Solid Shapes and Symmetry
Introduction to Solid Shapes and Symmetry
Solid shapes are all around us. They surround us and are an integral part of our everyday lives. However, it is due to their ubiquitous nature is the reason we stop paying them much attention. This can lead to some problems visualizing them and representing them on a 2-D piece of paper when asked to do so by the math teacher. Well, worry not, we’ve got you covered. As you can see below, there is a fairly straightforward way to represent a 3-D object on a 2-D space using Nets. We will learn more about nets later.
An Introduction to Solids: The Types of Polygons
We have already seen that polygons are closed shapes which are named after the number of sides they have. So, the three-sided polygon is a triangle, a five-sided figure is a pentagon, an eight-sided closed figure is an octagon and so on.
Another interesting topic of discussion to focus on are curved surfaces. Imagine a battery, how many sides and faces does it have? What constitutes a side or a face or a vertex? Well, these are the questions that will get answered in a visual and intuitive manner over the course of the Cuemath program!
The Big Idea: Solid Shapes and Symmetry
A simple idea: how a solid shape is different from a polygon
A polygon is a closed shape. The lines by which a polygon is bound are known as its sides, and the points where two sides meet are called vertices. When we talk of solid shapes, we have these elements to understand:
- Edges – These are the equivalent of the sides of a polygon and are straight lines which make up the boundaries of a solid shape.
- Faces – The polygons formed by a set of edges of a solid shape is called a face. Imagine the Pyramid we spoke about earlier. The 4 triangles are the faces of the Pyramid, and if you lift the Pyramid off the ground, you also have a 5th face – the square base. We will refer to a face as F.
- Vertices – When the edges meet at a point, that point is referred to as a vertex. We will refer to the vertices as V.
- Polyhedron – Just like we used the word polygon to describe any closed shape in two dimensions, the corresponding word for a solid shape which is made by a combination of several plane faces is called a polyhedron. We will use the word polyhedron in the rest of this chapter to refer to solid shapes.
The reason we are starting off with this simple idea of comparing a polygon to a polyhedron is that every polyhedron can be created by rotation of a two-dimensional structure.
How is it important?
We saw the meaning of faces, edges, and vertices of a polyhedron earlier. Euler was the first mathematician to actually define a relationship between these elements of a regular polyhedron. This was the relationship he deduced:
Let us try this formula with a cuboid made from 6 rectangles (imagine a matchbox).
No. of vertices in the matchbox = 4 + 4 = 8 = V
No. of faces in the matchbox = top + bottom + 2 small faces + 2 larger faces = 1+1+2+2 = 6 = F
No. of edges in the matchbox = 4 each for top and bottom faces + 4 verticals = 12 = E
So LHS (left hand side) of Euler’s formula = V + F – E = 8 + 6 – 12 = 2
And RHS (right hand side) of Euler’s formula = 2
So, this formula does hold true for a cuboid.
Symmetry of a polyhedron
Most of the polyhedrons that are commonly studied enjoy several levels of symmetry. This means that their reflection is exactly similar to the original shape, and if the polyhedron is rotated on any axis, the views seen are all similar to the original shape. Simply put, a regular polyhedron can be said to be symmetrical if its look does not change on reflection, rotation or scaling (increasing its dimensions proportionately).
Lines of Symmetry
Let’s start off by imaging a two-dimensional polygon first, a rectangle. Which are the imaginary lines passing through the rectangle which will divide it into two exactly symmetrical parts?
So, you can say that a rectangle has 2 lines of symmetry. But if you consider a square, then there are 4 lines of symmetry, because the 2 diagonals also act as axes of symmetry.
Nets, Perspectives, and Cross Sections
There’s a common trend of revisiting 2-D shapes, that because they are easier to visualize. That is why we often use certain definitions to help us visualize solid shapes better.
Cross Section: When a plane polygon intersects a solid polyhedron, we get a cross section. Let us go back to the Pyramid again and imagine a huge sheet of paper being inserted vertically through one of the triangular faces. That will give the cross section.
Net: If a polyhedron is cut and unfolded into a flat representation, that flat representation is called a net. The graph perspective allows us to use the same nomenclature that we do for graphs. A net of a polyhedron would also help in determining whether a polyhedron would have rotational symmetry or not.