Europe in Two Dimensions

On this Expedition, we’ll tour Europe with an unusual purpose in mind: we’ll be looking for 2-dimensional, or flat, shapes—shapes that have length and breadth but no depth.

This story was created for the Google Expeditions project by ePublishing Partners, now available on Google Arts & Culture

Of course, Europe, like all the other places on Earth, actually exists in 3 dimensions. But on this expedition, we’ll focus on flat surfaces. Get ready to become a geometric tourist!

A Big Circle in London

We’re cruising on the Thames River in London, and there are curved shapes everywhere we look. The clouds and trees have natural shapes. If you drew them, you would use curved lines.

 The big wheel, which is called the London Eye, is a geometric shape: a circle. A circle is a curved line that is perfectly round. Every point around the outside of a circle is exactly the same distance from the centre. 

The London Eye’s wheel has a diameter, or distance across the middle, of 120 metres. The wheel of the London Eye is made of concentric circles, which are connected to each other with triangles. The base is an enormous triangle. 

Rectangles, Squares and Right Angles

This building’s windows are made up of squares and rectangles. Squares and rectangles are shapes with 4 sides that meet at right angles. Right angles have a measure of 90 degrees.

Palace of Westminster

The Palace of Westminster is where the UK Parliament meets. Its sides, windows, and towers are made of rectangles. The tower with the circular clocks is Big Ben. It is named after a huge bell inside the tower.

A Big Square in Paris

We’re in a square, or open space, in Paris, looking at the Grande Arche de la Défense. This giant monument is a cube, a 3-dimensional shape that has 6 square sides. 

Every side of every square in the cube measures exactly 110 metres. By looking through one of the square sides, you can see another square of sky. 

Arches

This building’s roof is an arch, a curved shaped that is symmetrical, or even on both sides. Some arches are fractions of circles, but some are not. Arches can be part of a circle or an oval.

Parallelograms

This building has walls covered in diamonds. The building to the left of it has walls covered in squares. Squares, rectangles, and diamonds are parallelograms. They have 4 sides, and the opposite sides are parallel to each other. 

Triangles and Pyramids in Paris

Here we are in Paris, France, visiting the Louvre. This enormous building was once a palace for French kings. Now it’s one of the world’s finest art museums.

We’re looking at a big glass triangle, or 3-sided shape. The triangle is one side of a pyramid, a 4-sided structure made of 4 equal triangles and a square base. The pyramid in the courtyard of the Louvre is the entrance to the museum.

Three Sides and Four Sides

Each side of the big triangle is 34 meters long. If you look closely, you can see that the triangle is made up of smaller pieces of glass. Most are shaped like diamonds. The shapes along the base are triangles.

Shapes with Four Sides and Right Angles

Right ahead of us, there are 3 squares over a row of 3 arches. Squares have 4 sides that are the same length. Most of the windows in this building are rectangles. Rectangles have 2 longer sides and 2 shorter sides.

Circles, Curves, and Arches

We’re leaving the Louvre on a road that curves around a huge circle. Ahead of us is the Arc de Triomphe du Carrousel, a monument with 3 arches. The tops of the arches are semicircles, or half circle.

Giant Trapezoids in France

The Geoffrey Guichard Stadium in Saint-Etienne, France, is a modern renovation of a stadium that first opened in 1931. The stadium hosts football (soccer) and rugby matches. The building has entrances that are giant trapezoids. Trapezoids are shapes that have 4 sides. 

Two sides are parallel but of different widths. The parallel sides connect to the other 2 sides with oblique angles. Oblique angles have measures greater or lesser than 90 degrees. In other words, they are not right angles.

Acute Angles for Strength

The angles formed by the base and the upright sides of the trapezoid are acute angles, which means they measure less than 90 degrees. Because the sides slant inward, the construction has extra strength.

Obtuse Angles

The angles formed by the top and the upright sides of the trapezoid are obtuse angles, which means their measure is greater than 90 degrees. Regular trapezoids have 2 acute angles and 2 obtuse angles.

Roman Shapes in Spain

The aqueduct in Segovia is a symbol of the country of Spain. It was built around 100 A.D. by the Romans, who ruled Spain at that time. The aqueduct’s purpose was to bring water to the city. 

The water flowed through a channel at the top of the structure. The Segovia aqueduct shows off the shapes that made Roman structures strong enough to stand for centuries.

Roman Arches

Look through the aqueduct to the sky to see the shapes. The top row has half circles (semicircles) on top of squares. The bottom row has half circles on top of rectangles. Each rectangle is about 3 squares tall.

Shapes that Made Roman Arches

The bricks that make the columns have rectangular shapes. The bricks that make the arches have trapezoidal shapes. The slanted sides of the trapezoids create the arched shapes.

Circles in San Lorenzo

We’ve come to Mantua, Italy, to examine the Rotonda di San Lorenzo, an ancient church. The word rotonda, or rotunda in English, should give you a clue about what we’re looking for: a rotunda is a building with a circular plan. 

Here in the Piazza della Erbe, there are circles and half-circles all around us. The wheels of the vehicles are circles. The arches on the buildings are half circles. We can find other shapes, too, including decorative shapes made of more circles.

Rotunda of San Lorenzo

The Rotonda di San Lorenzo, which was built in the 11th century, is a cylinder, a 3-dimensional shape that has straight parallel sides and a circle as its base. San Lorenza is decorated with half circles along its roof lines. 

Shapes on the Tower

At the top of San Lorenzo’s tower, there is a triangle with an oval inside it. Right below the oval is a square. Farther down we can see a big circle and, below that, another oval. The tower’s sides are rectangles.

Curves on the Street

If we look down at our feet, we can see there are even circles in the streets! The paving stones are arranged in circular patterns. We can see the curves of quarter circles.

A House of Rectangles

Longleat House in Warminster, England, was built in the 1560s and 1570s, during the reign of Queen Elizabeth I. The outside walls of the 100-room house are almost entirely covered by windows—and glass was a great luxury in Elizabethan times. 

Longleat House was opened to the public in 1949. It was one of the first of Britain’s great country houses to become a tourist attraction.

Shapes at Longleat House

At first glance, the front of Longleat House seems to consist entirely of rectangles and squares. But if you look closely, you can see circles above the roofline and a triangle above the entrance. 

Rectangles inside Rectangles

Each window of Longleat House is made of many small squares or rectangles. There are thousands of right-angled shapes at Longleat. Just imagine the work it takes to keep all its windowpanes clean! 

 The cathedral was built between 1093 and 1133, but some decorations, such as some of the stained-glass windows, were added in more recent times. A tour of Durham Cathedral is a dazzling way to see some not-so-common shapes and to review 2-dimensional shapes we know.

Durham Cathedral: Shapes for Worship

We’ve travelled to Durham England to visit the city’s famed cathedral, a beautiful example of the Romanesque style of architecture. Throughout the building, we can see arches and other shapes that appeared in the buildings of ancient Rome.

Shapes in the North Transept

There are black and white diamonds on the floor. There is a clock made of one large blue circle and 3 smaller circles. Circles and rectangles decorate the pulpit on the left. A large square tapestry hangs on the wall.

Pointed Arches and Diamonds in the Nave

Now we’re looking at the nave, where worshippers sit. Diamonds decorate the pillars and the floor. We can see half-circle arches. The pointed arches were a new style at the time Durham Cathedral was built.

A Rose above the Altar

This is a rose window, a round window with shapes that radiate from a central circle. The ‘petals’ of the rose are made of polygons that have 5 sides. A polygon is a shape with at least 3 straight sides.

A Rose in the Roof

Stand in the centre of the cathedral and look straight up to see another rose. This one is carved from stone. It has 8 5-sided shapes that radiate from a central circle. What other shapes can you see?

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