“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
Conclusion
References

Escher’s tessellation artworks speaks of beautiful and highly creative ways of manipulating tilings in the 2D plane. We ask ourselves questions on how Escher’s method of creating such interlocking patterns could be extended into the realm of architecture which is essentially the 3D world. Could Escher’s tessellation be done in 3D too ?

Tiling in Architecture

Tiling in architecture is a very common sight. You see them everywhere, from floor tiles, to brick laying to curtain wall facades of buildings. However such tilings are usually 2D and comprise of simple platonic or archimedian tiling patterns. Escher’s tessellations offer a more interesting way of tiling in architecture. However such an application of tessellations on architecture is purely decorative. For example, floor tiles could literally be an Escher’s tessellation artwork, as the picture below shows.

Another application of tessellation could be in the façade treatment of architecture. Instead of using rectangular building blocks of glass or granite to clad the exterior surfaces of buildings, the basic shape of the building blocks could vary (not necessarily must be an Escher artwork, could be an elegant shape that tiles very nicely). This would create an interesting textural quality and feel of the building, instead of the usual “boring” rectangular shapes. While we recognize that this is not economically viable in reality and could be structurally difficult to resolve, it is an idea that could be expounded on. Currently, buildings designed by architects such as Daniel Libeskind have very interesting tiling patterns on the building façade. An example of such a building would be the Federation Square in Melbourne, designed by Lab Architecture Studio of London in association with Bates Smart of Melbourne.

Lego and other 3D building blocks

Has it ever occurred to you that Lego is actually a 3D building block that could be tiled and “locked” in 3D space, simply by the tongue and groove protrusions in each basic Lego unit? Expanding on that idea, complex building blocks could be designed based on the simple addition and subtraction method we demonstrated earlier. Such buildings blocks could manifest in a new kind of brick that locks and tile better or even interesting architectural concepts on the arrangement of 3D space that interlocks with each other. The possibilities are endless.

 Escher’s artwork Flatworm which shows off tetrahedron building blocks Our computer render of a possible "brick" to be used for 3D tessellations

Conclusion

"At first I had no idea at all of the possibility of systematically building up my figures. I did not know ... this was possible for someone untrained in mathematics, and especially as a result of my putting forward my own layman's theory, which forced me to think through the possibilities."

It is difficult to determine how mathematical Escher’s works really are, especially since Escher himself did not design his tessellations based on strict theories of mathematics, but rather, he used his understanding of it to develop his art. His art seems to evolve past pure mathematics, and most of which, we feel, is created by his aesthetic judgement and at times, seems somewhat arbitrary.

By imposing mathematical theories onto Escher’s works, we wonder if the true meaning of his art has been trivialized to something mechanical. Escher’s art should be admired and appreciated for its intricacy, sensitivity to details, attractive forms and fascinating ideas and imagination. To look at it in a purely mathematical manner would be equivalent to enforcing the "golden rectangle" on the Mona Lisa.

Mathematical his art may be but upon our groups' embarkation on using mathematical principals to form our own tessellations, we understood how it was not purely maths that allowed Escher to come up with his artwork but rather a layman's understanding of it and application coupled with artistic talent which made him the genius that he was. Only with this new realisation on our part can we truly respect the genius of this artist.

To conclude, we end of with another quote from Escher which implies that mathematics is all around us, it is just men who needs to open their eyes and see it and ultimately make use of it.

"I can rejoice over this perfection and bear witness to it with a clear conscience, for it was not I who invented it or even discovered it. The laws of mathematics are not merely human inventions or creations. They simply "are"; they exist quite independently of the human intellect. The most that any man with a keen intellect can do is to find out that they are there and to take cognizance of them."

References

The Magic Mirror Of M.C. Escher, Bruno Ernst,Barnes & Noble, Inc., 1994
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Escher.html