3D geometry: truncated polyhedra

it can be interesting to explore what happens to the numbers of faces, edges, and vertices of solids when you (symmetrically) cut off the corners - less than half way along each edge

the new F, E, V numbers can all be related just to the old number of edges
reasons for these relationships can be considered

powerpoint: truncated polyhedra

3D geometry: faces, edges and vertices

I'm not sure that enough work is done on 3D geometry
a good end of term task - if not used elsewhere

looking at a mixture of (2D representations of) some solids
powerpoint

and some complex solids
powerpoint: potatoes

it is relatively easy to prove that Euler's rule is true for any prism
and also any anti-prism
powerpoint: prisms (and anti-prisms)

it is also relatively simple to prove that Euler's relationship is true for any pyramid
and dipyramid
powerpoint: pyramids and dipyramids

although Euler wasn't the first to develop the rule that has his name, he did attempt an interesting proof, by induction: the relationship works for a resulting basic shape (the tetrahedron) following a collapse and at all stages of all options when progressively 'collapsing' a solid

this proof was later shown to be flawed but is still interesting
powerpoint: Euler's rule

can be used to establish a rule for the sum of the interior angles of a polygon

the idea behind Euler's attempted proof by induction