Synergizing the Beauty of Mathematics and Art

A customized gift for a loved one is the most satisfying creation. This piece of intricate geometrical art, counter-intuitively, is simple and straightforward and can be made in just a few hours, reducing the hunch of 'I'm not creative enough' in one's mind to get the output of a truly awe-striking piece.

The word mandala arises from the Sanskrit and means sacred circle, coming from its radial geometry. Mandalas translate complex mathematical expressions into simple shapes and forms. The shapes that are formed from the uniform divisions of a circle are fractals that embody the mathematical principles found throughout nature.

Pythagoras described geometry as visual music, since music is created by applying laws of frequency and sound in certain rhythm. These same laws can be applied to produce visual harmony, the only difference being that instead of frequency and sound it plays with angle and shape.

This piece, which I like to call a 'LayMandala' (a layered-mandala that any layman can create); takes almost no 'artsy' or complex mathematical skills. No big tools or long hours of craftsmanship required, yet a fine finished product at the end which will amaze your loved ones for a wedding gift, a birthday gift, or to adore it as a highlight wall piece for an accent wall.

The visual complexities derived from the mathematical relationship of form, movement, space and time evoked by a Mandala form is what makes this an awe-striking intricate piece to look at and adore from different angles. As a gift, this is truly a memorable one and definitely will not remain in someone's cupboard collecting dust like a mass-produced, impersonal gift!

Supplies

• Laptop/ PC and mouse
• Vector software (Adobe illustrator, Inkscape, CorelDraw, etc.)
• 1.5mm thick Mdf/ birch plywood/ or any cardboard
• Liquid glue/ masking tape
• Paint (Optional)
• Acrylic/ glass/ OHP sheet (Optional)
• Access to laser cutter/ exacto knife to cut manually

Here is also the pdf file of the 6 layers for your perusal

Step 1: Create a Mandala

This step is where the basic visual mathematics comes in. However, it is pure geometry and a Mandala is very simple to follow since it is one fractal repeating or arraying a shape in a radial grid at regular intervals.

Install any vector software such as Adobe Illustrator or Corel Draw, or an open source free vector software such as Inkscape. In this instructable I am using Adobe Illustrator.

To create a basic mandala pattern, the pen tool is the primary shape-making tool required to be acquainted with.

Making the Mandala grid -

In this design, the final mandala pattern is divided into 24 equal radial sections, and the piece is a 12x12cm square.

1. Start by drawing one line of length 12cm.
2. Divide 360° by the number of sections to determine at what angles the line will repeat along the central axis. Here, 360°/24=15°. Hence, we need to copy and rotate this line at 15 degrees, 11 times.
3. In Adobe Illustrator, select the line we created. Right click and select Transform>rotate.
4. A new pop-up comes up, asking for the angle of rotation. Type 15 and press COPY (not 'OK').
5. On pressing COPY, we can see that a copy of our initial line has been created, which is rotated at 15 degrees. To repeat this step multiple times, simply click CTRL+D ten times to rotate-copy 15 degrees successively, or you can choose to repeat step 3-4.
6. To create a mandala, we first need to plan a base guide grid of concentric circles which allows us to create our geometries, with repeating segments from centre to outer.
7. The outermost circle in this design is of 95mm, inscribed in a 120x120mm square. For a base grid, we can divide into any number of concentric circles within this outer circle. For this, select the circle, and go to Object>Path>Offset Path. The pop-up asks for the offset width. Make offsets of -12mm (minus sign to make offsets internally) and click ok. Create successive offset circles of -4.5mm, -4.5mm, 9.5mm, -3.5mm, -11mm. An outer concentric circle of the offset 8mm and 15mm can be made outside of the 95mm initial circle for the corner triangle elements, however, it is optional.
8. Once we are ready with the grid, we can choose a grey stroke color for it and lock it using CTRL+2, so that it does not hinder with our mandala creation but acts as a base grid.

Creating a mandala

1. Now that we have the base grid ready, we can begin with the mandala. For this mandala, start from the innermost two circles, and start by using pen tool to draw a 'V' shape, formed by intersecting three of the lines and the two circles.
2. Further, select the 'V' shape, and use the Rotate tool (R) from the toolbar. Press the ALT key and click on the centre of the circles to make that as the axis along which the shapes will rotate radially. A pop-up appears to ask for the angle on which it must be rotated, type 30degrees (not 15 degrees since we created a shape using two segments. select the preview option to check if its rotating as desired). Click CTRL+D to repeat this in all the segments and create a circular pattern of individual V's.
3. Go to the next circle segment and repeat the above steps for a reverse V this time, and further for each shape repeating in each segment. We can change these individual shapes to form more complex patterns and add more intricacies.

Note - It is important to make sure each shape drawn by pen tool in each round is joined. For this, check if each vertices are meeting and click CTRL+J. If joined properly, all the shapes must select together.

Joining to form a singular closed shape

1. After the whole mandala is formed, we can see that our mandala is made out of single lines with some line thickness. To make these into shapes with outer and inner lines which can be cut as a shape, select all the shapes and go to Object>Path>Outline Stroke. We can further add variation to stroke thickness for different shapes beforehand to get varying thickness of each shape in the consecutive concentric circle as per choice. In this design, I have kept the stroke thickness from a minimum of 3pt to maximum of 4pt for the diamond shape. To understand this better, this is similar to drawing block alphabets (alphabets with outlines) as compared to a single lined alphabet.
2. To join all the shapes into a singular closed shape, select the shapes and go to Window>Properties and click on the 'Unite' option under Pathfinder in the properties panel.
3. Clear out some extra corners and edges using the Direct Selection Tool (A) to get rid of extra anchor points and clean out the mandala.

Step 2: Create Offsets to Form the Layers

Step two is pretty straightforward! - creating offsets of our created mandala pattern to form the successive layers.

For this, follow these steps -

1. Select the shape, and go to Object>Transform>Path Offset and type 1mm (positive sign to make offsets on the outer side).
2. Choose a different fill color (preferably a darker shade of the first layer). Select this new layer and press CTRL+[ to make this layer go behind the top one. Now you can see how the two layers superimpose.
3. Repeat step 1 and 2 three more times for this design to get 4 successive layers in total.
4. Next, we need to clear out the extra undesired offsets being formed outside of the first layer's 12x12cm square. For this, select all the layers together and click on the Shape Builder tool (Shift+M). Now, while pressing the ALT key, drag or click on the undesired shapes outside of the 12x12cm square.
5. In this example, I have used a fifth layer which is a blank 12x12cm square placed in the bottom, and a sixth layer that comes on top, which is a 1mm outer offset of only the outermost 9.5mm circle and the four corner triangles. These two layers are optional, however they add to the visual interest.

Separate these 6 layers and place them next to each other, then save it as .svg by going to File>Saveas and select .svg in 'save as type'. We can also export it as .dxf, .dwg or .pdf.

Step 3: Cutting the Layers

Laser-cutting

The beauty of laser-cutting is not just the sharp accurate edges, but the capability to make multiples of your digitized art in minutes! Adding to that, laser cutting allows you to scale up your design into a large wall piece or scale down into a miniature version of your art in case you want to try it out in a small version to save material or enjoy the complexities of the design in a miniature handy piece.

The steps to laser cut is as follows-

1. Copy your .svg or .dwg (or other) file in the PC connected to the lasercutter.
2. Open the software (Lightburn or whichever software is associated with the laser cutter).
3. I used a 1.5mm birch plywood to cut the 6 layers, and used the settings of speed - 30mm/s and power - 60% for a 80 watt laser-cutter.

Manual Cutting

If in case you do not have access to a laser-cutter, we can also use an exacto knife on a softer cardboard like Sunboard or PVC board. For this, all we need to do is print out the created mandala on a paper and tape it on the cardboard. With an exacto knife, we can cut through the desired holes. Little time-consuming, but worth it in the end.

As you go deeper into the layers, the amount of intricacies decrease, hence the topmost layer would be the most cumbersome, but the holes become smaller, which might need a little more patience, however, faster to finish.

Step 4: Assembly and Finishing

Finishing

Now that we are ready with each layer of our desired product, we can go ahead with finishing the layers and assembling them to secure it in place.

You are free to paint, stain, polish, to finish the layers in a desired way, or leave the piece as is. The beauty of this piece is that it does not necessarily need paint or finish to enjoy its visual appeal.

A great way to add to the visual contrast and depth of the layered complexities is to paint the layers in the same color but the shade gets darker as you go deeper into the layers, i.e. first (top-most) layer being lightest and last layer (bottom-most) being the darkest. (as shown in the Adobe Illustrator piece)

Assembly

Once the desired finishing is done, we can go ahead with gluing the layers together.

For this, we can use regular Fevicol (or any liquid glue) and spread it evenly in a thin layer UNDER each layer. Care should be taken to not put too much glue as it might leak from the edges on pressing the piece together and because of the intricate shapes it would be hard to clean them.

Another option is to simply apply a masking tape around the corners or edges of the outer surface to secure it in place.

Further, to prevent dust accumulation and the piece from getting dirty, we can enclose it in an acrylic casing all around, which again can easily be lasercut as per the size. You can also choose to simply put a clear OHP sheet on top of the piece, or a glass if you choose to cut it in a larger scale.

Thats it!

I hope you enjoy creating this mathematical art. The output is truly mesmerizing and is a perfect intersection of geometry and visual art, opening up endless possibilities once you make one of the pieces. Go ahead and explore further mandalas designs with a guaranteed awe-striking final piece!

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