One of such solids is a polyhedron in the shape of a tetrahedron! In our case, three inner faces of the polyhedron should be slightly lower than those of the regular tetrahedron, as you can see in the figure. A regular triangle remains based on the polyhedron. Just imagine how 20 totally identical polyhedra meet at a single point and make up a completely new polyhedron — icosahedron!
You will need 20 models to make an icosahedron. Upcoming Events. Featured on Meta. Now live: A fully responsive profile. The unofficial elections nomination post. Related 4. Hot Network Questions. Question feed. Mathematics Stack Exchange works best with JavaScript enabled. How many vertices corners and edges are there? See Euler's formula. Here is one net for an icosahedron. Print it out, stick it on thin card, score along the lines and fold them, form the shape, then stick it together with small amounts of glue.
You might like to think of a colour scheme for your finished shape. It's a lot easier if you colour it in before you stick it together, or even before you cut it out.
Try to imagine what the finished shape will look like when colouring it in. You could try to draw lines that run over edges. That's easy if the faces are together in the net, less easy if there are gaps! Starting from the initial position a , the solid generated by the intersection of red and yellow tetrahedra is shown in b after an appropriate rotation, in order to evidence the presence of a vertical 3-fold axis and three 2-fold axes perpendicular to the 3-fold axis, characterizing the 32 point group.
In c the same colour has been given to all the faces belonging to each of the two crystallographic forms: trigonal trapezohedron 6 brown faces and pinacoid 2 fuchsia faces. Left : view, along the [] direction, of the faces of the polyhedron corresponding to the intersection of two tetrahedra and relative stereographic projection; in conformity with the 32 point group, the forms are a trigonal trapezohedron and a pinacoid here and in all the following stereographic projections, circles indicate the positions of faces belonging to the North hemisphere, crosses indicate faces belonging to the South hemisphere Right : orthographic view of the only trigonal trapezohedron.
Three alternative colourings of the faces of the polyhedron resulting from the intersection of two tetrahedra. The first two colourings refer to: the faces deriving from red and yellow tetrahedra the trigonal trapezohedron and the pinacoid constituting the polyhedron In order to evidence the single faces, the sharing of edges by faces having equal colouring should be avoided; since the triangular faces share their edges with three pentagonal faces, at least four colours need to be used third image : one for the two triangular faces and three other colours for the two sets of three pentagonal faces related by the 3-fold axis.
Moreover each pentagonal face of a set of three faces is related by two 2-fold axes to two pentagonal faces, coloured by the other two colours, belonging to the second set of three faces. Therefore, in case of this 4-coloured polyhedon, the relative symmetry results to be 3'2' , where 3' and 2' indicates chromatic 3-fold and 2-fold axes, respectively. The geometric features of the pentagonal faces are visible clicking here : the side lengths are relative to an unit distance of the face from the centre of the polyhedron.
Compound polyhedron made of 3 tetrahedra and solid generated by their intersection The following animated GIF , consisting of four frames, shows sequentially the images along different directions, included 2-fold and 3-fold rotation axes, relative to the pairs of: chiral compound polyhedra made of red, yellow and green tetrahedra first row chiral solids generated by the intersection of the three tetrahedra second row The polyhedra in the second column have been properly orientated to evidence their mirror symmetry concerning both shape and colouring with respect to the polyhedra in the first column.
Animated GIF of chiral compounds of three tetrahedra and their intersections, seen by ortographic viewing and along 2-fold and 3-fold axes. Starting from the initial position a , the solid generated by the intersection of red, yellow and green tetrahedra is shown in b after an appropriate rotation, in order to evidence the presence of both a vertical 3-fold axis and three 2-fold axes, perpendicular to the 3-fold axis, characterizing the 32 point group.
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