Mathematical Art With Islamic Geometric Patterns

Mathematics is the science that deals with the logic of shape, quantity, and arrangement. Math is all around us, in everything we do. It is the building block for everything in our daily lives.

In Islamic culture geometry is everywhere. You can find in mosques, madrasas, palaces, etc., Islamic craftsmen were prohibited from making representations of people and animals started the art of using geometric shapes to make repetitive patterns. Islamic geometry mixes art, history, and mathematics. Islamic geometric patterns occur in a variety of forms such as Islamic architecture, ceramic designs, stained glass painting, Carpets designs, Jali designs, etc.,
In this Instructables, we are going to understand the basics to design Islamic geometric patterns and will implement them on a few easy projects.

Step 1: History and Where Can We Find These Patterns Now?

The expansion and development of geometric patterns through Islamic arts can be dated back to the 8th and 9th centuries. This period of history was a golden age of Islamic culture.
Currently, these patterns can be found in countries like India, Iran, Pakistan, Syria, Spain, Afghanistan, etc. on Islamic architectures such as Madrasa, Mosques, Gates, Domes, etc.

Step 2: Construction Lines/Grid

Islamic geometric patterns can be created using just a compass to create a circle and a ruler to make lines within them. And from these simple tools emerges a repeating pattern.

Everything starts with a circle and is divided into equal parts and each division gives rise to distinctive pattern. The construction lines or grid is invisible in the final pattern design but is necessary for every pattern.

Every pattern has their distinct set of construction lines.

Step 3: Basic Constructions

Before proceeding to draw construction lines a circle needs to be divided into equal parts.

Here are few examples how to divide circles into equal parts without the need of a protractor.

4 and 8 equal parts

5 equal parts

6 equal parts

10 equal parts

12 equal parts

Step 4: Your First Basic Pattern

Let's draw our first Pattern.

Start with a circle of diameter 10 cm on the center of a4 paper.

Step 5: Tesselation

Tessellation is covering the surface using one or more geometric shapes with no gaps or overlaps.

Tessellation can also be found in nature for example; The arrangement of hexagons in honey comb. While honeycomb has only one geometric shape tessellated a good example of tessellation of two different shapes is soccer ball.

When we tessellate a geometric pattern a new pattern is revealed. For example if we take our basic design and tesselate a new plus signed shape is generated.

For a pattern to tessellate and look good it is necessary to have at least one axis of symmetry.

Step 6: Exact Vs Approximate

The thing which differs Islamic geometric patterns from other geometric patterns is they use the Exact method to construct the patterns. For an untrained eye, both of the geometric patterns may look the same but a small difference gives rise to large errors when tessellated.

An easy way to understand if the pattern is drawn with the exact method or approximate method is to look for the petals. In the exact method, four sides of the six-sided petal are the same while in the approximate method 2 sides of the six-sided petal are the same. This makes the petal of the approximate method look squished.

The reason why Islamic geometric patterns use the exact method is because the revealing pattern after tessellating is symmetric which upholds the symmetry of the whole tessellated pattern.

You can observe in the images that the star pattern revealed after tessellating the 8 fold geometric pattern in the approximate method has unequal corners. While the star revealed in the exact method has equal corners.

Here is both way a 8 fold pattern can be drawn. I want you to find the difference in constructing the pattern.

Approximate

Exact

Step 7: Folds

Patterns are categorized into folds. There is an easy way to determine the folds of the pattern. Most patterns contain star surrounded by petal-shaped structures. Counting the number of petals gives us the folds.

A pattern with 6 petals belongs to a six-fold category and a pattern with 8 petals belongs to 8 fold category.

Similarly a pattern with 8 star rays(red numbers in the image) belong to 8 fold category.

Many a time the pattern we draw does not contain petals or star here we have to look for the equal division the circle has in the construction grid. A circle with 10 divisions, the pattern falls into 10 fold category.

Step 8: Tracing and Copying

An easy way to tessellate a hand-drawn pattern is to make use of tracing paper.

Start by placing the tracing paper onto the pattern and trace the pattern using a dark pencil. Use masking tape to hold the tracing paper in place. Once the pattern is traced remove the tracing paper. Flip the tracing paper, now the side on which we have traced is facing the pattern paper side. place the tracing paper such that at least one point of the traced pattern is touching the original pattern. Align the tracing paper properly using construction lines of the original pattern. Here I have aligned it using a horizontal construction line. Tape the traced paper on the original pattern paper. Use the convex side of the spoon and scratch the traced pattern. A faint pattern will appear on the paper. Use a pencil to darken this faint pattern.

Step 9: Weaving Pattern

Most of the Islamic geometric patterns have thick borders and these borders are designed in such a manner that they seem to look like it is weaved or intertwined. The line which comes above is continuous while the line which is below is separated by a small gap.

An example of a weaved pattern is attached. This particular weaved pattern is on a prayer mat in a mosque near my house. You may have noticed this is the same basic geometric pattern with a thick border.

Here is an easy example of how you can weave your basic geometric pattern.

Step 10: Draw and Paint

This particular pattern can be found in Alhambra, Granada, Spain

Supplies:

• Acrylic color
• Flat brush
• Paper
• Thin tipped marker pen

Procedure:

1. Start by drawing the pattern, here I have used 8 fold rosette. you can draw the rosette by following the video in the previous step(exact vs approximate) or you can print the provided pdf.
2. If you have drawn the rosette erase all the construction lines.
3. I tried all the colors available to me on a piece of paper and chose acrylic colors for its brighter complexion.
4. You can choose any other source for painting the rosette such as colored pencil, crayons, etc.
5. Brown, black, blue, green, orange is the most dominant colors used in Islamic motifs.
6. I have painted the rosette with a flat brush.
7. Once the painting is complete and dry draw the borders using a thin tipped marker.

Step 11: Paper Layered Art

In this project we are going to use a 12 fold rosette found in Al Nasir mosque, Cairo, Egypt

Supplies:

• Ruler
• Hobby Knife
• Masking Tape
• Cardstock paper assorted colors
• Glue

Procedure:

1. Start by cutting the template into a square.
2. Place the square colored card stock of the same size on a table or a cutting mat and use masking tape on the corners of the square cardstock to hold it in one place. The masking tape used needs to be of light adhesive we do not want to tear our patterned paper when we remove it from the table.
3. Now place the square template on the colored card stock aligning all four corners.
4. Use a ruler and a sharp hobby knife to cut the template. Apply enough pressure to cut through both template and card stock.
5. Remove the template when all of the patterns are cut by carefully removing the masking tape. Remove the masking tape from the cut cardstock as well and store the pattern in a hardbound book.
6. Repeat the same steps for the other 9 square templates and card stock.
7. Align all the cut patterns serially and you are done. You can use glue to hold all the cut patterns in one place.
8. Stick the pattern on cardboard and enclose it in a plexiglass case.(optional)
9. Do not throw away any cutouts these will be useful for our next projects.

For this project, we have drawn a 12 fold rosette found in various places. You don't need to draw the pattern pdf is provided just watch the video and you can use it for any other project.

Step 12: Minimalist String Art

This project will help the student understand how weaving of a pattern works.

This is a 10 fold pattern which can be found in Mustansiriya Madrasah, Baghdad, Iraq

Supplies:

• Wooden ply 30 cm*30cm*0.5cm
• Nails
• Woolen thread/string
• Black Chart paper/Aerosol Spray paint
• Hammer
• scissor

Procedure:

1. Paint the wooden ply using a aerosol spray paint or glue black chart paper. I have glued black chart paper since it helps hide any imperfection on the wooden ply. Alternatively you can use cardboard if you don't have wooden ply.
2. Stick the geometric pattern paper provided using masking tape onto the wooden ply and make sure the paper is aligned center to the wooden ply. The provided geometric pattern pdf is in a3 so make sure you are printing it in proper size.
3. Start hammering the nail onto the wooden ply. The point where nail will come is shown in the image by a dot on the geometric pattern.
4. Make sure the nails are perfectly perpendicular to the wooden ply and firmly fixed as well.
5. Make a knot on the nail denoted by number 1 using woolen thread. I have used red woolen thread you can choose any .
6. Move towards the second nail denoted by number 2 and make a loop.
7. Make a loop as you move towards other nails Follow the number pattern to complete the art
8. Once you reach the end nail i.e. denoted by number 40 move towards the first number and tie the final knot.
9. Cut the thread using a knife.
10. Remove the masking tape.
11. Carefully tear the paper using a knife or scissors.
12. Your string art is ready .

To easily understand how weaving works follow two strings in the pattern i.e. string 1-2 and 4-5. String 1-2is below the string 4-5 and hence should be separated by a small gap and the string above i.e. 4-5 should be continuous. This goes on until the final string 39-40.

Here is how you can draw the geometric pattern.

Step 13: Coaster/tile

This project will help students understand the real implementation of tessellation of geometric patterns.

One unique thing of the geometric pattern used in this project is when we tesselate the geometric pattern the revealing pattern is also of the same shape.

This pattern is from my home country Agra fort, India

Supplies:

• Two part epoxy clay
• Hobby knife
• Ruler
• Butter paper
• Tracing paper
• Plastic film
• Acrylic colors
• Barbecue skewers
• Rolling pin

Procedure:

1. Trace the provided geometric pattern onto the butter paper.
2. Mix two part epoxy clay .
3. Stick plastic film onto the table /cutting mat using masking tape.
4. Place the mixed epoxy clay and start rolling with a rolling pin.
5. Roll till the epoxy tile is of 3 to 5mm thickness.
6. To make sure the thickness is uniform I have made use of popsicle sticks as limits.
7. Place the traced butter paper on the epoxy tile and mark the outer square of the geometric pattern using pin or nail by poking through the butter paper.
8. Cut the tile into shape using a hobby knife and remove any extra epoxy.
9. Store the extra for our next project.
10. Place the butter paper again on the tile and poke through the butter paper at every corner of the geometric pattern this will be helpful after the tile has dried.
11. Let the epoxy tile dry.
12. Sand the edges if needed with coarse sand paper.
13. Once the epoxy has dried draw the pattern on the tile using the poked holes as guide.
14. Paint the tile using Acrylic colors.
15. Start with light paint and then move for darker ones.

Step 14: Keychains/ Earrings

Here most of the steps remain same as the tile except for geometric pattern we will be using cut outs from the Paper Layered art project.

1. Roll the leftover epoxy from the tile project with a uniform thickness of 2 to 3 mm.
2. Place the cutout onto the epoxy tile and cut away the extra epoxy.
3. Poke a hole using barbecue skewers at the top position.
4. Let the epoxy dry.
5. For Keychains I have used petals and star of the 12 fold rosette and for earrings I have used star rays of the 12 fold rosette.
6. Paint the dried epoxy Earrings and keychains using acrylic colors.

Step 15: Class Room Activity

Objectives:

At the end of this activity students will be able to

• Divide circles into 4,5,6,8,10,12 equal parts.
• Identify and differentiate between Exact and approximate patterns.
• Identify Islamic geometric patterns.
• Identify the folds the pattern falls into.
• Understand what tessellation is.
• Draw construction lines/grid for basic pattern.
• Create their own geometric pattern.
• Tesselate the geometric pattern.
• Incorporate the pattern into a project.

Audience:

6 grade and above.

Although this can be enjoyed by everyone even children as small as 3 years age by providing coloring sheet with tessellated geometric patterns and asking them to color it.

Teacher can use a projector to explain the basics and to show images from real historical sites.

I am providing a sample presentation file, teachers can use this presentation by editing further or using the same in the classroom

Step 16: Class Room Activity :Level 1

Teacher should start by introducing students with basic geometric shapes such as rectangle, square, triangle, circle etc. and explaining what geometric pattern is.

Then teacher should explain how circles are equally divided without the use of protractor. This will not only be helpful for this activity but also for any projects in their life. Then teacher should move forward and explain every other thing in short depending on the audience.

Step 17: Class Room Activity : Level 2

Teacher should distribute the activity sheet among the students and ask them to draw a design using their imagination on the grid line provided. Students can introduce more construction lines if needed. Let the students use pencils so that they can erase any mistakes.

For students below grade 5, I have provided a coloring sheet which the students can trace the outline using a ruler and pencil and color the pattern.

Step 18: Class Room Activity : Level 3

Teacher now should ask the students to draw the basic construction grid using a smaller diameter on a blank paper. The students need to replicate the pattern they had drawn in the previous activity. If possible students should Tessellate the pattern on a blank sheet.

Step 19: Class Room Activity : Level 4

Teacher now should ask the students to incorporate either the singular pattern or the tessellated pattern drawn in the previous activity in a project (Drawing, coaster, earing recommended since they require less tools as well as less work. )

Step 20: Few Images From Mosques Near Me

Although the shapes appear to be complicated many times simple shape such as rectangles, squares, circles etc. are tessellated to make a geometric pattern.

Challenge:

Identify the patterns in the images, draw and post it in the I made it section