Geodesic polyhedron

A geodesic polyhedron is a convex polyhedron made from triangles. They usually have icosahedral symmetry, such that they have 6 triangles at a vertex, except 12 vertices which have 5 triangles. They are the dual of corresponding Goldberg polyhedra, of which all but the smallest one (which is a regular dodecahedron) have mostly hexagonal faces. Geodesic polyhedra are a good approximation to a sphere for many purposes, and appear in many different contexts. The most well-known may be the geodesic domes, hemispherical architectural structures designed by Buckminster Fuller, which geodesic polyhedra are named after. Geodesic grids used in geodesy also have the geometry of geodesic polyhedra. The capsids of some viruses have the shape of geodesic polyhedra, and some pollen grains are based on geodesic polyhedra. Fullerene molecules have the shape of Goldberg polyhedra. Geodesic polyhedra are available as geometric primitives in the Blender 3D modeling software package, which calls them icospheres: they are an alternative to the UV sphere, having a more regular distribution. The Goldberg–Coxeter construction is an expansion of the concepts underlying geodesic polyhedra.

3 constructions for a {3,5+}6,0 File:Geodesic icosahedral polyhedron example.png File:Geodesic icosahedral polyhedron example2.png File:Geodesic icosahedral polyhedron example5.png An icosahedron and related symmetry polyhedra can be used to define a high geodesic polyhedron by dividing triangular faces into smaller triangles, and projecting all the new vertices onto a sphere. Higher order polygonal faces can be divided into triangles by adding new vertices centered on each face. The new faces on the sphere are not equilateral triangles, but they are approximately equal edge length. All vertices are valence-6 except 12 vertices which are valence 5.

Construction of {3,5+}3,3 File:Geodesic dodecahedral polyhedron example.png Geodesic subdivisions can also be done from an augmented dodecahedron, dividing pentagons into triangles with a center point, and subdividing from that.

Construction of {3,5+}6,3 File:Geodesic icosahedral polyhedron example3.png Chiral polyhedra with higher order polygonal faces can be augmented with central points and new triangle faces. Those triangles can then be further subdivided into smaller triangles for new geodesic polyhedra. All vertices are valence-6 except the 12 centered at the original vertices which are valence 5.

Geodesic notation

In Magnus Wenninger's Spherical models, polyhedra are given geodesic notation in the form {3,q+}_b,c, where {3,q} is the Schläfli symbol for the regular polyhedron with triangular faces, and q-valence vertices. The + symbol indicates the valence of the vertices being increased. b,c represent a subdivision description, with 1,0 representing the base form. There are 3 symmetry classes of forms: {3,3+}_1,0 for a tetrahedron, {3,4+}_1,0 for an octahedron, and {3,5+}_1,0 for an icosahedron. The dual notation for Goldberg polyhedra is {q+,3}_b,c, with valence-3 vertices, with q-gonal and hexagonal faces. There are 3 symmetry classes of forms: {3+,3}_1,0 for a tetrahedron, {4+,3}_1,0 for a cube, and {5+,3}_1,0 for a dodecahedron. Values for b,c are divided into three classes:

Class I (b=0 or c=0): {3,q+}_b,0 or {3,q+}_0,b represent a simple division with original edges being divided into b sub-edges.

Class II (b=c): {3,q+}_b,b are easier to see from the dual polyhedron {q,3} with q-gonal faces first divided into triangles with a central point, and then all edges are divided into b sub-edges.

Class III: {3,q+}_b,c have nonzero unequal values for b,c, and exist in chiral pairs. For b > c we can define it as a right-handed form, and c > b is a left-handed form.

Subdivisions in class III here do not line up simply with the original edges. The subgrids can be extracted by looking at a triangular tiling, positioning a large triangle on top of grid vertices and walking paths from one vertex b steps in one direction, and a turn, either clockwise or counterclockwise, and then another c steps to the next primary vertex. For example, the icosahedron is {3,5+}_1,0, and pentakis dodecahedron, {3,5+}_1,1 is seen as a regular dodecahedron with pentagonal faces divided into 5 triangles. The primary face of the subdivision is called a principal polyhedral triangle (PPT) or the breakdown structure. Calculating a single PPT allows the entire figure to be created. The frequency of a geodesic polyhedron is defined by the sum of ν = b + c. A harmonic is a subfrequency and can be any whole divisor of ν. Class II always have a harmonic of 2, since ν = 2b. The triangulation number is T = b² + bc + c². This number times the number of original faces expresses how many triangles the new polyhedron will have.

PPTs with frequency 8

File:Geodesic principal polyhedral triangles frequency8.png

Elements

The number of elements are specified by the triangulation number $T=b^2+bc+c^2$. Two different geodesic polyhedra may have the same number of elements, for instance, {3,5+}_5,3 and {3,5+}_7,0 both have T=49.

Symmetry	Icosahedral	Octahedral	Tetrahedral
Base	Icosahedron {3,5} = {3,5+}1,0	Octahedron {3,4} = {3,4+}1,0	Tetrahedron {3,3} = {3,3+}1,0
Image	Icosahedron	Octahedron	Tetrahedron
Symbol	{3,5+}b,c	{3,4+}b,c	{3,3+}b,c
Vertices	$10T+2$	$4T+2$	$2T+2$
Faces	$20T$	$8T$	$4T$
Edges	$30T$	$12T$	$6T$

Construction

Geodesic polyhedra are constructed by subdividing faces of simpler polyhedra, and then projecting the new vertices onto the surface of a sphere. A geodesic polyhedron has straight edges and flat faces that approximate a sphere, but it can also be made as a spherical polyhedron (a tessellation on a sphere) with true geodesic curved edges on the surface of a sphere and spherical triangle faces.

Conway	u3I = (kt)I	(k)tI	ktI
Image	File:Conway polyhedron flat ktI.png	File:Conway polyhedron flat2 ktI.png	File:Conway polyhedron K6k5tI.png	File:Kised truncated icosahedron spherical.png
Form	3-frequency subdivided icosahedron	Kis truncated icosahedron	Geodesic polyhedron (3,0)	Spherical polyhedron

In this case, {3,5+}_3,0, with frequency $\nu =3$ and triangulation number $T=9$, each of the four versions of the polygon has 92 vertices (80 where six edges join, and 12 where five join), 270 edges and 180 faces.

Relation to Goldberg polyhedra

Geodesic polyhedra are the duals of Goldberg polyhedra. Goldberg polyhedra are also related in that applying a kis operator (dividing faces into triangles with a center point) creates new geodesic polyhedra, and truncating vertices of a geodesic polyhedron creates a new Goldberg polyhedron. For example, Goldberg G(2,1) kised, becomes {3,5+}_4,1, and truncating that becomes G(6,3). And similarly {3,5+}_2,1 truncated becomes G(4,1), and that kised becomes {3,5+}_6,3.

Examples

Class I

Class I geodesic polyhedra

Frequency	(1,0)	(2,0)	(3,0)	(4,0)	(5,0)	(6,0)	(7,0)	(8,0)	(m,0)
T	1	4	9	16	25	36	49	64	m2
Face triangle	File:Subdivided triangle 01 00.svg	File:Subdivided triangle 02 00.svg	File:Subdivided triangle 03 00.svg	File:Subdivided triangle 04 00.svg	File:Subdivided triangle 05 00.svg	File:Subdivided triangle 06 00.svg	File:Subdivided triangle 07 00.svg	File:Subdivided triangle 08 00.svg	...
Icosahedral	File:Icosahedron.svg	File:Pentakis icosidodecahedron.png	File:Conway polyhedron K6k5tI.png	File:Conway polyhedron k6k5at5daD.png	File:Icosahedron subdivision5.png	File:Conway polyhedron kdkt5daD.png	File:Conway dwrwD.png	File:Conway dcccD.png	more
Octahedral	File:Octahedron.svg	File:Tetrakis cuboctahedron.png	File:Octahedral geodesic polyhedron 03 00.svg	File:Octahedral geodesic polyhedron 04 00.svg	File:Octahedral geodesic polyhedron 05 00.svg	File:Octahedral geodesic polyhedron 06 00.svg	File:Octahedral geodesic polyhedron 07 00.svg	File:Octahedral geodesic polyhedron 08 00.svg	more
Tetrahedral	File:Tetrahedron.svg	File:Dual chamfered tetrahedron.png	File:Tetrahedral geodesic polyhedron 03 00.svg	File:Tetrahedral geodesic polyhedron 04 00.svg	File:Tetrahedral geodesic polyhedron 05 00.svg	File:Tetrahedral geodesic polyhedron 06 00.svg	File:Tetrahedral geodesic polyhedron 07 00.svg	File:Tetrahedral geodesic polyhedron 08 00.svg	more

Class II

Class II geodesic polyhedra

Frequency	(1,1)	(2,2)	(3,3)	(4,4)	(5,5)	(6,6)	(7,7)	(8,8)	(m,m)
T	3	12	27	48	75	108	147	192	3m2
Face triangle	File:Subdivided triangle 01 01.svg	File:Subdivided triangle 02 02.svg	File:Subdivided triangle 03 03.svg	File:Subdivided triangle 04 04.svg	File:Subdivided triangle 05 05.svg	File:Subdivided triangle 06 06.svg	File:Subdivided triangle 07 07.svg	File:Subdivided triangle 08 08.svg	...
Icosahedral	File:Conway polyhedron kD.png	File:Conway polyhedron kt5daD.png	File:Conway polyhedron kdktI.png	File:Conway polyhedron k5k6akdk5aD.png	File:Conway u5zI.png	File:Conway polyhedron dcdktkD.png	File:Conway dwrwtI.png	File:Conway dccctI.png	more
Octahedral	File:Tetrakishexahedron.jpg				File:Octahedral geodesic polyhedron 05 05.svg				more
Tetrahedral	File:Triakistetrahedron.jpg								more

Class III

Class III geodesic polyhedra

Frequency	(2,1)	(3,1)	(3,2)	(4,1)	(4,2)	(4,3)	(5,1)	(5,2)	(m,n)
T	7	13	19	21	28	37	31	39	m2+mn+n2
Face triangle	File:Subdivided triangle 01 02.svg	File:Subdivided triangle 01 03.svg	File:Subdivided triangle 02 03.svg	File:Subdivided triangle 01 04.svg	File:Subdivided triangle 02 04.svg	File:Subdivided triangle 03 04.svg	File:Subdivided triangle 01 05.svg	File:Subdivided triangle 02 05.svg	...
Icosahedral	File:Conway polyhedron K5sI.png	File:Conway polyhedron u5I.png	File:Geodesic polyhedron 3 2.png	File:Conway polyhedron K5k6st.png	File:Conway polyhedron dcwdI.png				more
Octahedral	File:Conway polyhedron dwC.png								more
Tetrahedral	File:Conway polyhedron dwT.png								more

Spherical models

Magnus Wenninger's book Spherical Models explores these subdivisions in building polyhedron models. After explaining the construction of these models, he explained his usage of triangular grids to mark out patterns, with triangles colored or excluded in the models.

Example model

File:Order in chaos Magnus Wenninger.jpg An artistic model created by Father Magnus Wenninger called Order in Chaos, representing a chiral subset of triangles of a 16-frequency icosahedral geodesic sphere, {3,5+}16,0

File:Magnus Wenninger Order in Chaos virtual model.png A virtual copy showing icosahedral symmetry great circles. The 6-fold rotational symmetry is illusionary, not existing on the icosahedron itself.

File:Magnus Wenninger Order in Chaos virtual model2.png A single icosahedral triangle with a 16-frequency subdivision

See also

• Conway polyhedron notation

References

Bibliography

• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A source book of Design. pp. 142–144, Figure 4-49, 50, 51 Custers of 12 spheres, 42 spheres, 92 spheres.
• Pugh, Antony (1976). "Chapter 6. The Geodesic Polyhedra of R. Buckminster Fuller and Related Polyhedra". Polyhedra: a visual approach.
• Wenninger, Magnus (1979). Spherical Models. Cambridge University Press. ISBN 978-0-521-29432-4. MR 0552023. Archived from the original on July 4, 2008. Reprinted by Dover (1999), ISBN 978-0-486-40921-4.
• Popko, Edward S. (2012). "Chapter 8. Subdivision schemas, 8.1 Geodesic Notation, 8.2 Triangulation number 8.3 Frequency and Harmonics 8.4 Grid Symmetry 8.5 Class I: Alternates and fords 8.5.1 Defining the Principal triangle 8.5.2 Edge Reference Points". Divided spheres: Geodesics & the Orderly Subdivision of the Sphere.