Origami Heaven

A paperfolding paradise

The website of writer and paperfolding designer David Mitchell

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Rhombic and Semi-Rhombic Polyhedra
 
A note on the meaning of the terms Rhombic Polyhedra and Semi-Rhombic Polyhedra can be found at the foot of this page.
 
Rhombic Dodecahedra
 
  Name: Kite Pattern Rhombic Dodecahedron

Modules / Paper shape / Folding geometry: 24 modules from silver rectangles.

Designer / Date: David Mitchell, 1990.

Diagrams: On-line diagrams are available on the Modular Designs page of this site.

 
  Name: David Mitchell's Rhombic Dodecahedron (so named to distinguish it from Nick Robinson's Rhombic Dodecahedron).

Modules / Paper shape / Folding geometry: 12 rhombic triangle modules folded from silver rectangles.

Designer / Date: David Mitchell, 2004.

Diagrams: In Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169

 
Rhombic and Semi-Rhombic Tetrahedra
 
  Name: The Mirror-Image Rhombic Tetrahedron.

Modules / Paper shape / Folding geometry: 2 mirror-image modules folded from silver rectangles.

Designer / Date: David Mitchell, 1996.

Diagrams: In Mathematical Origami - Tarquin Publications - ISBN 1-899618-18-X.

 
  Name: The Variable Rhombic Tetrahedron. The positions of the flaps and pockets can be varied to create five tetrahedra, all of identical appearance, but each made from a different combination of modules.

Modules / Paper shape / Folding geometry: 2 mirror-image modules folded from silver rectangles.

Designer / Date: David Mitchell, 2017.

Diagrams: As the Rhombic Petrahedron in Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169.

 
  Name: The Semi- Rhombic Tetrahedron

Modules / Paper shape / Folding geometry: 2 modules folded from silver rectangles.

Designer / Date: David Mitchell, 2017.

Diagrams: In Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169

 
The Rhombic Pyramid and the Semi-Rhombic Pyramid
 
  Name: The Rhombic Pyramid

Modules / Paper shape / Folding geometry: Made by combining a rhombic four-pockets module with a rhombic hat / Both folded from silver rectangles.

Designer / Date: David Mitchell, 2000.

Diagrams: In Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169

 
  Name: The Semi-Rhombic Pyramid

Modules / Paper shape / Folding geometry: 4 modules from silver rectangles.

Designer / Date: David Mitchell, 2017.

Diagrams: In Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169

 
The First Stellated Rhombic Dodecahedron
 
  Name: 24-part Stellated Rhombic Dodecahedron

Modules / Paper shape / Folding geometry: 24 modules in two sets of twelve from silver rectangles. Made by adding rhombic hat modules to Nick Robinson's Rhombic Dodecahedron.

Designer / Date: David Mitchell, 1997.

Diagrams: In Mathematical Origami - Tarquin Publications - ISBN 1-899618-18-X.

 
  Name: 6-part Stellated Rhombic Dodecahedron

Modules / Paper shape / Folding geometry: 6 modules from silver rectangles .

Designer / Date: David Mitchell, 1989.

Diagrams: On-line diagrams are available on the Modular Designs page of this site. Also in Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169

 
Rolling Ring of 8 Rhombic Tetrahedra
 
  Name: Rolling Ring of 8 Rhombic Tetrahedra

Modules / Paper shape / Folding geometry: 4 modules from silver rectangles.

Designer / Date: David Mitchell, 1995.

Diagrams: In Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169

 
A note on the meaning of the terms Rhombic and Semi-Rhombic Polyhedra
 
When I was writing the first edition of my book, Mathematical Origami, I found that I wanted to include several polyhedra for which there did not appear to be an established mathematical name. The characteristics of these polyhedra were that their faces were either 109.28/70.32 rhombuses or 70.32/54.84/54.84 (1:sqrt3/2:sqrt3/2) triangles, which can be obtained by folding the 109.28/70.32 rhombus in half lengthways, or a mixture of the two.

For want of better, I stole the word 'rhombic' from the name of the rhombic dodecahedron, whose faces are, of course, 109.28/70.32 rhombuses, and applied it to the other polyhedra as well, thus creating the terms rhombic tetrahedron, rhombic octahedron etc and the general term rhombic polyhedra to describe them as a group. I justified this usage to myself on the grounds that the triangular faces occurred in pairs, which if flattened would come back to a 109.28/70.32 rhombus. Somewhat to my surprise, my publisher raised no objection to this.

I was expecting that I would receive a barrage of criticism for this usage and that someone would soon point out to me a better terminology to use for these forms, but, so far, neither of these things have happened. I still, however, expect a better terminology to emerge at some stage.

Pro tem then I shall continue to use the term rhombic polyhedra, although in a slightly developed sense, to mean the set of those polyhedra which can be combined with other identical polyhedra to build a rhombic dodecahedron and which have at least some faces that are either 109.28/70.32 rhombuses or 70.32/54.84/54.84 (1:sqrt3/2:sqrt3/2) triangles. The members of this interesting set of polyhedra are illustrated below. All of them will fill space.

This definition deliberately excludes many polyhedra whose faces are 70.32/54.84/54.84 (1:sqrt3/2:sqrt3/2) triangles, and particularly those that can be made by putting together three-sided pyramids, or sunken pyramids, whose faces are such triangles and similar designs whose faces are or include 109.28/35.36/35.36 (sqrt2:sqrt3/2:sqrt3/2) triangles, which can be obtained from the 109.28/70.32 rhombus by folding it in half in the other direction. If these forms were to be included within the definition of rhombic polyhedra then we would be in a situation where there were, for instance, three separate forms that could quite correctly be called rhombic hexahedra and three more that could quite correctly be called rhombic dodecahedra. This is clearly nonsensical and so I have adopted the narrower definition given above. I have not invented any general name for the excluded forms.

It is worth pointing out that although the 109.28/70.32 rhombus has diagonals in the proportion 1:sqrt2 I have avoided using the term silver rhombus since that would suggest that this rhombus has properties similar to the silver rectangle and silver triangle, which it does not.

The classic book Mathematical Models, by H M Cundy and A P Rollett contains a net for the rhombic pyramid, although the authors do not give it a name. They do, however, use the term rhombic polyhedra in a different sense from mine as a general term for polyhedra whose faces are rhombuses, ie the cube, the rhombic dodecahedron and the triacontahedron. I apologise for introducing confusion by using this term in another way.

The third edition of this book, published in 1960, also contains the information that a Mr Dorman Luke, otherwise unknown to me, has found that all three stellations of the rhombic dodecahedron can be arrived at by adding what I call rhombic pyramids to the rhombic dodecahedron in successively larger numbers. These forms fall outside my definition of rhombic polyhedra but this is not a difficulty as they already have their own established names. I have, incidentally, found that by continuing the process of adding rhombic pyramids to the third stellation it is eventually possible to arrive at a much larger rhombic dodecahedron than the one used to initiate the process.