N-gonal Substituted Polyhedra

Heptagonal variant from seed of small rhombicosidodecahedron

Some years ago, I discovered a number of other families of polyhedra that could be generated by taking a pair of opposite m-gonal faces of a regular polyhedron and replacing them with a different polygon. All of the lateral faces, which had the original m-gonal prismatic symmetry, were duplicated as necessary to achieve the new n-gonal symmetry. This invariably led to distorted faces, so prior to the duplication, all faces were triangulated by replacing them with pyramids. This triangulated polyhedron could then be relaxed^{[Fn 1]} such that the triangles became equilateral. This then resulted in a polyhedron with all regular faces and two specified n-gonal faces.

A simple visual example involves the icosahedron, starting with a seed face of a triangle, mark this and the opposite seed face in grey. The remaining lateral triangles can be subdivided into three equal sets of six triangles, coloured red green and blue. For the n-folding to n=4, replace the seed triangles with squares, leaving one free edge of each square. Now insert a fourth set of lateral triangles (yellow) and relax.

The act of n-folding can also be applied to an edge of the seed polyhedron. The above images show a visual representation of a seed edge on an icosahedron being 3-folded into a triangle.

It has since become apparent to me that this process of polygon substitution, or 'n-folding',could be applied far more generally than I had originally envisaged. All of the Platonic and a number of the Archimedean polyhedra can be n-folded around either a face or in certain cases an edge (which can be regarded as a digon or 2-gon). All faces are replaced by pyramids, hexagonal faces are replaced by a complex of six co-planar triangles. Archimedean polyhedra containing octagonal or decagonal faces are excluded from consideration.

For the n-folding to take place, the original (or 'seed') polyhedron must have some symmetry about the axis about which it is being n-folded. For edges this means that only homogeneous edges with the same polygon on each side can be considered. For 'seed ' faces there must also be some level of symmetry, so for example the snub triangular faces of a snub-dodecahedron which are surrounded by triangle triangle pentagon, cannot be used as seed faces. A number of polyhedra can also have m/2 dihedral symmettry about an m-gonal face. In this case to replace the m-gon with an odd n-gon the lateral faces have to wrap twice about the symmetry axis meaning that the n-gon itself is not required and the result is a toroid with an n-gonal hole, these are marked in the table below with a '(T)'.

Examples can be created with the pyramids pointing outwards ('exo') or inwards ('endo'). Where pyramids have been placed on hexagonal faces they have been left coplanar ('cop'). In most cases below, both options are displayed. Where an example cannot be created it is shown in red and struck out.

The level of distortion in the resulting model depends to a large extent on the difference between the original m-gonal faces and the replacement n-gon. If the difference is small (e.g hexagon to heptagon) then the distortion is low. For large scale changes (e.g. triangle to heptagon) the distortion can be significant. Edge replacement always results in significant distortion as it involves replacing a 2-gonal face with an n-gon.

For the table below, I have linked to the heptagonal member of each family. In a number of cases where the heptagonal example does not exist this may be due either (i) the range for n for which a vaild solution exists does not include n=7, (ii) no valid solutions exist other than the seed value. There are also very often isomorphs and examples with star polygonal faces, these have not as yet been explored. Where further exploration does exist, separate pages are linked by 'More'.

Notes for the table below:
- Colours in the main follow the colouring of the seed polyhedron, unless certain faces are highlighted. Pyramids are coloured based on the colour of the unaugmented polygon.
- where the n-folded polyhedron has lateral square faces this is due to these faces not distorting under relaxation and augmentation not being necessary
- Links marked 'C-exo' etc are included where the n-folded polygon has coplanar sides. These 'C' links are to models where these coplanar sides are highlighted.
- Where a superscipt appears after a face number it is to distinguish between two distinct sets of faces in the polyhedron, e.g. 3⁸ refers to the octahedral triangular faces in the snub cuboctahedron, 3²⁴ refers to the snub faces.
- '(T)' denotes a toroidal polyhedron - see above

Seed Polyhedron	Seed Edge	Seed Face
Tetrahedron	3-3 tri ^{(notes 1 and 2)}
Octahedron	3-3 tri ^(note 3)	3 tri ^{(notes 1 and 2)}
Icosahedron	3-3 tri ^(note 2)More	3 tri ^(note 2)More
Cube	4-4 exo endo More	4 prism ^{(note 4)}
Dodecahedron	5-5 exo endo	5 exo endo More
Cuboctahedron		3 exo endo ^{(note 5)}; 4 exo endo More
Icosidodecahedron		3 exo endo More; 5 exo endo More
Truncated Tetrahedron	6-6 cop	3 cop ^{(note 6)}
Truncated Octahedron	6-6 exo C-exo endo C-endo	4 exo endo ; 6(T) cop
Truncated Icosahedron	6-6 exo C-exo endo	5 exo endo ; 6(T) exo endo
Rhombicuboctahedron		3 exo endo 4¹²(T) ; exo endo ; 4⁶ exo endo
Rhomobicosidodecahedron		3 exo endo ; 4(T) exo endo ^{(note 7)}; 5 exo endo^{(note 8)}
Snub Cuboctahedron	3²⁴-3²⁴ exo endo ^{(note 9)}	3⁸ exo endo ; 4 exo endo More
Snub Icosidodecahedron	3⁶⁰-3⁶⁰ exo endo	3²⁰ exo endo ; 5 exo endo More

Notes:
(1): The edge substituted tetrahedon and face substituted octahedron are the antiprisms.
(2): There are no endo or exo modes here as the original faces are already triangular
(3): The edge substituted octahedrons are the augmented prisms.
(4): The face substituted cubes are the prisms, no triangulation of the lateral square faces is required for these to be regular.
(5): The square faces need not be augmented, leading to n-gonal orthobicupolas, with an upper limit of n<6.
(6): The faces substituted in the truncated tetrahedron are a triangle and the opposite hexagon. The resulting figures can also be obtained by n-folding a pentagonal rotunda.
(7): The n=6 cases can be generated (exo endo), the n=8 case can not. n=7 is close to being regular implying an upper limit to this family is 6 < limit < 7, a near miss for the n=7 exo case is here.
(8): The n=6 exo case is a very close near miss. Despite appearances the square pyramids are not regular. The n=7 cases cannot be generated.
(9): The n=7 endo case is degenerate with the square pyramids collapsing into two coincident triangles (here). These can therefore be removed. An example without those faces is here. Despite appearances the square pyramids adjacent to the heptagonal faces are not regular.

Other N-Folded Polyhedra

In addition to the Platonic and Archimedean polyhedra above, a number of prismatic families exist which can also be N-folded to generate an n=7 case. These include the following:

Cupolas and semicupolas
Spheno-prisms and their elongated and gyro-elongated relatives
n-4-3-3 polyheda
Cingulated anti-prisms
Spheno-coronas
Spheno-mega-coronas
Hebe-spheno-mega-coronas
Edge Expnded Bi-prisms and Bi-anti-prisms

There are also families basic on rhombic polyhedra which are under investigation.

Credits

The augmented seed polyhedra were made using Great Stella. Heptagonal nets were then generated using the to_nfold functionality of Antiprism. These were then made into HEDRON input files using Roger Kaufman's VRML2OFF utility, then relaxed to regularly faced forms using HEDRON.

Footnotes

Fn 1: Relaxing a polyhedron refers to the process of iteratively adjusting the locations of a polyhedron's vertices such that they meet a set of predefined criteria (e.g. all edge lengths are equal and they form regular faces).

Back: to main page