Basic 2D design, a toolbox for artists, Part 2

This unit, looking like a book or an arrow’s feathers, is made up from two diamonds. It can build patterns in many ways, where the negative spaces create another pattern. Here is just one example.

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This is a photo from the Acropolis Museum in Athens showing a part of a snake’s tail from about 450 BC. Notice the spiraling rhombs decorating the snake’s skin.

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From square to circle

The square and the circle are often thought of as opposites. The squareness of a circle has been a synonym for a paradox similar to the one of creating a “tulip-rose”.

This illustration is taken from the German Fluxus artist Thomas Schmidt, who in 1972 made a number of humorous drawings showing ways of changing a square to a circle.

However there are a number of things that unite the circle and the square. In this case it is the change of the corner radius that is the clue. More about this on the next page.

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Corner radius The concept of corner radius is based on a replacement of the straight angle of a square or rectangle with a quarter segment of a circle.

In the case of a regular square, when the radius of the quarter circle is the same as half the side of the square the square has become a circle and the corner radius is 100 %. This means that there is a parameter that can be changed. The example below shows four steps from 10 % to 50 %. The example at the right shows a 100 % corner radius at different corners of a rectangle.

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As I just mentioned, the circle and the square can be thought of as opposites. One thing that unites them is the cross. The vertical and horizontal axes are visually and geometrically latent in both the circle and the square.

This is an ancient symbol found in early cultures all over the world. It has been called “the wheel”, “the sun-wheel” or “the sun-cross”. It is found carved out in rocks from early Stone Age as well as in Bronze Age tombs.

These combinations of circle and square can be seen as some kind of archetypes in the world of form. It is the basis of the Indian Mandala and it is the fundamental structure in most religious buildings, from the Buddhistic Stupas, to the Mosques, from the Synagogues to the Christian Domes and the Hindu Temples.

The cross seems to indicate the two directions, South/North and East/West or from a more subjective view: Front/Back, Left/Right. In short the essentials of the 2 D world.

Throughout history in all cultures the use of the combination of the square and the circle is predominant when it comes to the so called floor plans of sacred buildings.

We can see it below in the different plans of the cathedral St Peter’s in Rome, one of the greatest buildings ever built. These plans here worked out from 1505. First by Bramante, then by Michelangelo and lastly by Maderna who finalized these plans.

The floor plan made by Michelangelo for St Peter’s.

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The floor plan made by Bramante for St Peter’s in 1505. © All copyrights reserved

From the two grids below, where circles are drawn around the corners of squares, a number of crosslike images can be made.

The same ideas have been used for other purposes too.

Castles, gardens, parks, ponds etc. have been designed from the concept of combining circles and squares.

To the right a plan for the Royal Garden in Copenhagen from1606, which again builds its geometry on squares and circles To the left, a plan from the same age for an observatory, “Uranienborg”, planned by the Danish astronomer Tycho Brahe.

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Below are shown some designs combining the square and the circle used in for example floor plans.

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This archetypal symbol dates back to at least five centuries BC. It exists in a lot of variations within different traditions in Hindu and Buddhist religions.

It often frames the famous Tanka paintings in the Tibetan tradition. Like other religious symbols it is supposed to open connections to subconscious levels.

How it may be constructed from a 10 x 10 square grid is shown on the right. We see again the fusion of the circle and the square with a clear indication of four directions, front/back and left/right or north/south and east/west.

Perhaps the use of this archetype comes from the need for finding a point of reference, a fixed location in subjective space.

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This is copy of a page of one of Kasimir Malevitj’s sketchbooks from 1913. Malevitj is perhaps one of the most remarkable painters of the last century. Between 1913 and 1918 he made a series of drawings and paintings working on how visual tensions between abstract shapes could energize the space of the image.

In this work he came to the conclusion that the three abstract shapes that create the maximum of visual energy in the picture´s space is the square, the circle and the cross. He called these forms “the basic suprematist elements”.

This was a part of a larger theory contained in the idea of suprematist art. These concepts have had a

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radical impact on working with form not only in art but also in design and architecture. The discovery of how simple abstract shapes carefully related to the surrounding space still exercises a major influence on the world of design.

The works of “Suprematism” and of other Russian avant-garde artists at the beginning of last century made a break with all the imitations of style and all decorative elements that until then had limited the perspectives for designers.

In the context of this book it is interesting to notice that these “basic suprematist elements” are the same as the ones on the previous pages although based on completely different ways of thinking.

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A “round square” may also be created by replacing the straight lines that connect the corners of the square by segments of circles.

In the example to the right the sides of the square have been replaced by half a circle, one third, a quarter and one fifth of a circle.

Here is a connection to the central angle of different polygons, see page 126.

Below four segments of 240 degree centre angle are reflected.

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So… is it possible to make a square using only circles?

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This is a symbol from Zuni, a tribe of native Americans living south of the Navajo Nation in Arizona. Again an image which fuses the circle, the cross and the square.

It is interesting to see that cultures that have not had any kind of contact, like the Tibetan Tanka paintings with their mandalas, the architects of the cathedrals in medieval Europe or the American Indians with their sand paintings, still base their works upon the same visual geometry.

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When we send a post card with a photo of the place where we are staying to friends at home, sometimes we put a cross or a circle with a pen on a hotel window to say “I am here”.

Maybe this is the essence of these symbols. When you are “here” you are where the east/west axis crosses the north/south or from a more subjective view where the left/right crosses the front/behind.

This is the point of reference of both subjective and objective space.

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Working with triangles

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Working with the regular triangle, inscribing circles, connecting points with straight lines, seems to create an endless number of new regular triangles. There are some strange symmetries in this triangle. We have earlier spoken of vertical and horizontal reflection, that is to say, reflections by 90 or 180 degrees. In the development of regular triangles the angles of 60 and 120 degrees are predominant.

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One could also understand the regular triangle as a rotation of the 30, 30, 120 degree triangle as shown below.

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The smaller regular triangles that are contained in a bigger have a potential to be scaled. To the left is shown one descending and one ascending scaling.

Below is shown the first three down scaling possibilities. These options with the triangular form have been used in innumerable patterns and designs.

Three triangles placed like this form another semi regular shape, sometimes called “parallel trapezium”.

This shape is a unit that gives a lot of possibilities. It has found a practical application when shaping table tops to further a flexible way of putting together tables.

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This traditional development of a triangular shape is found in many cultures. It is sometimes called the Celtic flower or the Celtic knot but it is also found for instance in traditional Japanese family symbols.

In the example below a half circle is used. It is repeated three times, each time revolved 120 degrees.

These three elements are then fused so that the end points of the circle segments meet.

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Below to the right is shown a variation when a third of a circle is used. This gives a more pointed shape.

When using lines in a mesh creating the impression of one line going under the other, this impression becomes more striking when a third line is added between the outlines.

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This is a traditional way of working with a combination of triangles and squares. It is found in textiles, ceramics and wall paintings in Guatemala and Mexico, and may be inspired by the great stair pyramids in Tikal in northern Guatemala.

You can build triangles with regular squares in series like 1, 3, 5, 7 etc. The higher you go in these numbers the more triangular the shape. This ancient way of creating images resembles the modern use of pixels!

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1

1

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You can also close the series by going back again, like 1, 3, 5, 7, 5, 3, 1. In that case the final shape always fits into a regular square regardless of how long a series you make.

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By replacing the straight lines that by 120 degrees divide a circle in three equal parts with segments of the inscribed equal circles, a unit is formed that, paired with another similar unit twisted 60 degrees, gives the grey shape shown below. This new unit fills a surface as well as triangles do.

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This is an interesting way of using polygons where only the touching points count. Any line, straight or bent, that connects these points will give patterns or units that have the same properties as the original polygon. You can read more about this in “Surface filling” on page 164.

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Design developments with a rotating triangle

The following pages show how triangular grids can generate a wide variety of design developments. This can of course be done with all other grids and polygons. To save some space I choose here to limit these examples to the triangle to show some general principles. The regular triangle is maybe the most concentrated and potent shape for developments like this.

The general principles of visual rotation you can read about in the section “Rotation and movement” on page 41.

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About the polygons When studying 2D design, the polygons occupy a central place. Working with 2D means working on a flat surface and most surfaces can often somehow be defined as polygons. But more important is that each polygon, single or packed over a surface, creates structures for specific grids and opens up possibilities for a certain setup of points, lines, angles, proportions, surfaces and segments.

Each polygon has its own setup of all these features, which alone or combined with each other create a vast field of visual possibilities. You can say that each polygon has an anatomy of its own. As we have said earlier, the triangle, the square and the six sided hexagon are special that they are the only shapes which when combined, can cover a surface completely. All other shapes create interspaces.

Polygons are found in nature. Crystals and salts are organized with polygonic surfaces. The most common molecular structure in organic chemistry is built upon the hexagon. Flowers and fruits are based upon polygon grids. The family of Primula is based on a pentagon, the family of tulips is based on a hexagon etc. (see page 133.)

All star and flower shapes are derived from polygons. An interesting thing is that the polygon star shape, for instance the five pointed star, represents the concept "star" stronger than any photo of a real star would do!

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Stars based on the hexagon or the pentagon have been symbols for nations, armies, religions and multinational corporations, from the Red Army of China to the US Air Force, from Jewish to Muslim religions, from Texaco Oil to the European Union.

If we look at the regular polygons with three to twelve sides, we notice that they all have an integer for their degrees of number for the angle of their central segment except for the seven and eleven sided polygons. The whole turn at the centre of a circle is 360 degrees. This means that the triangular segment’s angle is 120 degrees, the squares 90, the five sided pentagons 72, the six sided hexagons 60, the eight sided octagons 45, the nine sided nonagons 40, the ten sided decagons 36 and the twelve sided dodecagons 30 degrees.

One can also claim that the triangle, the square and the pentagon are the most important because through them the six-, the eight-, the nine-, the ten- and the twelve sided polygons can be constructed. The six sided hexagon can be understood as a double triangle, the eight sided octagon as a double square and the ten sided decagon as a double pentagon etc.

We are now going to take a look at the regular polygons, starting with the five sided pentagon since we have already covered the triangle and the square in earlier sections. (See page 89 & 113.)

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The polygons, three to seven sides 3 Triangle

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4 Square / Tetragon

5 Pentagon

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Hexagon

Septagon

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The polygons, eight to twelve sides

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Octagon

Nonagon

Decagon

Endecagon

Dodecagon

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Here are all regular polygons from the three sided one to the twelve sided one inscribed with one side, the base, the same for all.

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All polygon segments with a whole number for the degree of their centre angles are listed to the right.

Three sides Triangle: 120 degree segment

These degrees multiplies by the number of the polygons’ sides always amount to 360.

Knowing these angles not only helps you to create the polygons but also to turn them when you want them to stand on the top or a side, when you join them etc.

Four sides Square: 90 degree segment

It is worth mentioning that the number 360 means that at least eight polygons have integers as numbers of their angular degrees.

If, as is sometimes suggested, the degrees of the circumference of the circle were to be changed to 100 (the metrical system), only three of the polygons’ angular degrees would be integers, namely the squares, the pentagons and the decagons.

Five sides Pentagon: 72 degree segment

Six sides Hexagon: 60 degree segment

Eight sides Octagon: 45 degree segment

Nine sides Nonagon: 40 degree segment

Ten sides Decagon: 36 degree segment

Twelve sides Dodecagon: 30 degree segment

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From polygons to stars In the column to the left we have all polygons from three to eight sided. The transformation from a polygon to a star shape can be understood as if the midpoints of the sides of the polygons are pushed towards the center. Thereby series of star shapes are made possible in a simple way.

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Five sides The pentagon

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Now we take a closer look at each polygon, starting with the five sided pentagon as we have already looked at the triangle and the square.

The pentagon has been associated with secrets and occultism maybe because of the difficulties of drawing a perfectly regular pentagon with only the help of compass and ruler. There are several ways to do this.

(One you can see on page 158 in “Proportions”.)

One of the most remarkable things about the pentagon is that its diagonals cross each other exactly at the point where the diagonal is divided in the proportions of the golden section. (See page 157.)

The pattern below from the Islamic tradition is based on a variation of the pentagonal star, the pentagram, a design that as shown above can be drawn as a single closed line.

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Looking at the star that is formed by the diagonals of the pentagon fools our perception. There seems to be an arrowlike shape crossing over a similar shape but the eye can't quite catch hold of them. In fact there are five shapes like this but each one is overlapping the other.

Notice also the number of symmetrical triangles created.

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This design is built with one central pentagon. Around that five other equal pentagons are fitted. Five copies of this new unit are placed in the same way. This can be repeated any number of times. In this final design you can see how the outline is still a pentagon which can be copied five times and placed around the five sides…

It is interesting to see the development of irregularities.

There is a space in this design that has a fractal evolution.

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These are the spaces between the pentagons in the previous design as they develop from the middle. This is an interesting example of a fractal development that is connected to the geometry of the pentagon. The similarities with "Koch’s curve" are quite obvious.

(See page 62. )

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Six sides The hexagon

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The hexagonal grid Shown here are some of the most common shapes that originate from a grid built upon hexagons. This grid enables very dense patterns, as the hexagon is one of the surface filling polygons. It is one of the most common molecular structures in organic life. And as you know, even the bees use this design to contain their honey!

In the figure below some of the possible shapes from this grid are cut out.

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As you have seen, there is a close relationship between the triangular and hexagonal grids.

Any hexagonal development can be broken down into triangles.

Another way of using the regular triangles’ surface covering possibilities is shown in this traditonal pattern. It exists in many traditions and its “3D” effect is quite astounding.

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The pattern below can be found on pottery from the Minoan period in Greece (about 1500 BC).

It can be seen at the museum in Heraklion, Crete.

It is based on a grid of hexagons, where the diagonals are replaced by wavelike lines.

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In this kind of development of the hexagon all diagonals plus lines connecting the half side points are used as a grid to create a new unit.

Below this unit is put together into a pattern. A new kind of surface covering unit has been created shown in grey below. This unit can also be understood as a combination of three rhombs of the kind discussed earlier on page 100.

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Below is another intricate design based upon the hexagonal grid. It was used in the Moorish culture dominating the south of Spain around 1100 AD.

Shown on the right is a development based on the rotation of the rhomb. The unit that constitutes the pattern above is marked with a thick outline in the grid below.

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Shown here is another principle for working with the hexagonal grid. The circles and the placement of the lines are shown beneath. From this starting point many developments are possible. In this case every other unit is reflected.

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If you have a coin and put coins of the same kind around it, exactly six coins will fit in. This is one of the strange things in the symmetries of the hexagon, which appears when you join the centers of the surrounding coins.

There are innumerable possible patterns and regular and semi reglular designs built upon hexagonal grids.

Most common is of course the connection of different centre or contact points with straight lines. Shown here is how segments of larger circles also fit and regularly pass through series of centre points. This can also be done in many other ways.

The whole development that can be seen to the right can be understood as some kind of scaling.

The symmetries seen in the last hexagon are also latent in the first.

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Eight sides The octagon

This pattern, based on an octagonal development, was used in the Moorish culture in the south of Spain about 1200 AD.

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As mentioned earlier (see page 72) the octagon can be seen as a doubled square. Since the diagonal of the regular square is longer (root of 2) than the square’s sides, each new square will be larger in an accelerated movement.

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To the right there is a photo of one of the mosaic walls of the mosque in Alhambra in the south of Spain from about 1200 AD. This complicated mesh work pattern builds upon octagonal development.

Below is a far simpler version showing a delicate mix of squares and octagonal stars. This kind of grid has also been used for window frames.

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As there is a relationship between the square and the octagon, this can be utilized in images and patterns.

Shown here is how the sides of the octagon are replaced by squares. In the traditional patterns below the star and the cross visually dominate over the squares. These patterns can be compared with patterns from “Working with squares”.

See page 89.

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This pattern was found on African pottery. It can be understood as a grid of octagons. Since the octagons are not surface filling units, the standing regular squares appear by themselves in the interspaces.

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Nine sides The nonagon

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These illustrations show some features of the regular nine sided polygon, the nonagon. Below is shown the construction of the nine pointed star that appears when you take a line from one corner to the fourth next. You can also see the figure that appears when you go from one corner to the third.

In the pattern at the bottom of the page you see the triangle packed in the surface filling way showed on page 85. As with the pentagon many regular patterns are possible just because of these polygons’ “irregularities”.

In the second row of designs you see this strange figure that at a first glance looks like two squares - but it is a spiral. Finally you see how a regular triangle appears when you go from one corner to the third.

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Ten sides The decagon

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Adding pentagons in a regular succession like this creates a segment with a 36 degree angle at its base. Copying a segment like this ten times around a centre gives a decagon. The principle of adding pentagons of decreasing sizes to each other, as shown here, is sometimes called “Plato’s harp”.

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Twelve sides The dodecagon

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As pointed out earlier most flowers seen from above fit into the symmetries of polygons. In the dodecagon, flowers with three, four, six or twelve petals find corresponding lines.

The circle to the right shows one twelfth of the full dodecagon diagram. See the previous page.

With this circular grid it is possible by taking away lines to create designs of a profile of flowers with two to six petals visible.

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This is a remarkable design based on the twelve sided dodecagon grid. Regular squares are laid out so that the diagonal of the square fits the side of the polygon.

This is repeated in smaller and smaller circles touching each other. In the shaded image below one can more easily see the spiraling movement latent in this geometry.

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Proportions 1:1 One of the basic parameters when working with design is the one of changing proportions. Within a complex of forms size is something relative. A shape of a certain size may look “big” compared with a smaller size but look “small” compared to a bigger size.

2:3

When the relation between sizes is altered the visual impact of the form is changed dramatically.

If, to begin with, we limit ourselves to the relation between height and width of rectangles we have these four most common relations or proportions.

In the illustrations here you can see four different rectangles with the proportions: 1:1, the regular square 2:3, close to the “A” format used in Europe and Japan, the 3:5, close to the golden section and last 1:2 the so called “double square” used in China and Japan for many purposes.

3:5

Here all of them has been given the same width so you can more easily compare there proportions visually.

Later we will go into their proportions of each of them.

1:2

1:1

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1:2

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The “A” format

A0

A2 The “A” format is created by making the longest side the length of the diagonal of the regular square that makes the shorter side. This enables this format to keep its proportions when divided into halves as the illustration to the right shows. The exact proportions are 1 to the root of 2 (approximately 1.41) or very roughly 2:3.

A1 A4 A6

A3 A5

The Golden rectangle

A golden rectangle with the sides proportional to the division of the golden section is created from a line from a point at the middle of a side of a regular square, to a point at an opposite corner. This line is used as the radius of a circle, as shown.

The golden rectangle keeps its proportions when a regular square is taken out as shown on page 157.

The proportions are 1 : 0.618.

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In the examples below there is also added a rectangle of the proportions 4:5, a common format for paintings and drawings. These different formats form a scale from the square to the double square. Each one of them has its own setup of visual possibilities with their own point locations, angles, diagonals, ellipses etc.

1:1

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4:5

They can be used as the whole surface of an image or be just a part of it. As the visual impact changes if a rectangle is standing or resting you can turn this page 90 degrees to see the resting versions.

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3:5

1:2

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An example of the use of the 1:2 rectangle in traditional Chinese woodwork. Within this strict use of the “double square” there are innumerable designs used for windows, fences, balconies and other items in interior and exterior architecture.

At the top right there is a floor plan from a Japanese villa. The similarity of these designs is striking. These proportions are also used for the standing walls. The floor plan is based on the traditional Tatami mat which measures 35.5 x 71 inches (0.93 x 1.86 meters); again the double square. These mats were originally meant to give enough space for one person to sleep. The size of a room is measured by the number of mats it can accomodate.

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The golden section The concept “the golden section” was established during the classical period (about 500 BC) in Greek culture. Some claim that the golden section was known in ancient Egypt and used for instance when building the great Cheop’s pyramid, but this is a disputed issue.

a

A quotation from Plato says “Beauty is a question of the right proportions”. The formula for these ”right” proportions in words says: “The smaller relates to the bigger as the bigger to the whole (the sum).”

If we take a line as an example, we divide it into two parts and name the smaller part “a” and the longer part “b”. Then the mathematical formula looks like this:

a b

b

=

b a+b

When calculated as a number this point appears at a little more than 61.8% of the line. As a number it is sometimes called “fi” or “phi” as a parallel to “pi”, which is also a number with an endless row of decimals. The “pi” is usually abbreviated to 3.14 and the “fi” as 0.618.

The most common geometrical solution for the golden section is shown to the left.

The sides of the rectangles that are formed also have the proportions of “a / b”, both the small and the big.

This rectangle is therefore called “the golden rectangle”. This format has been used for flags, books, bank notes and a lot of other items.

In the bottom illustration you can see how different “golden rectangles” can be derived from the space left over when a regular square is fitted into the rectangle. The three golden rectangles in this example are shown with dotted lines.

These concepts are nowadays by no means undisputed and few artists or designers find a practical use for them.

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a

The ideas were much appreciated from the beginning of the Renaissance but seem to have lost there appeal around 1900. In a way this concept rests upon an idealistic philosophy that has been challenged by other approaches, for instance the phenomenological approach.

Nevertheless the connection of the golden section to the Fibonacci series and the connection to growth patterns in nature, and to the modern concept of fractals has renewed interest in this ancient formula.

The new mathematical branch of fractals has again brought to focus “the relation of the smaller to the bigger”.

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The proportions of the golden section and the golden rectangle traditionally have been constructed from a triangle where one side is half the length of the other (abc). This construction is shown in the first diagram. 1. A circle with radius ac crosses ab at d. 2. A circle with radius bd crosses bc at e.

In the drawing below the construction lines used earlier are added. Then we made the construction from a regular square (egbf) and added a line from a point at half of one side to an opposite corner. In this illustration you can see the relation between these two ways of construction.

There are at least four different ways of constructing a regular pentagon with the help of a pair of compass and a ruler, but in the context of this book this way is maybe the most interesting since it shows the relations to the construction of the golden section from a regular square. This construction uses the fact that the diagonals of the pentagon cross each other at the dividing point of the golden section.

There are relations to the golden section in many geometrical constructions. The relations to the pentagon were mentioned earlier. Shown here is some relations between the 60 x120 rhomb (see page 100) and the golden rectangle.

The diagram at top right shows how to construct a regular pentagon out of the knowledge that the pentagon’s diagonals cross at the point of the golden section.

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So far we have spoken of the proportions of rectangles in terms of integers (1:2, 3:4 etc.). However, as shown below, there is a possibility of creating rectangles with the proportions 1: root of 2, 1 : root of 3, etc.

The geometrical foundation of these constructions comes from Pytagoras’s formula: a2 + b2 = c2 These rectangles keep their proportions when folded.

The 1 : root of 2 rectangle keeps its proportions when folded into two equal halves. (This is the “A” format.)

The 1 : root of 3 rectangle keeps its proportions when divided into three equal parts as shown below. The 1 : root of 4 rectangle (which is the “Chinese” double square) keeps its proportions when divided into four equal parts etc.

The 1 : root of 5 rectangle is interesting because the proportions of 2 : root of 5 is exactly the proportions of the golden section.

1

root of 2 root of 3 root of 4

root of 5

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Fibonacci In the year 1202, Fibonacci, an Italian mathematician, also known as ”The Lionardo of Pisa”, published his book Liber Abaci. In this work he presented his famous numerical series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 and so on, where the sum of the two first numbers make up the third and so on.

Later it has been shown that this series links the proportions of the golden section, used by classical Greek artists and architects, with the process of organic growth which can be seen in flowers, leaves, fruits etc. If you want to delve deeper into this issue there are some sites on the Internet covering it.

The coneflower is one of thousands of examples of fibonacci numbers appearing in nature. Here the first ring has 55 petals and the second 34.

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In this chart you can see nine rectangles created by two following numbers in the Fibonacci series. The diagonal is drawn in the largest which is closest to the “golden” rectangle. Notice that already from 5:3 the difference is hardly noticeable. So for any practical use 5:3 or 8:5 is good enough if you want to create this format.

This way of arranging rectangles also shows one possibility of a development based on the golden section. It also can be said to show a growth pattern according to the Fibonacci series.

55:34

21:34

21:13

13:8

8:5

5:3 3:2 2:1 1:1

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Another peculiar thing with the Fibonacci series is that when you put regular squares tight, with the sides 1, 1, 2, 3, 5, 8,13 and so on, they automatically fit as shown below.

The next regular square that will fit under this will be 21 x 21.

This is also the grid for a spiral that can be found in nature in for example the Nautilus seashell. In this case the pointer of a compass is placed in a corner and draws a quarter of a circle with the same radius as the square’s side.

At the top right four spirals like this are put together. Since they are formed from quarter circles all spirals like this will form a circle when joined this way, regardless of how many units they contain.

8

13

1 2 1 5 3

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It has been argued that the dynamic shapes of organic life can never be fully explained by the use of simple geometry or by the use of Fibonacci numbers. There is however a classical example where you can judge for yourself: the Nautilus sea shell. Here a copy of the spiral from the previous page is superimposed on a photo.

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Surface filling

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The concept of surface filling units builds upon the fact that only some shapes have the quality that they alone, when duplicated, can cover a surface completely without leaving any interspaces.

This has a practical application when working with patterns or mosaic, tiles or any other unit to be used for covering the ground, floors, walls etc.

Shown here is how the polygon is replaced with lines connecting the central point with the points of the corners. The same line is repeated rotating around the centre.

Basic 2D design

Among the regular shapes only three are surface filling: the triangle, the square and the hexagon.

Shown below are some principles of how to create an innumerable number of shapes that have this quality.

A wave line is used here for the triangle, a broken line for the square and a segment of a circle for the hexagon, but of course any other combination is possible. Shown below is the specific surface covering unit.

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Here are some examples showing surface covering with the units from the previous page.

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The shapes that can serve as surface covering units are of course innumerable. Here we see one such shape used for many purposes, for instance shoe soles to take an odd example. This design starts with a square inscribed in a circle. It is the point locations of the square grid that enable this tight design.

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Everybody knows that squares of the same size can fill a surface. But, if you have a regular square and add a smaller one of any other size, you can make a surface filling pattern with squares of two sizes.

The pattern below with the small square shaded is just one of several possibilities.

At the bottom of the page we can see how a single surface filling unit is created by integrating these two squares into one shape.

Basic 2D design

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At the top you can see a cross made up from five regular sqares. It is a surface filling unit and in this case the pattern forms identical negative spaces. However crosses like this can be put in other regular positions without creating this effect, as you can see in the second and third illustration.

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Word list A format 154

negative space 34

Aboriginal art 51

node 88

acceleration 43

nonagon 146

Alhambra 143

octagon 141

almond shape 48

packing 21

archetype 108

parallel trapezium 115

asymmetric 33

pentagon 128

concave 26

pentagram 130

convex 26

perception 15, 16

corner radius 104

point location 27

cross 105

points 45, 14

decagon 148

poly symmetry 32

density 38

polygons 122

dimensions 14

proportions 153

dodecagon 150

radiation 47

double symmetry 32

Rangavalli 52

ellips 88

Rangoli 52

Fibonacci 64, 160

reflection 28

floor plan 105, 156

rhomb 46, 99

fractal 62, 132

rotation 41

golden rectangle 154

samurai 24

golden section 157

Shinto 75

grid 20

spiral 63, 162

hexagon 133

stars 127

hexagonal grid 135

sun cross 105

horizontal reflection 29

superellips 88

intersection 39

surface filling 164

kolam 52

swastika 98

literature 170

symmetry 32

Malevitj 109

tatami 156

mandala 83

triangles 113

meander 34, 61

wavelines 59

movement 41

vertical reflection 29

Mykene 66

wheel 105

nautilus 162

yin yang 74

negative form 36

Zuni 112

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Literature Kandinsky, Wassily, Punkt und Linie zu Fläche, 1926.

Klee, Paul, Pedagogical Sketchbook, 1953.

Klee, Paul, Pädagogisches Skizzenbuch, 1925.

Akner-Koler, Cheryl, Three-Dimensional Visual Analysis, Konstfack, Stockholm,1994.

Bevlin, Marjorie Elliot, Design through Discovery, 1977.

Bloomer, Carolyn M, Principles of Visual Perception, 1990. de Sausmarez, Maurice, Basic Design: The Dynamics of Visual Form, 1992.

Hartung, Rolf, Textiles Werken, 1960.

Hesselgren Sven, On Architecture, 1985.

Itten, Johannes, Design and Form, 1978.

Lawley, Leslie W., A Basic Course in Art, 1962.

Röttger, Ernst & Klante, Dieter, Werkstoff Papier, 1966.

Röttger, Ernst & Klante Punkt und Linie, 1967.

Röttger, Ernst, Klante, Dieter & Sagner, Alfred, Werkstoff Holz, 1967.

Ullrich, Heinz & Klante, Dieter, Werkstoff Metall, 1967.

Wong, Wucius, Principles of Form and Design, 1993 Andersson, Donald M., Elements of Design, 1961.

Bayley, Stephen, In Good Shape. Style in Industrial Products 1900 to 1960, 1979.

Paul Jacques, Form, Function & Design, 1975.

Lingstrom, Freda, The Seeing Eye, 1960.

Neill, William, By Nature's Design, 1993.

Rowland, Kurt, The Development of Shape, 1964.

Whitford, Frank, Bauhaus, 1984.

Willcox, Donald J., New Design in Jewelry, 1970.

Dower W. John, The Elements of Japanese Design, Weatherhill, 1979.

Barrucand Marianne, B. Achim, Maurische Architektur in Andalusien, Taschen 1992.

Critchlow, Keith, Islamic Patterns, Thames & Hudson, 1976.

Critchlow, Keith, Order in Space, Thames & Hudson, 1967.

Lawlor, Robert, Sacred Geometry, Thames and Hudson, 1989.

Field, Michael and Golubitsky, Symmetry in Chaos, Oxford University Press, 1992.

Liungman, Carl G., Thought Signs, IOS Press, 1995.

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Basic 2D design

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