dualityIf you take a polyhedron and switch faces for vertices you will get its dual. More specifically, to create the dual, you place a vertex at the centre of every face and join them up. The tetrahedron is its own dual. What are the duals of the other regular polyhedra? And other polyhedra? The zometool models on the right are what my year 10 students made with very little guidance when asked to find any connections between the cube and the octahedron. Following this, I asked questions such as: how are the side lengths, surface areas and volumes related? Is there a link between the number of edges, faces and vertices on each shape? |
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Below is a slideshow of the construction of a model that demonstrates this which I made using a pattern from the book Dual Models by Marcus Wenninger (click to play and note comments below).
In the book, which you can see in some of the photos, Marcus Wenninger does not describe exactly how long each part must be. In fact, he leaves that to the reader, suggesting that the duality can be best seen by constructing a series of these models with different sizes of cubes.
I wanted to make mine so that the vertices of the cube were exactly at the centre of the faces of the octahedron. This requires some mathematics and visualisation and is worth doing from scratch. However, if you want some hints, my workings on this are below.
I wanted to make mine so that the vertices of the cube were exactly at the centre of the faces of the octahedron. This requires some mathematics and visualisation and is worth doing from scratch. However, if you want some hints, my workings on this are below.
I thought it interesting that the radius of the octahedron was 3 for a cube with side length 2.