multiverse . How to stack up shapes

Imagine the construction of a grand brick structure, say the ancient bath at Mohenjo-daro. Architects and builders would have planned out the geometry, construction methods and logistics of the process well in advance. They would have assumed that their building method would be used repetitively to give rise to the final grand structure. The primary structural unit in the Indus Valley Civilisation is the brick — somewhat different in size from the present-day ones but similar nonetheless in idea. At its simplest, a brick is a solid geometric object bounded by six faces all of which are rectangles. We call these cuboids. What happens when you stack cuboids on top of each other to make a structure? The process of designing a structure can be conceived of in two ways — of building a space with elementary blocks and also as imagining how spaces can be divided by geometric shapes packed next to each other without gaps.

Imagine a wall. A large rectangular swathe of whitewashed area in your room. The plaster and paint obscure the bricks underneath. Now try to imagine what kinds of shapes stacked on top of and next to each other could fill up such an area. The simplest choices are squares and rectangles. These are essentially the shapes of simple brickfaces. Humanity could have stopped here but artists, designers and mathematicians wondered if other shapes could fill up space. It turns out that a surprising number of shapes can be stacked to fill up a given space. Triangles will do, as will hexagons. Even arrows facing opposite directions work just as dumbbell shapes. These experiments led medieval designers to wonder about the decorative potential of tiling shapes. Islamic designers were particularly interested in these ideas as the depiction of human form was forbidden in their religion. The desire for ornamentation needed to be given material expression in the construction of grand castles and mosques. So began their experimentation with geometric shapes to tile not just plane surfaces but also domes, arches, pillars and vaults.

The designers of the Alhambra, a Moorish castle in Andalusia, Spain, experimented with numerous kinds of tiling: Various shapes and stacking patterns, patterns with gaps where the space is filled with yet another shape, and others with overlapping geometric shapes.

The Alhambra tiling is representative of the great variety of wall tiling methods that the Andalusian Moors experimented with. The castle boasts a wealth of tiling patterns and has influenced many artists through the ages and continues to be the object of study by many mathematicians, architects and designers.

In sync: Escher constructed tiling with human and animal forms which were previously thought to be impossible

Artists such as the Dutchman Maurits Cornelis Escher wondered if the plane could be tiled with not just abstract geometric shapes but with also those of recognisable forms of humans and animals. Artists, astronomers and mathematicians wondered about using certain ‘impossible’ shapes to tile the plane. No gaps allowed. An elusive shape though was the simple pentagon. Try to cover a space with it and you will end up leaving gaps. Could it be done otherwise? The German Renaissance artist, Albrecht Dürer, attempted his own set of pentagonal tiling as did the Polish astronomer Nicolaus Copernicus. An investigation into these ideas by the English physicist Roger Penrose led to a formal understanding of tiling the plane surface with a variety of shapes.

Shape of art: A Penrose tiling

Tiling of the plane using bricks are what we may call periodic. Shift the whole diagram by one brick unit and it will match perfectly with the unshifted pattern at that point. Shift it by two units and the result is the same. Penrose investigated ways of tiling which lacked this property. Shifting and matching was impossible in the other instances.Many of these ideas reappear in the study of crystals as they must be stacked together at a microscopic level. It turns out that even at the microscopic level, the packing of atoms and molecules follow very similar principles. Matter can be packed in ordered, symmetric and periodic fashion but also in the aperiodic fashions we’ve described.

Escher visited the Alhambra palace and copied the designs on the walls. While immensely impressed with the work of the Moorish masters, Escher took a different approach to tiling a space. He set himself the challenge of constructing tiling with human and animal forms, and achieved images which were previously thought to be impossible.

He also attempted the task of depicting the notion of infinity. Many of his images deal with structures such as stairs that you can ascend and descend at the same time, and waterfalls that go on in perpetual motion.

To produce such work, Escher combined the imagination of an artist with inputs from some of the best mathematicians of his era —Penrose and Harold Coxeter. Diverse set of ideas converged to set forth some of the most original work; Escher united art and science into one whole. Take a look at his work and you will be amazed.

Santanu Chakraborty

Santanu Chakraborty is a Bengaluru-based engineer, scientist and photographer