.


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

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

Two ways of packing circles

This simple difference of stacking circles gives twodifferent kinds of grids. The first way gives the structurefor the 90 and 45 degree angles, for square andoctagonal developments. The second gives the structurefor 90, 60 and 30 degree angles, for triangular andhexagonal developments.

45

60


Some basic linear possibilities with 90 degree circle packing.


Some basic linear possibilities with 60 degrees dense circle packing.


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


A development of the previous grid, found in both thePersian, the Moorish and the Japanese traditions. Ithas been used as it is, with lines, or further developedwith differently coloured surfaces.


Convex and concaveIf we define geometrical forms, regular or irregularpolygons, as a structure of point locations, these pointsare normally connected with straight lines.

As mentioned later (see page 164) the properties of theforms, i.e. its symmetries, its ability to cover a surfaceetc. will remain even if the straight line is replaced byanother kind of line.

In this simple illustration we choose the square as anexample of a form, and a segment of a circle as anexample of a line with which you replace the straight line.

As you will understand, this is only a very simple develop-ment of a principle that can be developed endlessly.

minimum maximum

Some variations. Of course the same thing can be donewith any shape and any wave line, zigzag line orsegment of an ellipse. Notice the changes of theinterspaces.


Point locationThe concept of grids is closely related to the concept ofpoint location. Using primary grids means connectingpoints with lines. These points may be centre points orcontact/touching points. What the grid does is that itfixes these points in a spatial relationship. To clarifythis we can take the following example.

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

These point locations are related to the regularhexagon (See page 133). One of the properties of thehexagon is that it is one of the “surface filling” units.(see page 164) When copied these units can cover asurface without leaving any interspaces.

The interesting thing is that if you connect thesepoints with any other line, curved or irregular, thisproperty will remain.


Reflection


As we have seen earlier, reflection is fundamental for allkinds of symmetries. A reflected form is like the image ina mirror. Shown here is a 90 degree (vertical) reflectionaxis that makes left to right and right to left.

A 180 degree (horizontal) reflection axis makes top tobottom and bottom to top. This can also be understoodas if a transparent sheet of paper is turned, like in abook. In fig. 1 to the left the first type of reflection isshown. In fig. 2 the horizontal reflection.

Vertical and horizontalreflection

Development by reflection with the letter “R” as anexample.


Another way of creating a pattern is to take any designcopy it and reflect it. In the example below the grey imageis a copy and a reflection of the black. In the fourfoldpattern these two images are copied and reflectedaround a 180 degree axis. Obviously many patterns innature that appear on for example snakeskins, butterflywings, furs etc. can be understood this way.


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


Symmetries

The principles of reflection as shown on page 29are the foundation for understanding the concepts ofsymmetry. Symmetry, as reflection, can be understoodfrom folding a piece paper. Cutting with a pair ofscissors in a piece of paper that is folded one or severaltimes creates different kinds of symmetries.

There are three different kinds of symmetries. Thesingle symmetry, where one half of the shape isreflected once. The double symmetry where onequarter of the shape is reflected vertically andhorizontally. And poly symmetry where morethan four reflection axes are used.

Single symmetry

Double symmetry

Poly symmetry


Even if the diffrence between single and doublesymmetries seems to be simple, it in fact results ininnumerable possiblities.

In the examples below an asymmetric unit (shown hereto the right) is used four times to create either a singleor double symmetric shape. There are of course manymore combinations possible than those shown here.


Negative andpositive space

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

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” patternnamed after a winding river, nowadays situated inTurkey. In this pattern the positive form is a reflection ofthe negative. This concept can be developed endlessly.

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

When working with shapes laid out on a surface,“a background”, one sooner or later realizes thatthis “background” also is made up from forms.One can as well work and change and developthese negative forms, these interspaces. It willeffect the positive forms to the same degree asif we where working directly upon the positiveforms, “the objects”.

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

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


An example of the relation between negative andpositive space can be seen when working with letters.

In this example the negative space is always the same.


The use of black paper and a sharp knife can givesome insights in the relationships of space andinterspace or negative and positive form.

In the simple example above you can see how cuttingout a face from a square black paper leaves all thepieces necessary to create an identical negative copy.

This is a good example of the fact that, whenever youcreate a positive shape, you automatically create itsnegative equivalent. Between negative and positiveshape there is not only a relationship but an absolutedependency.

Below is, let us say, a still life: “Pear on table”. Theconventional way of looking is object centered,which in this case means that the outline aroundthe pear is the outer border of the pear. But if youexperience this outline as the inner border of thesurrounding space the visual result is the same.

To work creatively with form you need to developthe visual receptivity that allows seeing both theseaspects simultaneously.


When working with 2D design the relations of thepositive and negative surfaces must be regarded asparameters, things that can be gradually changed. Asurface, negative or positive can shrink or grow and thebalance that is chosen might be the one that gives thestrongest visual impact when looking at the image.

Shown here are some simple developments withgradual changes which end in an inversion. It isstriking how the visual impact changes althoughthe shapes themselves are basically the same.


The relation between the negative and the positiveform can be related to symmetries, to reflection orto some kind of visual interchange. Another area forthis relation is the change of density. In the exampleto the right the same rhomb has been used for allfour images.

The development shows a change in density. Thenegative space becomes denser and the six pointedstar appears. The change of density can also beunderstood as a change of proportions. (See page153.) This is one of the main parameters whenworking with form. In this example you can seehow there is a radical change in visual impactalthough only this parameter is changed.


Add, subtract, intersect

One result of treating the negative space as of equalvisual importance to the positive space is thepossibility of using the logics of addition, subtractionand intersection to create or modify shapes. Hereare some examples showing the basic principle.If you look at the columns from top to bottomthey work like this:

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 theintersection can be useful in creating single images.Shown here are some simple developments usingthree circles. Three circles can be combined in thefollowing three regular positions. Each combinationgives a number of possibilities for new shapes.

A series of irregular and semi regular shapes aregenerated from the intersection of three circles.If the centre points and touching points are alsoconnected with lines, new sets of forms are madepossible. From this simple starting point manyarchetypal symbols, the drop, the star, the flower,the moon etc. are geometrically derived.


Rotation andmovement


Before going into further develop-ments of the primary shapes wecan look into some principles ofvisual movement. Forms can givethe impression of rotating move-ment. It is worth noting that thevisual impact differs accordingto what turn it takes. A clockwiserotation gives a different impres-sion from a counter clockwise.Both in Taoist and Buddhist textsit is supposed that the counterclockwise movement, going tothe left, is passive and receptivewhereas the clockwise is expres-sive and active.

The two last images on this pageshow how the illusion of depthcan help creating movement.Here overlapping or mesh workis used to create the illusion.


Visual acceleration

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

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


Part Two

Point, line and surface


The PointThe point is that graphic unit which can be describedwith the least possible information. It is the foundationof all other developments of form. Sometimes the pointin this sense has been called “the morphic point” fromthe 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 beused to define any form in all dimensions.

When we look at the two dimensional world of formsthe point serves as the starting point for two different,theoretically interesting, developments. The first is thecreation of a circle, which can be developed as a rowof equal intersecting circles, and the second is whenthe point serves as a centre of radiation. Here we aregoing to look into these two different developments.

These things may at a first glance seem too simple!But nevertheless these simple beginnings are ifrightly understood the basis of innumerable intricatedesign developments.

A

B

So, here is the point. Next to nothing but the onelittle thing which makes everything possible…

We move a distance from this first point “A” andstop at point “B”. This length is the only informationwe have added.

We let the first point serve as a centre and movethe second point all the way around. This way wehave created a circle.

Now we just change the points. Point “B” serves asthe central point and point “A” moves around.Notice that no more information about locationor distance is added!

When these two equal and intersecting circles havebeen created, we just connect the points wherelines cross with the original centre points. You cansee the number of new shapes this generates onthe next page.

A

B

A

B


This is a detail from a Latin translation of a manu-script written by the Arab mathematician al -Khwarizmis in the ninth century A.D. He based it onEuclid's Elementa. Al - Khwarizmis worked in whatwas called “the Academy of Baghdad”. In this aca-demy many new algebraic and geometric relationswere discovered. Descartes studied these discoveriesin the seventeenth century and integrated theminto Western science.

It is remarkable what amount of surface shapes thisintersection of two equal circles generates. Here weonly pick out the most common. Agains it is worthnoticing that they all originate from a minimum ofinformation, i.e. two point locations.

The regular triangle appears in two sizes. This unitis the basis of grids and other developments whichyou can read more about on page 20.

Here is a regular triangle with the three heightsinscribed.

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

Rectangles are also to be found within this first grid.

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


Another basic development of form is the principleof radiation. This can be seen in nature as the basisof several growth patterns and it also plays a rolein forming crystals etc.

From a central point lines, so to speak, beam out indifferent directions. From the point of view of eco-nomy of information this is interesting since all lineshave one point in common. For the regular polygonsyou can find the numbers of the central angles onpage 126.

If you put the center of a compass at this centralpoint and make a circle crossing the radiating linesyou will have point locations for every possiblepolygon, regular or irregular.

Shown here is a simple way of finding the pointlocations of a square, a pentagon and finally ofan eighteen sided polygon where these points areconnected in a regular way.

The center of an agave plant from Arizona, photo MS


This simple primary shape, the “almond” or “eye”shape, generated by two equal circles whoseperipheries touch each other’s centre points, canbe developed in many ways.

Shown here is crossings of the shapes. This can beseen as a repetition of the shape in a horizontaland a vertical position.

At the bottom is shown how the rhomb achievedby connecting the points in the first design developsa regular octagon when crossed.


The Bull or the Ox

Orion

Leo

Virgo

As shown in the section “The Dimensions”, the line,the surface and the solid body or the space arecategories that can be defined by connecting pointswith lines. A simple application of connecting pointsis found in books for children where points, some-times numbered, when connected result in an image.This way of creating forms has an ancient applica-tion in the way almost every culture have seenimages in the sky by connecting stars with lines.

To the right are shown four of the most commonimages that are still in use although their roots datefar back in the cultures of Greece and India.

It is worth noting that, at the time these imageswhere conceived the stars were supposed to befixed on a flat surface, as if painted on a ceiling.Later we have come to know that enormousdistances in depth separate the stars. This meansthat the flat image in these cases is a representationof a thre edimensional reality.

This shows an interesting difference between 2Dand 3D where two dimensional images can beunderstood as projections on a flat surface of a3D reality.

Again we can see how the world of images, the twodimensional world, is an artifact, something man-made. The image is the result, both when createdand conceived, of human mental activity. One ofthe few cases where 2D exists in nature is whenthree dimensional objects cast a shadow on a flatsurface.


So, let us say we have four points and want to see whatimage is created when we connect them with lines. Asshown here we easily get eleven completely differentimages. And still more are possible.

Already at this level we are faced with the complexityof possibilities that simple forms generate. Connecting afew points with lines generates an unexpected numberof images. Here is one of the clues to why “less is more”when working with graphic design.


Painting: Uta Uta Tjangala, without title, fromLouisiana Revue

The Aratjara, the art of the Aboriginal people, is perhapsthe oldest tradition still alive. Some of its designs dateback tens of thousands years. As with the sand paintingsof the American Indians these designs are supposed tobe connected to the subconscious or other worlds.

Without delving into this it is worth mentioning thatimages, 2D designs, have always been somethingdifferent from the “real world” which is three dimensional.These, both when created and perceived, are alwaysdependant of human consciousness.

In the context of this book the design above is anotherexample of the relation of points and lines, where linescan be understood as tranportations between points.


This is a tradition still alive in India. It is practised byvillage women by pouring rice flour on the groundoutside the door. This tradition is called “Rangoli”or “Rangavalli” or in South India “Kolam” and ismentioned in the literary epic Ramayana whichdates back to maybe eight centuries BC.

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


Theopenline


When working with open lines we come into wediscover the phenomenon that lines tend tocreate surfaces, as points tend to create lines.

Here we see how an open line visually successivelycreates a more and more complex play of surfaces.

The final design was found on a piece of African pottery.


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 interestingdevelopments. The pattern below appears in the richMexican tradition and is constructed by the creation ofsurfaces from the lines above. Again it is striking how,from a very simple beginning, the positive and negativeshapes are woven together in a visually complex way.


Here is another development of a single, unclosedline. The unit that is reflected and repeated isshown here to the right. The first part of the lineis reflected 90 degrees and the next part is reflected180 degrees. (See page 29.)

All open line developments can be used asoutlines for surfaces, for example as shown below.


Shown here are two developments of a single open linefound for instance in the traditions of Mexico andGuatemala. It is common both in weaving and decorativepainting. There is a clear relationship to the classicmeander line, but instead of using 90 degree anglesthere is in these cases a use of 45 degree angles.

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


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


Yet another development of a single open line is thewave. Wave lines appear in nature and have beenused in many different design developments. Wavelines of a certain regularity can be described mathe-matically and are called sinus waves.

Here are two examples of simple developments, thefirst using half circles and the next quarter circles.


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


This is still another development of open lines with a strongconnection to the classical Greek meander design. (See page34.) However this pattern is copied from carvings in bone frompaleolithic times (8000 BC).


Another development of the open line can be under-stood from branching. The first example below showsa development when each line is divided in two.In nature this is called dichonomy branching. The nextexample shows a branching pattern where every newunit is a downscaling of the first. The third exampleis a variation of the same principle.

An interesting thing is that these growth patternsrelate to the formula of the “Golden Section”, whichsays, “the smaller relates the the bigger as the bigger

to the whole”. (See page 157.) The same goes for themodern discovery of fractals. Below is shown “Koch’sCurve”, a fractal development that can develop fromleft to right endlessly. Fractals are built by a mathe-matical formula which so to speak “gives a recipe”for how a line (graph) should evolve.

Fractals were mathematically developed by ProfessorMandelbrot. Below is a photo from a fractal developedby Jurgen, Peitgen and Saupe. Recently fractals found apractical application as antennas in cell phones.


Spirals


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

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

A spiral with an accelerated movement can bedescribed as a mathematical graph, i.e. a logaritmicspiral. Such a spiral is based on the formula:

R= 2(L ) 360

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

Another kind of accelerating spiral is the Fibonaccispiral or the spiral of the golden section. Based onquarter circles with radiuses increasing accordingto the numerical series of Fibonacci. (See moreabout this on page 162.)


The spiraling movement appears in growth patternsin nature. Below is a photo from a sprouting fern.As you can see this is a logarithmic spiral develop-ment.

Another way of working with spirals graphicallycan be done from the starting point of concentriccircles. In this example the smallest has half theradius of the next. The next circles areincreased by the difference of the first two.

Now, if all these circles are cut into two halvesvertically and all the circle segments of the righthalf are moved downwards one step, this kind ofspiral is created.

The spiral above is constituted by two lines thatdo not meet at the centre. If you go on and moveall the halves on the right still one step furtherdown, this kind of spiral is created.

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


This spiral design, carved out in red marble about1500 BC. for Atreus treasurehouse, Mykene, Greece,is supposed to build upon designs that originated from

the earlier Minoan culture on Creete. From there itspread slowly north, to appear in Scandinavianbronze designs about a thousand years later.

Yet another kind of spiral development is foundin the geometrical fact that the diagonal is slightlylonger than the side of a rectangle. In this examplewe use a rectangle with the proportions 2:1, and fitanother rectangle with the length of the first dia-gonal as the length of the new one’s sides, alongthe diagonal of the first. This way a complex gridfor a spiral development is created.


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


Round, square, triangularWhen discussing form one always comes back to thesebasic principles: the rounded shapes, the squareshapes and the triangular shapes. These three seemto be a kind of “ABC” when one tries to analyzeimages, symbols, signs, patterns etc.

Each one of them has a significant character of theirown as you can see in the column to the left. To theright you can see that they acquire another visualsignificance when packed in patterns.

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

The triangle represents the minimal area that canbe enclosed by one line.

Squares and triangles belong to the family ofsurface covering units. This is a concept of a formthat, when repeated, can cover a surface completely.See more about this on page 164.

These three different principles of form also givethree different developments of lines as shown onthe following page.


Here are some freehand developments of linesemanating from the circle, the square and the tri-angle. You can see how the significant character ofthe original form is also retained in these develop-ments.

The first row is an open line without crossings. Thesecond row’s open lines allow crossing. Notice theclose step from lines to surfaces.

Without going too far in interpretation most peopleagree that the circular developments create an im-pression of something soft, organic, friendly. Thesquare of something regular, balanced, constructiveand the triangle of something moving, energetic,sharp.

So, these are the three extremes from which endlessvariations and combinations can be drawn.


Here are three examples of closed lines built upon thethree different primary shapes. These are designed toshow that the negative form and the positive form havethe same character. There is a visual interchangebetween them so that the positive changes place withthe negative and vice versa.

After this introduction of the different charactersof the primary shapes we will go on to a closerlook at the possibilities of each of them.


Working with circles


From one to six

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


This may be a more than 3,000 year old symbol, created byan anonymous person in China, one of the most timelessvisual designs ever created by man. It is still used as somekind of free logotype.

We are not going to look into its esoteric significance asa symbol of the unification of the two opposite forces thatconstitute our universe, but only look into its visual qualities.

Anyhow it is worth noting that what the symbol literallymeans: two in one, it also “says” geometrically withits two circles housed in one.

Here is the main example of a design playing with thecontrast of positive and negative form, where one formis a reflection of the other and of visual movement.Below is shown how the grid works and how the curvedline comes from jumping from one periphery to theother at the touching point.

At the top right you can see how the two smaller equal circlesalso give the proportions of the 1:2 rectangle or the “doublesquare” widely used i China and Japan. (See page 156.)


The same principle can be used to inscribe any numberof equal circles. Shown below is a symbol from theJapanese Shinto religion, where three circles are used.

When more than two circles are to be inscribed like thisyou can use a regular polygon (see page 122). The dia-meters of the touching circles should be of the samelength as one side of the polygon in use.


In the grid for this classic design two circles areinscribed in two larger circles. This leads to a com-parison with the grid for the Yin Yang symbol, builtup in a similar way, where the negative space is aturned reflection of the negative.

This design is usually associated with the Ionian culturein Greece but appears in different variations also inmany other times and places. It is common in floormosaics in Roman baths and in the Aescyloustemples from the Greek Roman period.


This is another classic design found on pottery fromthe Greek islands from 1500 BC and onwards. Thisparticular pattern is copied from a bronze vase tobe seen in the National Museum in Athens.

As you can see on the grid, it is a development ofthe so called “dogtail” design just going on includ-ing more circles. More turns in the spiral comefrom adding pairs of the smaller circles.


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


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


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


If we make three circles where the bigger ones increaseby the measure of the radius of the smallest, we willhave 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 theknot. Designs like these appear in many cultures. TheNordic Viking, the Celtic, the Indian Rangolla traditionhave all been working with grids like these.


In this design, based on using two quarter segments oftwo equal circles, the same principle is used as that usedin constructing the Yin Yang symbol is used. At the pointwhere the circles touch you move from the outline ofone circle to the other.

This grid is contained in a rectangle with the pro-portions 1:2. You can see more about this under“Proportions” on page 153. Here is still anotherexample of a classical shape which creates anidentical but reflected negative shape.


Again, if we take the wave created by two quarter circlesmany designs are possible with this unit. As a leafshapeit creates an identical mirrored negative form as seen inthe pattern below. Put in a circular grid (see more aboutthis under “Decagon” on page 149) it creates a lotusmandala common in the Indian culture where the lotuspetal is used in designs in both the Hindu and theBuddhistic traditions.

A special form of these images are the so called Yantras,images used as objects of meditation.


Here are two lotus mandalas from the Yantrictradition, one building upon the triangle and theother upon the square.


This is a variation of the grid wth of dense packing.Here we start with a row of circles where the peri-phery of each circle touches the centre of thefollowing circle.

On the left is shown the final design. And below isshown 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. Thetop and bottom parts of circles have to be removed.

A rectangular section is taken out and intersectswith a copy of the first grid to create the vividpattern shown below.

As you understand this is just one possible develop-ment of a very potent grid.


To the left is shown the grid of a design for aEuropean cathedral window from about 1500 ADwith another application of the dense packingof double circles shown on the previous page.

During the Renaissance the use of circular grids andthe use of half and quarter circles became apredominant feature in interior and exterior design.


This development of covering a surface withsegments of circles is one of the oldest patternsfound. It is sometimes called “fish scale” pattern.At right there is a freehand version carved in bonefrom the Stone Age found in many places. Belowfollows two different geometrical developmentswhere the last has a visual similarity to the “axeshape” found in the Minoan culture in Crete about1500 BC. The photo at the bottom shows a detailof a snake sculpture in the Acropolis Museum inAthens from about 450 BC.


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

1: 2

1:2

5:8

In a way the circle can be seen as a special case ofthe ellipse where the proportions of the vertical andthe horizontal axis are 1:1. Above you can see how anellipse can be created with simple tools. The needlesmark the nodes, two points inside the ellipse withalso a visual interest. The longer axis X is sometimescalled the primary axis and the shorter Y the secondary.

The “Superellipse” above was created by the Danishpoet and mathematician Piet Hein. This shape issupposed to be the perfect balance of circle and square.

It’s formula is X2.5 + Y 2.5 = 1

Y

x

The two dots placed off centre onthe x - axis are the nodes of theellipse. They are placed at thosepoints on the x - axis where thedistance is the same, vertical andhorizontal, to the circumferenceof the ellipse. This gives somegeometrical options as shown inthe drawing to the left.


Workingwith

squares


A traditional Persian pattern where the lines comingfrom the corners and from the middle of each sideprovides opportunities creating a regular new shape.This new crosslike shape gives space for a starlikenegative form. This is a striking example of a widelyused traditional pattern that with great simplicityplays with positive and negative form.


Another example of using the inherent lines in a regularsquare to build a unit that in a pattern creates twodistinct different shapes with an unexpected interplaybetween the negative and the positive surfaces.


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

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


Another way of working with squares is shown here.This grid is fixed to 5 x 7 regular squares. By fillingsome of them an interplay between negative andpositive surfaces is created.

This design is found, as below, in a single row or in acovering pattern in the Chinese tradition. Here theform that follows is a reflection and the followingrow is also a reflection of the first.

With this principle as a starting point it is easy toconstruct similar patterns or images. You can workwith 5 x 5 squares or any other number.

It is worthwhile noticing that there is a strong re-semblance between these Chinese patterns and allthe different “Meander” patterns from Greece. Oneof the differences is that the visual movement inthe meander patterns in the Chinese versions isoften completely “stable”.


A pattern built up by units of a five times five squaregrid, where each new unit is rotated counter clockwisea quarter of a circle.

This creates a pattern where the negative space isidentical with the positive in an intricate way.


Again another variation of the square grid. In thisexample the starting point is a grid of six times sixlines. The development starts by “avoiding” thecentral square from which a rotating movement iscreated. To build the row as in the pattern to theright, these 6 x 6 squares are simply stacked ontop of each other.

On the following page you can see a variationwhere the centre is closed.


This little sculpture is from a tantric tradition inTibet. The design on its chest is developed from asquare grid with seven times seven squares. Thewhite forms looking like a “T” are five squaresbroad and three squares high.

In this case, as in many other in this book, it be-comes obvious that the artistic quality, the visualstrength, gets more or less lost when analyzed in astrictly geometrical drawing. But as mentioned inthe foreword, these drawings are not there forartistic reasons but to help to understand how theimages are structured.


This is a design based on a 9 x 9 square gridinvolving a rotation (you can read more aboutthis on page 41). All these grids made up fromregular squares refer to the basic 90 degreegrid shown on page 21.


Still another development made possible by a grid ofsquares. The “swastika”, sometimes called the sun cross,is found in many early cultures. Shown here are, to theright, a traditional design from the Indian “Rangavalli”tradition which dates back several centuries BC. Andat the bottom a photo from the Acropolis Museum inAthens showing a design which dates back to the 5thcentury BC.

The fact that the Nazi party in Germany in the middleof the last century used this swastika as its symbolassociates it with negative feelings. Sometimes it isargued that the movement of the Nazi symbol is turnedclockwise. You can read more about these things in CarlG Liungman’s book Thought signs (see literature list).


The rhomb is used in several traditional patterns. Hereare some examples from the rich treasure of visualdesign in the African countries Mali and Senegal. Itis a striking fact that peoples living in or near a desertoften develop strong and simple geometrical designs.The patterns of Rajastan in India are another example.


So far we have been looking at regular squares. Butin 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 andtwo 120.

In the examples at the bottom of the page this unitis put around a regular square.

The intersection of two equal circles, where eachcircle’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”. Itcan also be seen as a reflected regular triangle.

This unit has a capacity to form a number of complexdesigns and patterns mostly belonging to the familyof triangular or hexagonal grids(see page, 21, 134).


basic 2d design, a toolbox for artists, part 1

Documents

visual point of view

visual traditions

visual context

visual ex

visual movement

visual qualities of

number of principles

surfacethe point