Geodesic polyhedron

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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.

Contents

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, [1] [2] and some pollen grains are based on geodesic polyhedra. [3] 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. [4] [5] The Goldberg–Coxeter construction is an expansion of the concepts underlying geodesic polyhedra.

3 constructions for a {3,5+}6,0
Geodesic icosahedral polyhedron example.png
Geodesic icosahedral polyhedron example2.png
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
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
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
Geodesic principal polyhedral triangles frequency8.png

Elements

The number of elements are specified by the triangulation number . 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.svg Octahedron.svg Tetrahedron.svg
Symbol{3,5+}b,c{3,4+}b,c{3,3+}b,c
Vertices
Faces
Edges

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)tIktI
Image Conway polyhedron flat ktI.png Conway polyhedron flat2 ktI.png Conway polyhedron K6k5tI.png Kised truncated icosahedron spherical.png
Form3-frequency
subdivided icosahedron
Kis truncated icosahedron Geodesic polyhedron (3,0) Spherical polyhedron

In this case, {3,5+}3,0, with frequency and triangulation number , 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)
T1491625364964m2
Face
triangle
Subdivided triangle 01 00.svg Subdivided triangle 02 00.svg Subdivided triangle 03 00.svg Subdivided triangle 04 00.svg Subdivided triangle 05 00.svg Subdivided triangle 06 00.svg Subdivided triangle 07 00.svg Subdivided triangle 08 00.svg ...
Icosahedral Icosahedron.svg Pentakis icosidodecahedron.png Conway polyhedron K6k5tI.png Conway polyhedron k6k5at5daD.png Icosahedron subdivision5.png Conway polyhedron kdkt5daD.png Conway dwrwD.png Conway dcccD.png more
Octahedral Octahedron.svg Tetrakis cuboctahedron.png Octahedral geodesic polyhedron 03 00.svg Octahedral geodesic polyhedron 04 00.svg Octahedral geodesic polyhedron 05 00.svg Octahedral geodesic polyhedron 06 00.svg Octahedral geodesic polyhedron 07 00.svg Octahedral geodesic polyhedron 08 00.svg more
Tetrahedral Tetrahedron.svg Dual chamfered tetrahedron.png Tetrahedral geodesic polyhedron 03 00.svg Tetrahedral geodesic polyhedron 04 00.svg Tetrahedral geodesic polyhedron 05 00.svg Tetrahedral geodesic polyhedron 06 00.svg Tetrahedral geodesic polyhedron 07 00.svg 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)
T3122748751081471923m2
Face
triangle
Subdivided triangle 01 01.svg Subdivided triangle 02 02.svg Subdivided triangle 03 03.svg Subdivided triangle 04 04.svg Subdivided triangle 05 05.svg Subdivided triangle 06 06.svg Subdivided triangle 07 07.svg Subdivided triangle 08 08.svg ...
Icosahedral Conway polyhedron kD.png Conway polyhedron kt5daD.png Conway polyhedron kdktI.png Conway polyhedron k5k6akdk5aD.png Conway u5zI.png Conway polyhedron dcdktkD.png Conway dwrwtI.png Conway dccctI.png more
Octahedral Tetrakishexahedron.jpg Octahedral geodesic polyhedron 05 05.svg more
Tetrahedral 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)
T713192128373139m2+mn+n2
Face
triangle
Subdivided triangle 01 02.svg Subdivided triangle 01 03.svg Subdivided triangle 02 03.svg Subdivided triangle 01 04.svg Subdivided triangle 02 04.svg Subdivided triangle 03 04.svg Subdivided triangle 01 05.svg Subdivided triangle 02 05.svg ...
Icosahedral Conway polyhedron K5sI.png Conway polyhedron u5I.png Geodesic polyhedron 3 2.png Conway polyhedron K5k6st.png Conway polyhedron dcwdI.png more
Octahedral Conway polyhedron dwC.png more
Tetrahedral 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. [6]

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

See also

Related Research Articles

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<span class="mw-page-title-main">Icosidodecahedron</span> Archimedean solid with 32 faces

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<span class="mw-page-title-main">Polyhedron</span> 3D shape with flat faces, straight edges and sharp corners

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<span class="mw-page-title-main">Truncated icosidodecahedron</span> Archimedean solid

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<span class="mw-page-title-main">Tetrakis hexahedron</span> Catalan solid with 24 faces

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<span class="mw-page-title-main">Disdyakis dodecahedron</span> Geometric shape with 48 faces

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<span class="mw-page-title-main">Disdyakis triacontahedron</span> Catalan solid with 120 faces

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<span class="mw-page-title-main">Spherical polyhedron</span> Partition of a spheres surface into polygons

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<span class="mw-page-title-main">Pentakis snub dodecahedron</span>

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<span class="mw-page-title-main">Goldberg polyhedron</span> Convex polyhedron made from hexagons and pentagons

In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (1902–1990). They are defined by three properties: each face is either a pentagon or hexagon, exactly three faces meet at each vertex, and they have rotational icosahedral symmetry. They are not necessarily mirror-symmetric; e.g. GP(5,3) and GP(3,5) are enantiomorphs of each other. A Goldberg polyhedron is a dual polyhedron of a geodesic polyhedron.

<span class="mw-page-title-main">Chamfer (geometry)</span> Geometric operation which truncates the edges of polyhedra

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<span class="mw-page-title-main">Order-5 truncated pentagonal hexecontahedron</span>

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<span class="mw-page-title-main">Hexapentakis truncated icosahedron</span>

The hexapentakis truncated icosahedron is a convex polyhedron constructed as an augmented truncated icosahedron. It is geodesic polyhedron {3,5+}3,0, with pentavalent vertices separated by an edge-direct distance of 3 steps.

<span class="mw-page-title-main">Goldberg–Coxeter construction</span>

The Goldberg–Coxeter construction or Goldberg–Coxeter operation is a graph operation defined on regular polyhedral graphs with degree 3 or 4. It also applies to the dual graph of these graphs, i.e. graphs with triangular or quadrilateral "faces". The GC construction can be thought of as subdividing the faces of a polyhedron with a lattice of triangular, square, or hexagonal polygons, possibly skewed with regards to the original face: it is an extension of concepts introduced by the Goldberg polyhedra and geodesic polyhedra. The GC construction is primarily studied in organic chemistry for its application to fullerenes, but it has been applied to nanoparticles, computer-aided design, basket weaving, and the general study of graph theory and polyhedra.

References

  1. Caspar, D. L. D.; Klug, A. (1962). "Physical Principles in the Construction of Regular Viruses". Cold Spring Harb. Symp. Quant. Biol. 27: 1–24. doi:10.1101/sqb.1962.027.001.005. PMID   14019094.
  2. Coxeter, H.S.M. (1971). "Virus macromolecules and geodesic domes.". In Butcher, J. C. (ed.). A spectrum of mathematics. Oxford University Press. pp. 98–107.
  3. Andrade, Kleber; Guerra, Sara; Debut, Alexis (2014). "Fullerene-Based Symmetry in Hibiscus rosa-sinensis Pollen". PLOS ONE. 9 (7): e102123. Bibcode:2014PLoSO...9j2123A. doi: 10.1371/journal.pone.0102123 . PMC   4086983 . PMID   25003375. See also this picture of a morning glory pollen grain.
  4. "Mesh Primitives", Blender Reference Manual, Version 2.77, retrieved 2016-06-11.
  5. "What is the difference between a UV Sphere and an Icosphere?". Blender Stack Exchange .
  6. Wenninger (1979), pp. 150–159.

Bibliography