Origami Heaven

A paperfolding paradise

The website of writer and paperfolding designer David Mitchell

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Modular Polyhedra - Flat-Faced Designs
 
There is a separate page for Flat faced Cubes and Silverhedra
 
Deltahedra
 
I use the term deltahedra to refer to solids whose faces are all equilateral triangles.
 
Regular Tetrahedra
 
  Name: Mirror-Image Tetrahedron

Modules / Paper shape / Folding geometry: 2 mirror-image modules from double bronze rectangles using 60/30 degree folding geometry.

Designer / Date: David Mitchell, 1996.

Reference: 131

Diagrams: Mathematical Origami - Tarquin Publications - ISBN 1-899618-18-X // In Portuguese - Origami matematicos - Republicao - ISBN 972-570-257-3
 
  Name: Robust Mirror-Image Tetrahedron

Modules / Paper shape / Folding geometry: 2 mirror-image equilateral modules from bronze rectangles using 60/30 degree folding geometry.

Designer / Date: David Mitchell, 1998

Reference: 162

Diagrams: Not yet available.
 
  Name: Variable Tetrahedra. 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 modules from bronze rectangles using 60/30 degree folding geometry.

Designer / Date: David Mitchell, 2017

Reference: 575

Diagrams: As Regular Tetrahedra in Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169
 
  Name: Striped Mirror-Image Tetrahedron

Modules / Paper shape / Folding geometry: 2 mirror-image Abe or Terada modules or the equivalent geometry embedded in squares. 60/30 degree folding geometry.

Designer / Date: David Mitchell, 2003.

Reference: 534

Diagrams: Not yet available.
 
  Name: Quartetra - four part modular assembly puzzle.

Paper type / shape: Hexagon / Any kind of paper.

Designer / Date: David Mitchell, 1988.

Reference: 030

Diagrams: Paperfolding Puzzles - Water Trade - ISBN 0-9534774-0-X // Paperfolding Puzzles (2nd Edition) - David Mitchell - Water Trade - ISBN 978-0-9534774-5-6
 
  Name: Reptile Tetrahedron

Modules / Paper shape / Folding geometry: 12 Reptile modules from squares using 60/30 degree folding geometry.

Designer / Date: David Mitchell, 1997.

Reference: 146

Diagrams: BOM 189 April 1998 // On-line diagrams are available on the Modular Designs page of this site.
 
  Name: 12-part Tetrahedron

Modules / Paper shape / Folding geometry: 12 modules from silver rectangles using 60/30 degree folding geometry.

Designer / Date: David Mitchell,1991.

Reference: 081

Diagrams: In BOS Convention Pack Spring 1992.
 
Equilateral Hexahedra
 
  Name: Robust Diamond Hexahedron

Modules / Paper shape / Folding geometry: 6 equilateral modules in 2 sets of 3 mirror-image versions folded from bronze rectangles using 60/30 degree folding geometry.

Designer / Date: David Mitchell, 2017.

Reference: 162

Diagrams: As Diamond Hexahedron in Mathematical Origami - David Mitchell - Tarquin publications - ISBN 9781911093169
 
Regular Octahedra and other Deltahedral Bi-Pyramids
 
  Name: Robust Triangular Equilateral Bi-Pyramid

Modules / Paper shape / Folding geometry: 3 equilateral modules folded from bronze rectangles using 60/30 degree folding geometry.

Designer / Date: David Mitchell, 2017.

Reference: 162

Diagrams: As Triangular Equilateral Bi-Pyramid in Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169
 
  Name: Regular Octahedron (2-part). Not robust.

Modules / Paper shape / Folding geometry: 2 modules folded from double bronze rectangles using 60/30 degree folding geometry.

Designer / Date: David Mitchell, 1997.

Reference: 144

Diagrams: Mathematical Origami - Tarquin Publications - ISBN 1-899618-18-X // In Portuguese - Origami matematicos - Republicao - ISBN 972-570-257-3
 
  Name: Robust Octahedron

Modules / Paper shape / Folding geometry: 4 equilateral modules folded from bronze rectangles using 60/30 degree folding geometry.

Designer / Date: David Mitchell, 1998.

Reference: 162

Diagrams: As Regular Octahedron in Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169
 
  Name: Robust Pentagonal Equilateral Bi-Pyramid

Modules / Paper shape / Folding geometry: 5 equilateral modules folded from bronze rectangles using 60/30 degree folding geometry.

Designer / Date: David Mitchell, 2017.

Reference: 162

Diagrams: As Pentagonal Equilateral Bi-Pyramid in Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169
 
Regular Icosahedra
 
  Name: Robust Icosahedron

Modules / Paper shape / Folding geometry: 10 equilateral modules in 2 sets of 5 mirror-image versions folded from bronze rectangles using 60 degree folding geometry.

Designer / Date: David Mitchell, 1998.

Reference: 162

Diagrams: As Regular Icosahedron in Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169
 
  Name: Striped Mirror-Image Icosahedron - can be made from Abe or Terada modules or from the equivalent designs embedded in squares.

Modules / Paper shape / Folding geometry: 10 modules in 2 sets of 5 mirror-image versions folded from bronze rectangles or squares using 60/30 degree folding geometry.

Designer / Date: Modules - Various. Assembly - David Mitchell, 2003.

Reference:

Diagrams: Not yet available.
 
The Equilateral Dodecahedron
 
  Name: The Robust Equilateral Dodecahedron

Modules / Paper shape / Folding geometry: 6 equilateral modules in 2 sets of 3 mirror-image versions folded from bronze rectangles using 60 degree folding geometry.

Designer / Date: David Mitchell, 2017.

Reference: 162

Diagrams: As the Equilateral Dodecahedron in Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169
 
The Equilateral Hexakaidecahedron
 
  Name: The Robust Equilateral Hexacaidecahedron

Modules / Paper shape / Folding geometry: 8 equilateral modules in 2 sets of 4 mirror-image versions folded from bronze rectangles using 60 degree folding geometry.

Designer / Date: David Mitchell, 2017.

Reference: 162

Diagrams: As Equilateral Hexacaidecahedron in Mathematical Origami - David Mitchell - Tarquin publications - ISBN 9781911093169
 
The Equilateral Heccaidecahedron
 
Details not yet available.
 
Regular Dodecahedra
 
  Name: Regular Dodecahedron - full-face module version - not robust.

Modules / Paper shape / Folding geometry: 12 modules from squares using mock platinum folding geometry.

Designer / Date: David Mitchell, 1996.

Reference: 126

Diagrams: Mathematical Origami - Tarquin Publications - ISBN 1-899618-18-X // In Portuguese - Origami matematicos - Republicao - ISBN 972-570-257-3
 
  Name: Robust Dodecahedron. Versions only differ in paper shape used.

Version 1: Faces stand slightly proud.

Modules / Paper shape / Folding geometry: 30 modules from silver rectangles.

Designer / Date: David Mitchell, 1992. Possibly invented at an earlier date by David Brill.

Reference: 576

Diagrams: Not published.

Version 2: Flat faces.

Modules / Paper shape / Folding geometry: 30 modules from mock platinum rectangles.

Designer / Date: David Mitchell, 2016.

Reference: 576

Diagrams: As Regular Dodecahedron in Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169
 
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.
 
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.

Reference: 129

Diagrams: Mathematical Origami - Tarquin Publications - ISBN 1-899618-18-X // In Portuguese - Origami matematicos - Republicao - ISBN 972-570-257-3
 
  Name: The Robust Mirror-Image Rhombic Tetrahedron.

Modules / Paper shape / Folding geometry: 2 mirror-image truncated Robinson modules folded from silver rectangles.Three versions possible as the set of modules can be configured and assembled in three different ways.

Designer / Date: David Mitchell, 1998.

Reference:129

Diagrams: Diagrams not yet available.
 
  Name: The Variable Rhombic Tetrahedron

Modules / Paper shape / Folding geometry: 2 modules from double silver rectangle strip. Three versions possible as the set of modules can be configured and assembled in three different ways.

Designer / Date: David Mitchell, 2017

Reference: 545

Diagrams: Diagrams not yet available.
 
  Name: Pocket Rhombic Tetrahedron

Modules / Paper shape / Folding geometry: 2 modules from 1x3/4 silver rectangles using silver rectangle geometry.

Designer / Date: David Mitchell, 1994.

Reference: 113

Diagrams: Diagrams not yet available.
 
  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.

Reference:

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.

Reference: 547

Diagrams: In Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169
 
Rhombic Hexahedra
 
  Name: The Rhombic Hexahedron - replihedron which is one third of a rhombic dodecahedron.

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

Designer / Date: David Mitchell, 2004.

Reference: 268

Diagrams: Diagrams not yet available.
 
 
 
Rhombic Octahedra
 
  Name: The Assymetric Rhombic Octahedron

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

Designer / Date: David Mitchell, 2004.

Reference: 266

Diagrams: Diagrams not yet available.
 
 
 
The Rhombic Pyramid, the Semi-Rhombic Pyramid and the Square Base Rhombic Pyramid
 
  Name: The 2-part 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.

Reference: 147

Diagrams: In Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169
 
  Name: The Square Base Rhombic Pyramid

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

Designer / Date: David Mitchell, 2017.

Reference: 543

Diagrams: In Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169
 
Rhombic Dodecahedra
 
  Name: Kite Pattern Rhombic Dodecahedron

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

Designer / Date: David Mitchell, 1990.

Reference: 067

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, from which it is derived).

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

Designer / Date: David Mitchell, 2004.

Reference: 267

Diagrams: In Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169
 
The First Stellated Rhombic Dodecahedron
 
  Name: 6-part Stellated Rhombic Dodecahedron

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

Designer / Date: David Mitchell, 1989.

Reference: 042

Diagrams: Reviewed in BOM 139 December 1989 // Wonderful World of Modulars - Tomoko Fuse - 4-405-07553-0 - published in Japanese // BOS Convention Pack Autumn 1989 - diagrams drawn by Francis Ow // Photo on Gallery section of Origami: The Complete Guide to the Art of Paperfolding - Rick Beech - Lorenz Books (Anness Publishing) - ISBN 075480782 // Photo on cover of Russian Origami Society magazine issue 3 1999 // Orison 6 2009 // Also in Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169 // On-line diagrams are available on the Modular Designs page of this site.

 
  Name: 24-part Stellated Rhombic Dodecahedron - made by adding 'hats' to Nick Robinson's Rhombic Dodecahedron.

Modules / Paper shape / Folding geometry: 24 modules in two sets of twelve from silver rectangles using silver rectangle folding geometry.

Designer / Date: David Mitchell, 1997.

Reference: 150

Diagrams: Mathematical Origami - Tarquin Publications - ISBN 1-899618-18-X // In Portuguese - Origami matematicos - Republicao - ISBN 972-570-257-3 // Quadrato Magico 101 Autumn 2011
 
Other Flat-Faced Polyhedra
 
  Name: The Robust Diamond Hexahedron

Modules / Paper shape / Folding geometry: 6 equilateral modules in 2 sets of 3 mirror-image versions folded from bronze rectangles using 60 degree folding geometry.

Designer / Date: David Mitchell, 2017.

Reference:

Diagrams: As the Diamond Hexahedron in Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169
 
  Name: The Square Base Equilateral Pyramid

Modules / Paper shape / Folding geometry: 4 modules from silverdouble bronze rectangles using 60 degree folding geometry.

Designer / Date: David Mitchell, 2017.

Reference:

Diagrams: Mathematical Origami (2nd Edition) - David Mitchell - Tarquin publications - ISBN 9781911093169
 
  Name: Compound of Cube and Octahedron - made by adding hats to the Twisted Square Cube. 12 squares.

Modules / Paper shape / Folding geometry: 6 modules from squares using standard folding geometry and 6 from squares using 60/30 degree folding geometry..

Designer / Date: David Mitchell, 2016.

Reference: 541

Diagrams: Not yet available.
 
  Name: Great Dodecahedron

Modules / Paper shape / Folding geometry: 12 modules from pentagons.

Designer / Date: David Mitchell, 1990.

Reference: 061

Diagrams: Diagrams not yet available.
 
  Name: Small Stellated Dodecahedron (30-part)

Modules / Paper shape / Folding geometry: 30 modules from squares using mock platinum folding geometry.

Designer / Date: David Mitchell, 1990.

Reference: 536

Diagrams: Diagrams not yet available.
 
  Name: Small Stellated Dodecahedron (60-part) - Design from squares that allows each of the pentagrams visible on the surface of the form to be made from paper of a different colour.

Modules / Paper shape / Folding geometry: 60 modules from squares using mock platinum folding geometry.

Designer / Date: David Mitchell, 2016.

Reference: 537

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

Modules / Paper shape / Folding geometry: 30 modules from 2x1 rectangles using standard rectangle folding geometry.

Designer / Date: David Mitchell, 2001.

Reference: 238

Diagrams: Diagrams not yet available.
 
  Name: Kite Pattern Triacontahedron

Modules / Paper shape / Folding geometry: 60 modules from custom rectangles using golden rectangle folding geometry.

Designer / Date: David Mitchell, 1990.

Reference: 068

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