Technicals – Making Hexagon Using a Ruler and a Compass

It seems like I still have problems with wordpress regarding uploading my pictures during my trip to KL -even the smallest, compressed to the very lowest quality ones- so I guess I have to postpone it, even the monthly Imagining Islamic Aesthetic posts had to be put on hold. Instead I am going to post about how to make a Hexagon (and subsequently, a perfect Hexagram) using two equipments a Mathematician should be familiar with and should have access to should he or she wants to make geometrical shapes – a ruler and a geometric set compass. I can post the instructional pics here because frankly the images are very basic and that’s all WordPress can download at the moment.

Start with two lines – horizontal and vertical. In the middle of the cross you just made put your compass on the point and draw a perfect circle.

Using your compass, point the sharp end to the point where the horizontal line meets the circle and your compass’ pencil point to the center. Then, still the sharp end at the point and the length from the circle and line meeting point unaltered, draw a semi circle. Referring to the figure above, the green line on the horizontal line is the length between the circle-horizontal meeting point and the centre and the semi-circle. Do the same step to the opposite side.

Then its just a matter of connecting the meeting points in the circle. Erasing all the guidelines you will have a regular hexagon.

However, joining each point of the Hexagon like the figure above, you will have a Hexagram, a six-pointed star. Note that this is a not a perfect representation of both Hexagon and a Hexagram since I am doing it on Paint.

Technicals – Different Shapes, Simillar Tessellations

There is an article that had been posted before about creating a pattern or tessellation using just one simple prism shape. (click here to go to the article) This article is a continuation of the prior posting.

Following are three different variations of the pattern. They are basically similar in tessellation and repeating pattern, however using different prism shapes produces fairly different results.

This is basically a simple modification of the prism by curving the sides , and the edges transformed from a sharp pointed edge to a single curve on each edge. Even though it is just a simple change, but the overall pattern changes significantly – the negative spaces between the prisms changes into curved stars as opposed to the sharp geometric shapes.

This is an even simpler shape than the prior article’s prism shape. A regular rectangle with soft curved sides are arranged into the same pattern, resulting in negative spaces in the shapes of thick stars with short, blunt points. This shape and pattern is pretty much fitting into a modern decor, perhaps as a modern Islamic inspired pattern.

Another simple modification to the prism with a triangular shape jutting out of the middle of the shape. An easy modification but yet creates a significant change. The negative space created with the shapes resembles four-pointed stars with the points cut. However, with the extra point jutting out, the overall pattern covers a larger space.

Technicals – Arranging the Six-Pointed Star

I had this in my laptop for quite a while now so I think it is best to post it here today. I had noted the technique of creating a proportioned six pointed star before and now the question is how to arrange them in traditional Islamic art.

The six-pointed star is a very common motif in Islamic art. The usage outside Islamic art can be traced in history. In religion it is mainly associated with Judaism, viewed as two triangles overlapping each other, while the six points represent the Twelve tribes, and the two interlacing triangles are the letters Dalet and Yud in Hebrew, the two letters assigned to Judah. It is also being repeatedly used in flags. The six-sided star or hexagram is a powerful symbol used in heraldry and occultism. but enough of that, let’s get to the point which is how to assemble the stars into motifs that commonly occur in Islamic art. Please note that I had rotated the stars so that it would not be associated with any religion, which in this article it is purely for artistic reasons, not for religious or occultism.

This is one of the common arrangement of the six-pointed star in Islamic art. The stars met with another star by the points and forming a negative space between them creating the shape of a hexagon. This may be added with details such as hexagons in the middle of the six-pointed stars (remember that the six-pointed star is made from the shape of a hexagon)

By arranging the stars with every points of each star touching one other you will end up with a design that leaves a negative space between the stars in the shape of a diamond. This is also a common motif in Islamic and again details can be added such as adding hexagons in the middle of each stars.

In this design, the stars are arranged in a linear fashion – both horizontally and vertically. This creates two different negative spaces between the stars. The vertical spaces formed a diamond shape polygon while horizontally the spaces are in another different polygon shape.

Technicals – Even More Ways to Arrange the Eight-Pointed Star

In the last articles under the Technical section I had posted some regarding how to build eight-pointed stars, one of the most common motifs in Islamic art, as well as how to arrange them in specific ways. This article will show you other methods of arranging this polygon, some with help of different types of polygons.

1. A variant of the Stars and Cross arrangement

This variation of the Stars and Crosses arrangement features a slight difference – the addition of a square in the middle of each cross. This arrangement is usually used for Jaalis or Masyrabias because of the empty spaces in the cross parts are filled, allowing more privacy.

2.With 2 different polygons

The eight-pointed stars are arranged that there are two different polygons are formed between the stars, here shown in cyan and dark blue. Two points on the upper part of the star is joined with the lower two points of another star to make a four sided diamond shape polygon, while two points of four stars are connected to create an octagon.

3. With Arrow-shaped polygons

The stars are arranged in a ladder-like way – one star is arranged lower than the other and goes on. The pointed sides in the cardinal direction (up, down, left and right) are connected with each other and creates arrow shaped spaces between them.

Technical – The Basics – Building a Tesellation with One Simple Shape

I once had posted about how to make a prism Geometrical shape under the Technical section (you can read it here). But if you are lazy to click the link, the following picture might spark your interest –

So for this post, it is about arranging simple shapes like this one, into a beautiful geometric pattern or a tessellation. Muslim artisans had founded quite a number of ways of creating patterns out of simple shapes, along with creating complicated patterns with a number of different but equally simple rhombuses, diamond shapes, squares and the like.

For this kind of shape (prism) there is two common ways in the Islamic world of aesthetics that this shape can be arranged-

Prisms, Squares and Crosses The prisms are arranged in a cross shape, and by arranging in such a way, it will produce a square shape in the middle. This is a very simple pattern, mostly utilized for tiling for floors in Islamic or Muslim environments such as palaces (even in non Islamic areas) , not as wall tiling as it is too simple and not impressive enough to be featured on walls.
Prisms and Stars – This pattern is more common and used more as an Islamic pattern, as wall tiling. The prism shapes are arranged in a row and column, vertical and horizontal alternately. By arranging the shape this way it creates four pointed stars space (resembling the ninja throwing stars, Shurikens, if I might add) between them (highlighted in blue in the figure above)

Technicals – Advanced – Finding The Eight-Pointed Star In a Grid

It’s the 100th post for the blog! Nothing to celebrate much, though.

Anyway, it’s another posting for making the eight-pointed star, this time on a grid! Don’t worry about the “Advanced” tag on the title ; It may be confusing because of the grid, but with a keen eye and careful observation, you can make Eight-pointed stars easier and more accurately (providing that your grids have correct measurements)

First, have your grid ready. One easy way to do this is by using grid paper, and creating extra diagonal lines for each of the squares. done correctly, you would have something that looks like this. Note that you have to do it accurately, since slight differences would make the eight pointed stars turn weird. –
Once you have an accurately measured grid to work on, you only have to find a suitable spot to create your Eight-Pointed stars! Start slowly at first by creating a square. Make it large – take a 3×3 grid –
Add another square, using the original square you have made. Look at the next picture and note where the corners of the next square is placed.

There is nothing “advanced” to this technique, actually. The only hard thing here is about the grid, and making correct alignments and measurements (as the rest of the techniques used to create the Eight-Pointed stars I had noted before.)

Technicals – Intermediate : Finding the Eight-pointed Star Using Circles and Lines

We have discussed about how to build Eight-Pointed stars using lines as guides. Now we will look on how to create these ubiquitous motif in the Islamic Art and Architecture using simple circle shapes and lines. Don’t worry about the ‘intermediate’ word in the title – it may be a little bit different than the basic star shape building technique I had posted before, but this method offers more flexibility, as well as accuracy.

For this technique, if you want to make this on paper (or other materials you prefer) you need a straight ruler and a drafting compass.

Start with drawing a straight horizontal line, and a vertical line through the center of the horizontal line, so it creates a cross.
Put your sharp point of your compass in the middle of the cross – as marked in red in the diagram, and create a circle around it , as so –
Create four more circles with the first circle as your main guide. Use the points where the circle meets with the lines to put your compass and draw the circles. Here in this diagram, I had drawn four different coloured (cyan, purple, yellow and green) circles with their respective points .
Draw two diagonal lines crossing the circles and the original cross lines. Here it is marked with two black lines going over the original diagram. This is the base of many other shapes that you can create, however I will only concentrate on Eight-Pointed Stars.
You can now create the Eight-pointed Star by drawing lines from the points where the four outer circle are made, as well as where the diagonal lines meet the connected circles, as shown in this diagram in dark blue –
Now you may clean the guiding circles and lines, and you will have an accurate and geometrically correct Eight-Pointed Star.

Technicals – The Basics : Arranging Eight-pointed Stars

In the first posts in the Technical category, we have learned how to make the ubiquitous motif in the Islamic aesthetics – the Eight-Pointed Star. Now let us look into how this motif can be arranged to make a pattern. There are many different patterns that can be derived from the Eight-Pointed Stars, however, we shall look into the basic ones, simpler patterns that we can distinguish the stars used in each.

The “Stars and Cross”One of the most common pattern done utilizing the Eight-Pointed stars. The design arranges the stars on a straight line, horizontally and vertically. The corners of each stars (in dark blue)connects with each other, creating a cross (in cyan) between them. Another version of this is by arranging the stars (and subsequently the cross) diagonally, as such –

Overlapped

The stars can also be overlapped with each other, creating a tightly packed pattern with small diamond shaped spaces in between, like the example given above. This pattern can also be recreated into another form by arranging the stars vertically and horizontally using the same idea of overlapping one side of the stars with another in succession.

Arranged Horizontally and Diagonally

The stars can also be arranged both horizontally and vertically in one pattern. It creates interesting pattern for the spaces between between the stars – as you can see in the example above, the spaces created an ‘I’ shape between the stars. However, there is very little versatility arranging the stars into this design because the stars can only be arranged either horizontally or diagonally in each row ; mixing the stars differently in each row would created a more strangely shaped spaces an does not create the effect of uniformity in the pattern.

Soon in another edition in the Technical category, we will look into more different kinds of patterns that can be made using the eight-pointed stars, its variations as well as creating a whole new star using different arrangements.

Technicals – The Basics : Builiding a Prism Geometrical Shape

When this shape is not really a star, it is one of the a common motif used for Islamic decoration. When one prism shape isn’t looking like a typical Islamic aesthetic art, a cluster of them arranged accordingly will feature a composition that will comfortably fit to be used as a tiling pattern in Islamic art. It also can be arranged in multiple ways.

When arranged, the prisms will form yet another shape – a four armed star, if composed correctly. We will discuss this later in another edition of the Technicals post. For now, let us see how this prism can be drawn.

First, draw an equal-sided diamond shape, like such.
Next, draw a line in the center, as this is also your guideline.
Half way from the center line to the end of the left (or right, whichever you prefer) corner, draw another line.
Repeat with the other side of the diamond shape, also half way from the center guideline to the right (or left, but in this case it is left) corner.
Erase the guidelines as such :
You basically have the shape now, but wait, there is more cleaning up to do…Clean the center line and recolour the whole shape. Now you have the basic prism shape. The point here is not to complicate things altogether, but to get a shape that is symmetrical and to achieve a perfect pattern, once this shape is arranged in a group accordingly.

Technicals – A Variation of the Eight-Pointed Star

Before I have shown you how to create one of Islam’s prominent geometrical motif, the Eight-Pointed-Star. In this edition I would like to show you another variation of the star.

I have forgotten to mention in my last post, the Arabic name for the motif is Rub’ el-Hizb (ربع الحزب) meaning ‘quarter of a group’ – Rub’ means one-fourth or quarter, and Hizb means a group or party. Originally it is used in the division of the Al-Quran. The Al-Quran is divided into 60 Hizb. The motif is used to mark a quarter of one Hizb, while a Hizb is half of a Juz.

It is also a common shape to be used on logos and flags, most usually in the Islamic or Arabic states. It is used in the coat of arms of Turkmenistan and Uzbekistan, logo of the Cairo Metro and the 2009 Arab Capital of Culture – Al-Quds.

There are two ways to create the alternate version of the eight-pointed star : –

1. Using the original eight-pointed star.

This method is more simple , so it takes the first spot. Start off by drawing an eight-pointed star. (you can refer to my last post).

Extend the lines from the original shape. Don’t worry for the excess, you can erase it later. Do it for the vertical lines first…

The extend the lines for the diagonal lines.

You should note that all the lines that are extending should meet each other.

Clean the excess lines as needed, and as usual, clean as needed, that is, you can erase the original eight-pointed star for some complexity or erase it for a cleaner look.

2. Start off from scratch.

This method is harder, but I will still tell you how to, in case you would like to try it.

Start off with a square. Be sure the dimensions of the sides of the square are equal, otherwise you would end up in an asymmetrical shape rather than symmetrical.

Like the above method, extend the lines from the square.

Draw a line from the extended line of the square from any side of the square (in this case, the left side) and draw it to the third line from your original line downwards or upwards (this instance, downwards) , as such.

Draw two lines where the line you just draw meet. It should look like a half of a square. this will be your guideline.

Continue drawing another line from the extended square line, and repeat drawing the guideline.

Repeat the last two steps until you find yourself having drawn the alternative form of the eight-pointed star. You can also note that you also drawn the original shape of the star.