This week looks at the properties of 3D shapes, plans and elevations and a few other interesting visual ideas worth exploring.
Solids and Nets
For a quick recap on 3D shapes and their nets, work through this interactive set of activities:
Try the Dice Net Challenge to further explore different nets of cubes and recap number properties.
Properties of 3D shapes
The properties of 3D shapes involve identifying faces, edges and vertices (corners). Try this Transum quiz, with three levels, to remind yourself about these attributes:
The 3D shapes contained in the cabinet are: cube, cuboid, cone, sphere, triangular prism, square based pyramid, tetrahedron, cylinder. The clues are as follows:
- The cuboid id not next to the shape with all square faces.
- The tetrahedron is adjacent to just one shape with a triangular face.
- The top front right hand corner holds the shape with only one vertex.
- The shape with six vertices is directly on top of the shape with five.
- The two shapes with the most faces are both at the front of the cabinet.
- The sphere is in the corner that you can’t see on the diagram.
- The cube is not next to the shape with eight edges.
In the activity, Cutting Corners, faces, edges and vertices are further explored by considering what happens when you change a solid by cutting off its corners, as shown:
Check that you can work out where all of these faces (F), vertices (V) and edges (E) are on the new shape. Now try work out the number of faces, vertices and edges, before and after, for a rectangular based pyramid:
Could you investigate this for other solids? What patterns and rules can you find?
Hint: You could look into groups of solids (prisms, pyramids, etc). For cubes and cuboids, try to explain what is happening each time one corner is cut. You could also investigate what would happen if you cut the corners for a second time…
Plans and elevations
Firstly, to introduce the ideas behind plans and elevations, work through the three levels of this activity to get a sense of what it means to consider different viewpoints of three dimensional objects.
This next section takes you through an example of plans and elevation from a diagram of a 3D shape.
Now try to draw the plan, and the three elevations for each of these two 3D shapes:
Note that the convention is to provide three (instead of four) 2D diagrams – the plan, the front elevation and the right side elevation.
An exam question can also be posed the other way around, where the plan and elevations are given and you have to work out what the 3D shape is. There is another example and a question like that here, from BBC Bitesize.
To practise more on plans and elevations, use the powerpoint:
And print off these worksheet questions:
The Penrose Triangle is an optical illusion. Here you can explore it in one form:
And the idea is extended here. Can you translate the Spanish?
One artist who produced many works using these kinds of optical illusion was M C Escher, a Dutch graphic artist who lived and worked during the twentieth century. You can see examples of his work here. In particular, have a look through the Mathematical and Impossible Constructions collections.
Can you design your own optical illusion? Or create your own design inspired by an Escher work? Send it in to the Student Showcase.
In this Oxford Mathematics Public Lecture, The Mathematics of Visual Illusions – Ian Stewart, some of these ideas are explored. It does go on to include some more complicated mathematical ideas but the first 15 minutes includes some other interesting examples of visual illusions.
ANSWERS – click below