Many of the drawings of Dutch artist Maurits Cornelis (M.C.)
Escher closely connect with the
mathematical concepts of infinity and contradiction. While these concepts lead
to many themes, *tessellations of the plane* appear particularly often in
Escher's work.

A tessellation (or tiling) of the plane is a construction that fills a flat surface completely with geometric shapes, usually called tiles. Escher often explored symmetric tessellations that were formed by repeatedly duplicating and rearranging only a single tile through translation, rotation and reflection. A simple example of such a tessellation is one which uses only squares — you can imagine the following pattern repeated to completely cover an arbitrarily large surface.

Or, we might have a slightly more complex construction using triangles.

From the mathematical perspective, these constructs are often quite rigid as in
the above examples, but Escher discovered more complex tessellations that bring
visual interest and aesthetic beauty. This complexity is found in, for example,
Escher's *Lizard*.

The lizard's shape, while on its own not particularly symmetric, combines with
copies of itself to create symmetry on a larger scale. Escher uses tessellation
to highlight the artistic relationship between positive and negative space. In
*Lizard*, there isn't a clear distinction between the subject and its
background. If you view the lighter-colored lizards as the subject, then the
negative space is comprised of the darker-colored lizards and vice versa. In a
sense, the subject is its own negative space through the symmetric tiling.

Escher was probably aware of the perceived rigidity of tessellations. In
contrast, Escher has a few works in which he depicts the transformation from
rigid geometric tilings into organic and natural imagery. *Liberation* shows a
triangular tessellation (similar to the example above) that slowly transforms
into an array of birds mid-flight.

Which are being liberated — the birds or the triangles? On the one hand, the birds seem to be breaking free from the rigid geometric form of the triangles. But on the other hand, the triangles exist in a world of mathematical perfection which has a beauty of its own.

The transformation demonstrates one technique for constructing complex tilings by making small perturbations to an existing tiling. This process is detailed at the end of the post.

Escher's Circle Limit drawings — *Circle Limit I* is pictured at the
beginning of this post and *Circle Limit III* is pictured above — display
a different kind of tiling than the ones we've talked about so far. Up until
this point, the tilings we've seen have been such that they could be extended
outwards toward infinity on a plane. The Circle Limit drawings instead seem to
converge toward a circle. In fact, these are still plane tilings, but in
different kind of geometry, namely hyperbolic geometry.

In Euclidean geometry, the parallel postulate roughly states: For any line $L$ and point $P$ not on $L$, there is exactly one line parallel to $L$ that passes through $P$. This matches our intuitive notion of parallel lines from our everyday experiences (which Euclidean geometry models). But it is not necessary for geometry to include the parallel postulate, and hyperbolic geometry is the consequence of modifying the postulate to allow multiple lines to be parallel to $L$ and pass through $P$.

Taking a look at both *Circle Limit I* and *Circle Limit III*, both images are
based on the Poincaré Hyperbolic
Disc. In this model
of hyperbolic geometry, we work within a circular disk. Lines are represented as
circular arcs that intersect the disc at right angles. *Circle Limit III*
highlights these arcs in the white stripes running along the fish. Two
hyperbolic lines are considered parallel as long as their arcs don't intersect.
Under this definition, we can confirm that multiple parallel lines can run
through the same point — consider the following annotated version of
*Circle Limit III*.

The red and pink arcs exemplify lines of this hyperbolic space. Call the red line $L$ and the labeled point where the fish mouths meet $P$ as in the modified parallel postulate. Then, we see that there are indeed multiple lines parallel to $L$ and passing through $P$ by observing the three pink arcs passing through $P$ yet not intersecting $L$.

Interestingly, all the birds in *Circle Limit I* are the same size with respect
to their hyperbolic geometry, and the same goes for the fish in *Circle Limit
III*. This is because the notion of measurement in the Poincaré model is
different from the notion of measurement in the Euclidean plane in such a way
that equally spaced objects (from the hyperbolic perspective) are closer to each
other (from our eyes' perspective) as you move toward the perimeter of the disc.

There are many other interesting and unintuitive facts that emerge in hyperbolic geometry which you can read more about here.

Want to create your own tessellation? One technique takes inspiration from
Escher's *Liberation*. Start with a simple tessellation (like the geometric
examples above), and make a small modification. Let's start with the
square-based one.

And then let's cut out a part of one of the tiles.

In order to maintain the symmetry, we must then copy our modification over to the other tiles (making sure to apply the appropriate translations, reflections and rotations).

Repeat steps 2 and 3 until you are satisfied with the result.

From this process, you can arrive at some interesting planar tessellations and gain more appreciation for Escher's art. If you're up for the challenge, you could try your hand at constructing a hyperbolic tiling in a similar way.

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