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Polygons and Polyhedra

Shapes and geometry is everywhere around us, is nature, architecture, technology or science. Here you will learn about angels, polygons, tessellations and polyhedra.

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Polygons

You’ve already learned about points, lines, angles, shapes and other objects in geometry. A polygon is any shape which is made up only of straight lines. A square, for example, is a polygon, but a circle is not. Select all of these shapes which are polygons:

If all the sides of a polygon have the same length, it is called a regular polygon. Select all of these shapes which are regular polygons:

Regular triangles are often called equilateral triangles, and reqular quadrilaterals (4-gons) are often called squares.

Finally, the sides of a polygon are called edges, and the points are called vertices. However these are enough words and terminology for now…

Angles in Triangles

Every triangle has three internal angles – angles at every vertex on the inside. In this diagram you can move the vertices and see what happens to the internal angles. Try making all angles as large as possible, or as small as possible, or as equal as possible.

${deg(c,a,b)}°
${deg(a,b,c)}°
${deg(b,c,a)}°
${selected(a,b,c)}°

In any triangle, the sum of the internal angles is always °.

This is one of the most fundamental results in geometry and it is not difficult to see why.

First we take one of the sides of the triangle and draw a parallel line through the remaining vertex.

Now we can shift the green and blue angles up one level, because they are angles.

Finally we can flip the red angle over, because they are angles.

Now the blue, red and green angle form exactly one semicircle, which is 180°. Like before, you can move the vertices in the diagram to see that this works for any triangle.

Since we know the sum of the angles in every triangle, we can also work out the size of a missing angle.

Find the missing angles

Exercises under development…

Angles in Polyhedra

Next, let’s have a look at quadrilaterals. 

${deg(b,a,d)}°
${deg(c,b,a)}°
${deg(d,c,b)}°
${deg(a,d,c)}°
${selected(a,b,c,d)}°

It seems that the sum of the internal angles is always °.

This is no only a full circle rotation, it is also twice the sum of triangles. And this is because we can split every quadrilateral into two triangles.

polygons 180 180

The same, of course, also works for larger polygons. We can split a pentagon into triangles, so its internal angle sum is °. And we can split a hexagon into triangles, so its internal angle sum is °.

In general, a polygon with ${fn1(x)} sides will have an internal angle sum of 180° × ${fn2(x)}.

Remember that in regular polygons, all sides have the same length and all angles are all the same size. Therefore we can calculate the size of a single internal angle in a regular polygon:

angle = = 180° × (x – 2)x = 180° – 360°x.

In particular, the sum of a regular polygon with ${x} sides is 180° – 360°${x} = ${Math.round(180-360/x)}°.

Quadrilaterals

Quadrilaterals are polygons with four sides. There are several special kinds of quadrilaterals:

quadrilaterals

If a pair of opposite sides are parallel, we have a Trapezium.

quadrilaterals

If both pairs of opposite sides are parallel, we have a Parallelogram.

quadrilaterals

If, instead, two pairs of adjacent sides have the same length, we have a Kite.

quadrilaterals

If all four angles are right angles (90°), we have a Rectangle.

quadrilaterals

Or if all four sides have the same length, we have a Rhombus.

quadrilaterals

Finally, if all angles are 90° and all sides have the same length, we have a square.

Any quadrilaterals could fall into more than one of these categories. Let’s have a look at a simple example:

quadrilaterals

This shape is clearly a , because all four of its have the same length.

But top and bottom pairs of adjacent sides also each have the same length. Therefore this is also a .

And both pairs of opposite sides are parallel, so this is also a .

Finally, if both pairs of sides are parallel, then one pair of sides must also be parallel. Therefore we also have a .

To avoid this kind of ambiguity when identifying quadrilaterals, we usually only use the most specific one – or the type furthest down in the tree diagram above. In the above example that would be a .

Identify the Quadrilateral

Exercises under development…

We often need to calculate the area of quadrilaterals. The area of a square is simply its width squared. The area of a rectangle is the of its width and its height.

For other quadrilaterals, we first have to try to “convert” them into rectangles.

quadrilateralswidthheight

Parallelogram

quadrilateralsbaseheight

Notice how the overlap on the left is exactly the same as the gap on the right. Therefore the area of the blue parallelogram is the area of the red rectangle. This means that the area of any parallelogram is simply

base × height.

Kite and Rhombus

quadrilateralswidthheight

Each the four smaller rectangles are exactly half blue. Therefore the area of the blue kite is half the area of the red rectangle. In other words, the area of a kite or rhombus is always

12 width × height.

Trapezium

quadrilateralstopbottomheight

Notice how the gaps on the left and right are each exactly as big as the overlap on that side. The width of the red rectangle is the average of the widths of the top side and the bottom side of the blue trapezium. Therefore the area of a trapezium is

top + bottom2 × height.

Area of Quadrilaterals

Exercises under development…

If we connect the centers of the four sides of a quadrilateral, we get . Try to move the vertices of the outer quadrilateral to change the shape on the inside.

It looks like the quadrilateral on the inside is always a !

Explanation coming soon…

Tessellations

Polygons appear everywhere in nature. They are especially useful if you want to tile a large area, because you can fit polygons together without any gaps or overlaps – these patterns are called tessellations.

Hexagonal honeycomb

Sinaloan milk snake skin

Cellular structure of leafs

Basalt columns at Giant's Causeway in Northern Ireland

Pineapple skin

Shell of a tortoise

And, of course, humans have copied many of these natural patterns in architecture and technology:

Rectangular pavement pattern

Greenhouse at the Eden project

Mosaic at Alhambra

Great Court Roof at the British Museum in London

Cellular tessellation pavilion in Sydney

Study of Regular Division of the Plane with Reptiles, M. C. Escher

Here you can create your own tessellations with regular polygons. See if there are any shapes that tessellate well or not at all, and try to create interesting patterns.

When creating tessellations with regular polygons, you'll have noticed that some kinds of polygons tessellate easily, while others don’t tessellate at all.

tessellate60°60°60°60°60°60°

Triangles because 6 × 60° = 360°.

90° 90° 90° 90°

Squares because 4 × 90° = 360°.

108° 108° 108° 108°

Pentagons because multiples of 108° don't add up to 360°.

120° 120° 120°

Hexagons because 3 × 120° = 360°.

This has to do with the size of their internal angles, which we leaned to calculate before. If you can make multiples of their internal angle add up to °, the shape will tessellate. Otherwise there will either be a gap or an overlap.

You can also make tessellations out of multiple different kinds of regular polygons, provided their combined internal angles make 360°:

tessellation

equilateral triangles and squares

tessellation

equilateral triangles and squares

tessellation

equilateral triangles and regular hexagons

tessellation

equilateral triangles and regular hexagons

tessellation

equilateral triangles, squares and regular hexagons

tessellation

squares and octagons (8-gons)

tessellation

equilateral triangles and dodecagons (12-gons)

tessellation

equilateral triangles, squares and dodecagons

Notice, however, that none of these tessellations contain a regular

And of course we can make tessellations out of non-regular polygons as well:

Interactive diagram coming soon…

Triangles always tessellate. Their internal angle sum is °, so if we use every angle twice at a vertex, we get 360°.

Interactive diagram coming soon…

Quadrilaterals always tessellate. Their internal angle sum is °, so if we use every angle at a vertex, we get 360°.

Diagram coming soon…

Pentagons sometimes tessellate, but only if they have particular shapes.

Many artists came up with much more complicated tessellations, most famously M. C. Escher. Here are a few examples:

Magic Mirror (top left), Regular Division of the Plane III (top right), Reptiles (bottom left) and Sky and Water I (bottom right) were all created by M. C. Escher.

All the tessellations we saw so far have one thing in common: they are periodic. This means that they consist of a regular pattern that is repeated again and again. They can continue forever in all directions and they will look the same everywhere.

In the 1970s, the English mathematician Sir Roger Penrose discovered non-periodic tessellations – they still continue infinitely in all directions, but it will never look exactly the same. These are called Penrose tilings, and you only need a few different kinds of polygons to create one:

Move the slider to reveal the underlying structure of this tessellation. Notice how you have the same patterns at various scales: the small yellow pentagons, blue stars, orange rhombi and green ‘ships’ appear in their original size, in a slightly larger size and an even larger size. This self-similarity can be used to prove that this Penrose tiling is non-periodic.

Polyhedra

Up to now we have looked at what you can do with polygons in a flat, two-dimensional world. We can also connect polygons to form three-dimensional solids – these are called Polyhedra.

The sides of a polyhedron are called faces, the lines where faces meet are called edges, and the corners where edges meet are called vertices.

Diagrams and Euler’s Formula coming soon…

Platonic Solids

Just like we defined regular polygons as particularly symmetric polygons, we can create particularly “regular” polyhedra – for example, only using just a single kind of regular polygon:

To create a 3-dimensional object, we need at least faces to meet at every vertex. Let’s start with equilateral triangles:

platonic

We can create a polyhedron where three equilateral triangles meet at every vertex. The result is called a Tetrahedron.

platonic

If four triangles meet at every vertex, we get a different polyhedron called Octahedron.

platonic

If five triangles meet at every vertex, we get another polyhedron called Icosahedron.

platonic

If six triangles meet at every vertex we don’t get a polyhedron: just a flat .

We always need a to create a 3-dimensional shape. Therefore seven or more triangles at every vertex also can’t produce new polyhedra.

We can do the same with squares:

platonic

If three squares meet at every vertex, we get a cube, sometimes also called Hexahedron.

platonic

If four squares meet at every vertex, we get another tessellation. Like above, five or more squares also won’t work.

Next, let’s try pentagons:

platonic

If three pentagons meet at every vertex, we get a polyhedron called Dodecahedron.

platonic

Like above, four or more pentagons because there is no gap we can fold.

And similarly for hexagons:

platonic

If three hexagons meet at every vertex we get a flat tessellation but no polyhedron. And again, four or more hexagons also don’t work.

The same also happens for all regular polygons with more than six sides. They don’t tessellate, and we certainly don’t get any 3-dimensional polygons.

Thus, there are different polyhedra consisting of just one kind of . These are called Platonic Solids, named after the Greek philosopher Plato who discovered them. Their individual names are derived from the Greek name for their number of faces: “tetra”, for example, means “four”.

Plato believed that all matter in the Universe consists of four elements: air, earth, water and fire, and that these correspond to different Platonic solids. The fifth, he thought, was the shape of the entire Universe. Today we know that there are more than 100 elements which consist of spherical atoms, not polyhedra.

Try to fill in the following table with properties about the platonic solids:

TetrahedronCubeOctahedronDodecahedronIcosahedron
FireEarthAirThe UniverseWater
Faces
Vertices
Edges
Faces
Vertices
Edges
Faces
Vertices
12 Edges
Faces
20 Vertices
30 Edges
Faces
12 Vertices
30 Edges

Notice how the number of faces and vertices are for cube an octahedron, and dodecahedron and icosahedron, while the number of edges . These pairs of Platonic solids are called dual solids.

We can even turn one into each other, by “replacing” every face with a vertex, and vice versa:

The tetrahedron is dual with itself. If we were to do the same kind of intersection, we would simply get two tetrahedra.

More on Polyhedra

Archimedean solids are slightly less regular. They are like Platonic solids, but they can consist of more than one type of regular polygon. They are named after another ancient Greek mathematician: Archimedes of Syracuse. In total there are 13 Archimedean solids, plus two mirror images.

Truncated Tetrahedron

Cuboctahedron

Truncated Cube

Truncated Octahedron

Rhombicuboctahedron

Truncated Cuboctahedron

Sub Cube

Icosidodecahedron

Truncated Dodecahedron

Truncated Icosahedron

Rhombicosidodecahedron

Truncated Icosidodecahedron

Snub Dodecahedron

Plato was wrong in believing that matter consists of Platonic solids. But regular polyhedra have many special properties that make them appear everywhere in nature – and we can copy these properties in science and engineering.

Viruses and Bacteria

Radiolaria skeleton

Icosahedral virus

Many viruses, bacteria and other tiny organisms are shaped like icosahedra. Viruses, for example, must enclose their genetic material inside a shell of many identical protein units. The icosahedron is the most efficient way to do this – because it consists of a few regular elements but is almost shaped like a sphere.

Molecules

Buckyball molecule

Montreal Biosphere

Many molecules are shaped like regular polyhedra. The most famous example is C60 which consists of 60 carbon atoms arranged in the shape of a truncated icosahedron. It was discovered in 1985 when scientists researched interstellar dust. They named it “Buckyball” (or Buckminsterfullerene) after the architect Buckminster Fuller, famous for constructing similar-looking geodesic domes.

Crystals

Fluorite octahedron

Pyrite cube

Most crystals have their atoms arranged in a regular grids consisting of tetrahedra, cubes, octahedra and many other polyhedra. When they crack or shatter, you can see these patterns on a much larger scale.

Construction

Octagonal Space Frames

Louvre museum in Paris

Tetrahedra and Octahedra are incredibly rigid and stable, which makes them very useful in construction. Space frames are polygonal structures that can support large roos and heavy bridges.

Games

Football

Polygonal role-playing dice

Because of their symmetry, Platonic solids are often used to create dice. Every side has the probability of landing facing up, so the dice are always fair.

The truncated icosahedron is probably the most famous polyhedron in the world: it is the shape of the football.