                     Things to try with !PolyDraw

Here are a few suggestions of explorations of polyhedra which can be done
with !PolyDraw. All terms used here are explained in file 'Docs.Glossary'

1. Polyhedra are made from polygons, see if you can find the polyhedron
   made from the smallest number of polygons, and which from the largest.
   Does it make a difference if all the polygons are the same shape?

2. Some polyhedra are made from identical regular polygons, see if you can 
   find which ones they are. How many are made from just triangles? and from
   squares? and polygons with more sides? Is there a limit to the number of
   sides of such polygons which can make a convex solid? If so, explain why.

3. Try saving the Planar Net of an interesting polyhedron as a Draw file,
   then use the Draw application to add tabs on the sides. Print the file,
   then cut it out and glue the tabs together to make your own 3D model.
    (If you have the !PolyNet application use that instead.)

   Start with a cube, with only 6 sides it is easier to stick together than
   some of the larger ones. To make it more interesting you can decorate the
   sides of the cube using Draw, before you print the Planar Net. You might 
   put your name on one side, or one line of poetry on each side, or use a
   library of ClipArt.

4. Can you make any other polyhedra into useful boxes apart from the cube?

5. Try making some measurements on the Platonic solids and the Kepler-Poinsot
   solids. Look at the dodecahedron, what is the angle between the opposite
   faces? Look at the stella octangula, it looks as if it is made from little
   tetrahedra stuck on to an octahedra. Try measuring the angles between the
   faces of the tetrahedra. Are they all the same?  Are any of the faces in
   the same plane?  

6. If you made several models of a cube, and placed them together they 
   completely fill the space.  How many other polyhedra can you find which 
   will do that just by themselves? Then see if you can find 2 different
   polyhedra which fill space together, prove it by making models of them.  

7. You can change the colours of the polygons making up the polyhedron by
   editing its data file, the colour numbers are the last numbers on the 
   lines in the file which follow the word 'solidf' (see file 'Formdata' for
   further details of the data file format).  These colours are the ones
   you see using the colouring option 'datacolour', and the ones which are 
   listed in the window you get by clicking 'select' over a face. 

   Look at the 3 files for the stellated octahedron, the difference between
   them is the way their sides are coloured. 
   Are there any other polyhedra which can be coloured in several ways?  

8. Try looking at several polyhedra and for each one decide what is the 
   smallest number of colours you need to use if you want all the adjacent
   faces to be different colours?  

9. Look at the text window which describes a polyhedron, try adding 
   together the number of faces and number of vertices, and compare the
   result with the number of edges. Try several more polyhedron.
   Is there a relation between the number of faces, vertices and edges?  
   Can you find a polyhedron for which this rule does not apply?

10. Look at the icon names in list of types in the 'Selections Window.'
   Try to understand what each one means by selecting one icon and then look
   at those polyhedra to see how they differ from the others. They are all
   defined in 'Docs.Glossary'    

11. Try making a new polyhedron data file and adding it to the PolyList.

12. If you start from one polyhedron and imagine taking a small slice from
    all of its corners you get another polyhedron in this set of data files,
    taking a deeper slice gets you yet another, and you can keep doing this
    until you find you have reached another well known polyhedron.
    Look at the file 'Sheet 1' for an explanation using diagrams, and follow
    its instructions to draw all the named polyhedra.

13. Now think about a different pair of polyhedra; find out how many there
    are between a dodecahedron and an icosahedron, and make a Draw file
    showing how the shape changes as the slices are cut deeper. 
    Can you find any other pairs?

14. Plot one of the Platonic solids. Then rotate it until you can see an axis
    of symmetry, save the 3D plot as a Draw file, and look for another axis
    of symmetry, save that, keep doing this until you think you have found
    all of them for your chosen solid.  Then use !Draw to add the positions
    of all the axes of symmetry.

15. Look at the Draw file 'Sheet2' which show 'wire' plots for several 
    polyhedra. Find out their names. Which ones are related to others?
    How are they related?

16. repeat 14. but using Draw file 'Sheet3'

17. Try making yourself some pleasing patterns using any of the display forms
    of any polyhedron, first try all possible patterns from one polyhedron,
    then try mixing patterns from several. 

18. a) Look at the Platonic solids and find which one has the smallest ratio
       of surface area to volume. (Hint use the 'text' option from the main
       menu).
    b) Do the same for the Archimedian solids.
    c) Which solid (not necessarily in the PolyDraw data) has the smallest
       ratio; how would you prove it is the smallest. 
    d) Advanced students can calculate the ratio for the solid in c).

19. Look at the prisms starting with the 3-sided and going up to the 10-sided.
    (Don't forget the cube is the 4-sided prism). See how their surface area
    decreases at first, but beyond the 5-sided prism increases again.
    Why is this? Would you expect the same effect with the antiprisms?