what are the lengths?
the lengths of 1 are the edges of the cube
the lengths of root 2 are on the faces, 2 on each
the lengths of root 3 connect the corners of e.g. the top face to the opposite corner of the lower face
28 lengths altogether (which is 8 choose 2)
only 3, perhaps surprisingly
56 triangles altogether = 8 choose 3
but you could, without loss of generality, have 1 as the first corner to obtain the (correct) probabilities
what is intriguing is that the numbers of different lengths
match the numbers of different triangles
there may be a good reason for this (but I can’t see it!)
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
if you sum all the angles of all the faces this is called the total angle sum (the TAS)
it was explored by Descartes (who obtained Euler’s rule for polyhedra (V + F = E + 2) quite a bit before Euler)
this exploration was suggested to me by Gordon Haigh when he was at Wolverhampton University
it can be a good task to work on following a consideration of (interior) angles in regular polygons
the relationship is quite easily established (proved) for any prism and any pyramid
the deltahedra can be reasonably easy to construct
all the faces are equilateral triangles
powerpoint:
platonic solids have the same regular polygons for all of the faces
it is possible to consider the options for how regular polygons can meet at a vertex to establish that there are only 5 platonic solids
it can be interesting to look at how edges join for the platonic solids
powerpoint: platonic solid nets
one interesting way to represent a solid is by means of a Schlegel diagram
powerpoint: Schlegel diagrams and hexahedrons
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
