“For me it remains an open question whether [this work]
pertains to the realm of mathematics or to that of art.”

- M.C. Escher

Contents

Page 1
Introduction
The Art of Alhambra
Our Area of Focus
Our Aim

Page 2
The Principals behind Tessellations
- Translation
- Rotation
- Reflection
- Glide Reflection

Page 3
Mathematics in Escher's Art
- Translation
- Rotation
- Glide Reflection

- Combination

Page 4
Our Original Tessellations
- The Catch (Translation)
- Under the Sea (Rotation)
- The Herd (Glide Reflection)
- Dumbo & Butterfly (Combination)

Page 5
Possible links with Architecture
Conclusion
References


Our Original Tessellations

After disecting a selected piece of Eschers work from the previous page, our group then embarked on a task to come up with our own tessellations based on the mathematical principals we discovered in his work. Each member who created the tessellation then penned down their thoughts after their attempts.

Before we begin the group members fully admit that we are no M.C. Escher as such our tessellations can be considered a far cry from what Escher has produced. Then again, it was our first try at something Escher had devoted his whole life to.

In addition, even though we knew the mathematical principals behind the tessellations, we realised that coming up with one is a lot more difficult than it seems, as we soon found out.



The Catch (Translation) by Doreen Koh

At first sight, my part of the project seems simple enough. I just need to draw a couple of grids, doodle some shapes and repeat the rest of the pattern. Escher’s work contained of several pieces on fish. Thus, I decided to let my subject, i.e. the form of tessellation, be the fish to try to experience and understand more of his concept on tiling using similar approach. Besides, fish seemed undemanding enough.

However, as I proceed on, I realize that it is easier said than done. Amidst the frustration of getting the right shape and trying to fit each individual tile with one another to prevent gaps, I began to appreciate the brilliance of his work. Many considerations and manipulations had to be made in order to reach the final piece. A good selection on the type of grid to base your work on is of utmost importance. The wrong first step will result in a painful journey later.

The creation of the final piece is mainly through the adjustment of shapes and our stylistic touch and deal not so much with any mathematic principals despite the end product deeming so otherwise.


Under the Sea (Rotation) by Janice Teo

When I first saw this tessellation, it seemed pretty straight-forward: simply a man under 3-fold rotational symmetry. However, upon closer inspection, I realised it was pretty amazing how everything fitted so perfectly into one another. For instance, the space beside one man's face is another man's hat! When you zoom into the individual units that make up this tessellation (triangles), they don't seem to make any coherent sense! However, through the rotations into hexagons, they miraculously start to form the shape of the men! It took me quite a while to digest the picture and after analysing it, i realised it was certainly more than simply rotating men! The amount of precision and detail that made up the tessellation astounded me.


I tried to create a tessellation similar to Eschers. The hardest part of which was choosing a motif that i could subject to rotation and not leave any gaps! This was a long and tedious process, and eventially, i had to settle for a tessellation slightly different from Eschers as i found out that having a 3-fold rotational symmetry at 3
points is really tough! I chose to do instead, something simpler, which is a 6-fold rotational symmetry about a point. I evetually settled on a dolphin motif, and rotated it.

Though i realise that my tessellation is infinitely simpler than Eschers', coming up with the tessellation itself was not easy Some difficulties involved was resolving how the tails of the dolphins would all come together, and also the conscious attempt to not leave any empty spaces in between. The tessellation finally evolved through a lot of patience and manipulation. It is through the creation of our own tessellations that we appreciate the beauty and genius of Eschers' work to a much higher level.


The Herd (Glide Reflection) by Ervine Lin

Looking at Escher’s example of glide reflection (the horsemen) I figured that it would turn out to be quite a simple task. Just draw the object, reflect it, translate and maybe manipulate it a little to fit. However, it turned out to be a lot more complex than that. I had to make sure that no gaps were allowed and that simply by glide reflecting didn’t automatically mean that there would not be any gaps. The animation greatly simplifies how I came about with this particular tessellation but all I can say that it was a lot of trial and error and very little mathematics.

Ironically, I had originally planned to tessellate a series of elephants but somewhere along the way when I was sketching the tessellation out, I realized I could not squeeze the outline of the elephant in correctly. As such it somehow changed into a ram instead which brings me to the next point.

I could have placed any other animal or object into that outline of the ram and it would still tessellate. In fact, it is the outline that is the most difficult to come up with, after that what ever details go on the inside are very much non-mathematical and purely artistic.

In fact I would say that the actual creation of the tessellation is more art inclined than mathematics despite the final outcome having mathematical properties.


Dumbo & Butterfly (Combination) by Tay Wee Lian

I honestly felt that I had the most complex tessellation of all to create. To dissect Angels & Devils was a heavy task let alone creating a tessellation that used its fundamental principals. To fit the elephant and the butterflies perfectly together was such a mind boggling experience that it took me ages to get to the final outcome. However, after much perseverance I finally managed to come up with something that was at least remotely decent.

To conclude, I feel that the most difficult part in obtaining this kind of tessellation is the process of finding out how a pattern is repeated such that there are vertical, horizontal and glide reflections are all involved. What Escher did was truely amazing and it is only after I had a go at it did I fully appreciate his work.