From Crystal Lattices to City Streets

From the air, parts of Barcelona read as repeated blocks punctuated by intersections. Their corners are cut back, leaving space around the crossing. Trace a few blocks on a piece of paper and you can shift the trace sideways to recover much of the same arrangement. A crystallographer uses a related operation to describe periodic structures: translate an ideal pattern by the right distance and the repeat unit lines up again.

That shared operation makes a precise comparison possible. It also tells us where to start looking for differences.

Define the unit

In a simple two-dimensional lattice drawing, every point has an equivalent neighborhood. The square between four adjacent points can serve as a repeat cell. Repeated in both directions, it describes the ideal pattern across the plane. A crystal structure adds a motif, such as atoms at particular positions, to its underlying lattice. Actual crystals are three-dimensional, and the drawing below isolates just one aspect of periodicity.

In a city plan, intersections could stand in for points and the block between them could play the role of the repeat cell. That choice depends on what we want to see. The block outline leaves out parcel lines, the direction of traffic and the layout of the buildings. Each of those details may follow a different rule.

Follow a diagonal

In the comparison, the regular array sits beside an idealized street plan cut by a diagonal. Use the annotations to locate the repeat cell and the block boundary. Both give the eye a reference for what repeats. Along the diagonal, blocks are interrupted and intersections change. In Barcelona, Avinguda Diagonal offers a real example of a route crossing the Eixample’s grid.

A crystal can also depart from an ideal repeat. A vacancy is a missing atom at a site that would ordinarily be occupied. A street running on a diagonal is a different kind of departure. It can connect destinations or serve other planning purposes. Identifying a regular pattern helps name the deviation in both cases, while the reason for the deviation has to come from the domain itself.

Ask what maintains the arrangement

An atomic arrangement is constrained by the interactions of its constituents and by conditions during and after growth. A street plan can be established through surveying, law, building and maintenance. What happens on either side of an intersection can change while the intersection stays in place. A block that looks familiar on the map might house a school, a courtyard or an entirely different mix of activities from the block next door.

Put the two diagrams next to each other and give each the same questions. What repeats? Where does it stop? What can vary while the geometry still reads as a grid? The answers show what translation can describe across scales and what the eye alone leaves unresolved.