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Answer:
Let’s begin by introducing the protagonist of this story — Euler’s formula:
V – E + F = 2.
Simple though it may look, this little formula encapsulates a fundamental property of those three-dimensional solids we call polyhedra, which have fascinated mathematicians for over 4000 years. Actually I can go further and say that Euler’s formula tells us something very deep about shape and space. The formula bears the name of the famous Swiss mathematician Leonhard Euler (1707 – 1783), who would have celebrated his 300th birthday this year.
What is a polyhedron?
Before we examine what Euler’s formula tells us, let’s look at polyhedra in a bit more detail. A polyhedron is a solid object whose surface is made up of a number of flat faces which themselves are bordered by straight lines. Each face is in fact a polygon, a closed shape in the flat 2-dimensional plane made up of points joined by straight lines.
Three polygons
Figure 1: The familiar triangle and square are both polygons, but polygons can also have more irregular shapes like the one shown on the right.
Polygons are not allowed to have holes in them, as the figure below illustrates: the left-hand shape here is a polygon, while the right-hand shape is not.
A polygon and a shape that isn’t a polygon.
Figure 2: The shape on the left is a polygon, but the one on the right is not, because it has a ‘hole’.
A polygon is called regular if all of its sides are the same length, and all the angles between them are the same; the triangle and square in figure 1 and the pentagon in figure 2 are regular.
A polyhedron is what you get when you move one dimension up. It is a closed, solid object whose surface is made up of a number of polygonal faces. We call the sides of these faces edges — two faces meet along each one of these edges. We call the corners of the faces vertices, so that any vertex lies on at least three different faces. To illustrate this, here are two examples of well-known polyhedra.
The cube and the icosahedron.
Figure 3: The familiar cube on the left and the icosahedron on the right. A polyhedron consists of polygonal faces, their sides are known as edges, and the corners as vertices.
A polyhedron consists of just one piece. It cannot, for example, be made up of two (or more) basically separate parts joined by only an edge or a vertex. This means that neither of the following objects is a true polyhedron.
Figure 4: These objects are not polyhedra because they are made up of two separate parts meeting only in an edge (on the left) or a vertex (on the right).
Answer:
Let’s begin by introducing the protagonist of this story — Euler’s formula:
V – E + F = 2.
Simple though it may look, this little formula encapsulates a fundamental property of those three-dimensional solids we call polyhedra, which have fascinated mathematicians for over 4000 years. Actually I can go further and say that Euler’s formula tells us something very deep about shape and space. The formula bears the name of the famous Swiss mathematician Leonhard Euler (1707 – 1783), who would have celebrated his 300th birthday this year.
What is a polyhedron?
Before we examine what Euler’s formula tells us, let’s look at polyhedra in a bit more detail. A polyhedron is a solid object whose surface is made up of a number of flat faces which themselves are bordered by straight lines. Each face is in fact a polygon, a closed shape in the flat 2-dimensional plane made up of points joined by straight lines.
Three polygons
Figure 1: The familiar triangle and square are both polygons, but polygons can also have more irregular shapes like the one shown on the right.
Polygons are not allowed to have holes in them, as the figure below illustrates: the left-hand shape here is a polygon, while the right-hand shape is not.
A polygon and a shape that isn’t a polygon.
Figure 2: The shape on the left is a polygon, but the one on the right is not, because it has a ‘hole’.
A polygon is called regular if all of its sides are the same length, and all the angles between them are the same; the triangle and square in figure 1 and the pentagon in figure 2 are regular.
A polyhedron is what you get when you move one dimension up. It is a closed, solid object whose surface is made up of a number of polygonal faces. We call the sides of these faces edges — two faces meet along each one of these edges. We call the corners of the faces vertices, so that any vertex lies on at least three different faces. To illustrate this, here are two examples of well-known polyhedra.
The cube and the icosahedron.
Figure 3: The familiar cube on the left and the icosahedron on the right. A polyhedron consists of polygonal faces, their sides are known as edges, and the corners as vertices.
A polyhedron consists of just one piece. It cannot, for example, be made up of two (or more) basically separate parts joined by only an edge or a vertex. This means that neither of the following objects is a true polyhedron.
Figure 4: These objects are not polyhedra because they are made up of two separate parts meeting only in an edge (on the left) or a vertex (on the right).
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