               Answers to the Questions in file 'Activities'

  These assume you are using only the standard distributed data files.

1.The smallest number of polygons is   4 in the tetrahedron     
  The largest number of polygons is  180 in the great icosahedron   

  If you want all the polygons to be the same the answer is different, then
  the smallest number of polygons is still 4 triangles in a tetrahedron, but
  the largest number of 3, 4 and 5 sided polygons is given in the table below. 

  No. sides of  no. of   polyhedron
    polygon      faces     name
       3         120     hexakis icosahedron
       4          60     trapezoidal hexecontahedron
       5          60     pentagonal hexecontahedron

2. All of the Platonic solids are made from identical regular polygons, to 
   see which they are use the 'Select' window. They are made from triangles,
   squares and pentagons.
 
   Number of polyhedra made from identical triangles (deltahedra)= 9

    They are: tetrahedron, octahedron, icosahedron
              1st stellation of the octahedron (compound of 2 tetrahedra)
              triangular dipyramid (J12)
              pentagonal dipyramid (J13)
              gyroelongated square dipyramid (J17)
              triaugmented triangular prism (J51)
              snub disphenoid (J84)

   Number of polyhedra made only from squares = 1 cube

   Number of polyhedra made only from pentagons = 1 dodecahedron

3. No answer to this, you just try making a box

4. You can make boxes from: dodecahedron, cuboctahedron, and others 

6. Other space filling polyhedra are:  rhombic dodecahedron, 
    truncated octahedron, triangular prism,  hexagonal prism.

5. All the tetrahedra are the same. Some faces are co-planar. You can find
   which ones by looking at the 3rd data file of the stella octangula.  

7. The 3 data files of the stellated octahedron are coloured to show the
   polyhedron is made from:
    a)small tetrahedra each fixed to the face of an octahedron
    b)it is a compound of 2 tetrahedra
    c)a set of intersecting planes

8. The smallest number of colours needed for a cube is  3

9. Here are the numbers of faces, edges and vertices for several polyhedra:
                              
                          Faces    Edges     Vertices
   dodecahedron            12       30          20
   tetrahedron              4        6           4
   truncated icosahedron   32       90          60
   cuboctahedron           14       24          12

  Faces + vertices - 2 = Edges 
  
  This is Euler's Theorem. All the polyhedra in the original distribution
  obey it. There are some in the extra datasets which do not.  

10. The glossary explains the differences between the named sections of 
    the data

11. the small file 'Dat0.MyTest' shows all you need to make a new cube.

12. Look at the file Worksheets.Answer1 for the names of the polyhedra

13. Polyhedra obtained by cutting off corners from a dodecahedron are:
     truncated dodecahedron
     icosidodecahedron
     truncated icosahedron
     icosahedron

14. Look at file 'Answers4 ' to see all the positions of the axes of 
    symmetry of a cube.

    Make a similar Draw file for an octahedron, and then compare the axes 
    of symmetry for the cube and the octahedron.   

15. the Draw file 'Anwsers2a3' shows the names and relationships

16. the Draw file 'Anwsers2a3' shows the names and relationships

17. Any pattern can be made.

18. a) The Icosahedron has and area of 5.148 for a volume of 1 unit
    b) The Snub dodecahedron has 4.925 for unit volume.
    c) The sphere; a free soap bubble has a fixed volume and takes up the
       shape with minimal surface area proving that the sphere is the
       optimal shape.
    d) The sphere has a surface area: 4 x pi x r^2 and a volume:
       4/3 x pi x r^3 so if the volume = 1, r=0.620351 and area=4.83596

19. The reason is that the prism most like a sphere is the one with 5 sides.
    With less than 5 sides they get long and thin; with more than 5 sides
    they get flat like a disc.  
    Antiprisms behave in the same way as prisms. The smallest area antiprism 
    is the square one. 