Europe in Three Dimensions—and More

Our ancestors all over the world built simple structures like these, combining cylinders, cubes, cones, and other basic 3-dimensional shapes. As we tour the monastery, we’ll find 3-D shapes and analyse how the shapes work together to make buildings better.

These one-walled huts are cylinders, shapes with circular ends and straight sides. The roofs are cones. A cone has a flat circular base and a curved side that tapers to a point.

The roof of the church has identical triangles for ends and 3 rectangular sides, all the same length. It’s a triangular prism. One side of this prism is the same size as the base of the rectangular prism it sits on.

These 4-walled structures have rounded roofs. The roof in the foreground is a rough dome. The roof behind it is an irregular pyramid with rounded edges. Smoke from a central hearth would rise through the centre point of the roof.

The Romans built straight streets and walls that met at right angles. Notice the flat, rectangular stones that paved the streets and created steps. Main streets were often lined with round columns, made by stacking smaller cylinders.

Polyhedrons are shapes that have faces, edges, and vertices (corner points). Cubes and rectangular prisms have 6 faces, edges at right angles, and 8 vertices. The bricks in an arch are trapezoidal prisms. Two of their 6 sides are trapezoids.

Roman architects followed an ancient Mediterranean tradition of using columns to create open walls, or colonnades. Long, straight rectangular slabs, called lintels, were laid across the tops of the columns to support upper levels or roofs. Lintels are rectangular prisms.

Hagia Sofia’s central dome is 55 metres above the floor and 33 meters in diameter. The dome is supported on the inside by an intricate system of arches and columns. The building is buttressed by cylindrical walls, half-domes, and arches. 

The Sultan Ahmet Mosque is also called the Blue Mosque, after its stunning cobalt and turquoise interior tiles. The mosque is composed of the same shapes as the Hagia Sophia. The minarets are cylinders topped by cones.

We have travelled to Ávila, Spain, to examine the Muralla de Ávila, the city’s great wall. This massive structure, which dates from the 11th century, encloses 31 hectares (about 76.5 acres) in a huge irregular rectangle. 

The walls combine the shapes of polyhedrons and cylinders. Arches help support towers and create gates.

Ávila´s great wall is a masterwork of masonry. The irregular stones were chiselled into polyhedrons and then carefully matched and levelled to form horizontal rows. A polyhedron is a shape with flat sides, edges, and vertices, or corners.

Turning our backs to Ávila’s great wall, we see buildings from both Renaissance and modern times. We also see basic 3-dimensional shapes. The church ahead is composed almost entirely of stacked rectangular prisms.

Cones and pyramids sit on top of cubes and cylinders. A cone has a flat base and a curved face that rises to a central point. A pyramid has a flat base and multiple triangular faces that rise to a point.

The huge stones of the outer walls are mostly rectangular prisms. But some stones are trapezoidal prisms that are used to compose even larger 6-sized trapezoidal shapes. A roof in the upper left is also shaped like a trapezoidal prism.

This is a common roof shape, but it is a special kind of triangular prism called a truncated triangular prism. The triangular ends truncate the prism at an oblique angle.

Motorists approaching or leaving the Guggenheim Museum drive through a bright red circle and past other curved shapes. They are sections of a torus, a circular shape that forms a ring. 

When you look at a Gehry building from the outside, it can be very hard to imagine what the inside spaces are like. This structure’s design plays with spiral, spherical, and trapezoidal 3-dimensional shapes that support each other in a manner that makes the whole building seem mobile.

Both the road and the sidewalk are long ramps that curve around the spirals, cubes, and trapezoidal prisms of the Guggenheim. The irregular 3-dimensional shapes that make up the building contrast with the very regular shapes of the buildings across the street.

London’s skyline includes very old buildings, made of the shapes you expect to find in buildings, and very new buildings that surprise the eye. One very large skyscraper is called the Gherkin because it’s shaped like . . . yup, a very large pickle!

This is the Tower of London, an ancient fortress composed of cubes, rectangular prisms, cylinders, and truncated spheres. To the left, you can see the top of the Gherkin, a slightly bulging cylinder that curves from base to top, where it tapers to a point.

Modern glass and steel structures along the Thames have variations of cylindrical and spherical shapes as well as polyhedrons. The triangular tower in the distance is an irregular pyramid known as the Shard, because it resembles a sliver of ice.