Make some close-packed rhombic dodecahedra

To make this model you will need:

• A printer
• Some method of cutting out paper or thin card
• Quick-drying paper glue
• PDF viewing software (e.g. Acrobat Reader)

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This model fills space, click here for an explanation of space filling polyhedra.

	This picture shows a stack of rhombic dodecahedra. They close-pack, i.e. they fit together with no gaps between them; here they are arranged to show how they fit in twos: 1 at the top lying in a set of 4 which sit in a set of 9.
	This picture shows the same rhombic dodecahedra but they are arranged to show how they fit in threes: 1 at the top lying in a set of 3 which sit in a set of 6.
	This picture shows the same rhombic dodecahedra but here they are arranged to show how they fit in fours: 1 at the top lying in a set of 4 which sit in a set of 9. How does this differ from the two-fold arrangement above?

To make these models, download the PDF (35K); then follow the directions below to make the models.

• First decide how many dodecahedra you want to make. The models above need:
  1+4+9 = 14 for the 2-fold model;
  1+3+6 = 10 for the 3-fold model;
  1+4+9 = 14 for the 4-fold model but you do not need to have them all built up at the same time so 14 are enough.
• Print the Draw file 'page1' [PDF page 1] so that you can see how to construct the models.
• Print the other 7 Draw files [PDF pages 2 - 8] onto stiff paper (we use 160 gm weight). The dodecahedra are printed 4-at-a-time on the Draw file 'dodeca' [PDF page 2] so you will need to print this 4 times on the same coloured paper. If possible print the 6 sheets containing the bases, 'base..' [PDF pages 3-8] onto different coloured paper to show off the models better.
• Do not scale the individual pages. They must all be the same scale to fit together correctly. We have made the bases a little bigger to allow for the thickness of the stiff paper.
• You may find it more interesting to make the models in stages; e.g. first make 4 dodecahedra and then make a base to see how they fit in; next make enough dodecahedra to finish this model etc.
• Cut out all the nets with scissors or a modelling knife around the outside solid lines.
• Score along the internal solid and dashed lines with a compass point and a ruler.
• First make a dodecahedron; the one on the left of the cut-out sheet has been numbered but the numbers should be inside the finished model and will not show. Notice that the rhombus at the top (12) of each 'cut-out' does not have any tabs; this should be the last one to glue.
  Bend up the card along all lines.
  Start at the bottom and glue the 3 rhombi (1, 2 & 3) together with the tab 'c' stuck to the top of face 1. In this way the outside will not show the tab nor the ink of the lettering or lines.
  Now continue up the chain of rhombi:
  • add faces 4 & 5 making a 3-fold vertex with tab d
  • add face 6 making a 4-fold vertex with tab g
  • add face 7 making a 3-fold vertex with tab h
  • add face 8 making a 4-fold vertex with tab f

    Your model should look like this just before gluing face 9 to face 7 with tab i. The inside is coloured yellow and the tabs white.

  • add face 9 making a 3-fold vertex with tab i
  • add face 10 making a 4-fold vertex with tab k
  • and a 3-fold vertex with tab b
  • add face 11 making a 4-fold vertex with tab m
  • and a 3-fold vertex with tab l
  • finally glue face 12 down onto tabs a, e & j making two more 3-fold and two more 4-fold vertices

  Here is your finished model. This figure shows a 3-fold and a 4-fold symmetry axis. Which is which?

• the 3-fold base is the smaller one having only 9 dimples so make this first, referring to the 'page1' [PDF page 1] to see what it should look like.
  

  There are 6 hollows with 3 faces each, 3 cross-over points with 6 faces each and a peak in the middle with 3 faces.

  After you have cut out and scored the model from 'base3I' [PDF page 5], identify these features and glue up the triangular base making sure that:
  • the dotted lines bend down (you're making the base upside down so as not to show the guide lines);
  • the solid lines (mostly on tabs) bend up;
  • you stick the tabs onto the side with the guide lines so they don't show on the top.
  Next glue on the surround for the base from 'base3X' [PDF page 6]; each of the 3 sides glues on to 6 tabs and finally glue the triangular base onto the sides.
• make the 4-fold base similarly from pieces 'base4I' [PDF page 7].

  There are 9 hollows with 4 faces each, 10 cross-over points with 4 faces and 4 peaks with 4 faces.

  Glue on the surround for the base from 'base4X' [PDF page 8]; each of the 4 sides glues on to 6 tabs and finally glue the square base onto the sides.
• make the 2-fold base with 9 hollows from 'base2I' [PDF page 3], joining the 3 rows together with the 4 diamond shaped pieces from 'base2X' [PDF page 4] from which you cut out the base and surround. Glue the cup structure to the base, add the triangular fillers and finally glue the edges.
  More explanations are on the pages themselves.