(True) Goldberg operations as a formal approach to operations on - - PowerPoint PPT Presentation

(True) Goldberg operations as a formal approach to operations on polyhedra

Gunnar Brinkmann, (Universiteit Gent) Pieter Goetschalckx, (Universiteit Gent) Stan Schein (Univ. of California, Los Angeles)

text and pictures by Helmer Aslaksen

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and many more. . . source: wikipedia

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The famous Goldberg-Coxeter construction

(Figure from Coxeter’s paper)

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(0,0) (3,1) "Goldberg parameters" equilateral triangle

Cut out this triangle. . .

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and glue a copy into each triangle of the icosahedron (or another triangulation) Old vertices keep their degree, new vertices have degree 6. For fullerenes: then take the dual.

This idea can also be used to decorate polyhedra where all faces are 4-gons with quadrangles cut out

• f a quadrangular net so that all 4 sides are

equivalent. (Tarnai, Kov´ acs, Fowler, Guest)

(3,1)

For other face sizes or a mixture of triangles and quadrangles it does not work.

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Coxeter (1971): “Independently of Michael Goldberg (1937), Caspar and Klug (1963) proposed the following rule for making a suitable pattern”. Does this imply or at least suggest that they proposed “the same” rule? So what exactly did Goldberg do?

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Let’s look at a figure from Goldberg’s paper:

That’s a right triangle – not an equilateral one. . .

Goldberg starts from the dodecahedron and states:

“Each pentagonal patch may be divided into ten equal (congruent or symmetric) triangular patches”.

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Reverse operation: subdivide the pentagons of the dodecahedron and glue copies (half of them mirrors) of the right Goldberg triangle cut out of the hexagonal tiling into it.

Convention: original edges, face center to edge center, face center to vertex

This approach can be applied to all polyhedra, as every n-gon can be subdivided into 2n coloured triangles:

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The result of applying Goldberg’s (7, 0)

• peration to this polyhedron:

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Why does it fit at the edges of the triangle?

The sides of the triangle are mirror axes! Except at the black dots the situation in the decorated polyhedron is like in the tiling!

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Let’s look at another figure from Goldberg’s paper:

No mirror axes – nothing fits!

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The sides of the rectangular Goldberg-triangle are mirror axes if and only if the parameters (k, l) fulfill 0 ∈ {k, l} or k = l:

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With two different triangles it would have worked

rotation by 60 degrees rotation by 180 degrees (5,3)

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Note that this triangle is – though equilateral – quite different from the one used by Caspar and Klug!

rotation by 60 degrees rotation by 180 degrees (5,3)

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So:

• The method known as Goldberg-Coxeter

construction is by Caspar and Klug and different from what Goldberg proposed.

• The method used by Goldberg is appli-

cable in a much more general context than that of Caspar and Klug.

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Also the Caspar/Klug (k, l)-operations with 0 ∈ {k, l} and k = l do not preserve all symmetries!

Let’s for now stick to

• perations preserving all

symmetries.

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Reminder: Why do rectangular Goldberg triangles fit at the edges?

The sides of the triangle are mirror axes! Let’s start from that!

Local symmetry preserving

• perations:
• Take any 3-connected periodic tiling T

in the plane.

• Take any three pairwise intersecting

mirror axes forming a right triangle.

• Take the interior as the

Goldberg right triangle.

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Reminder: Blue edges are glued to original edges.

An example:

60

• 90
• 30
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In order to be a good definition, it must:

• encompass all known operations that are

considered to be local symmetry preserv- ing operations

• transform polyhedra to polyhedra
• be useful and open new directions of

research

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Dual:

to face center to edge center to vertex

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Truncate:

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Ambo:

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Chamfer: . . . and so on . . .

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Theorem:

If a local symmetry preserving operation is applied to a polyhedron (a plane 3-connected graph), the result is a polyhedron.

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Usefulness and new directions

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In fact we have already seen one useful feature: Everything you prove for the class of local symmetry preserving operations needs not be proven for each element separately! E.g. One doesn’t have to prove separately that dual, truncation, ambo, chamfer, quinto, . . . produce (3-connected) polyhedra.

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The ratio Ea

Eb between the number of edges

(or equivalently coloured triangles) after the

• peration (Ea) and before the operation

(Eb) is independent of the polyhedron and is called the inflation factor.

Now it is possible to ask for all operations with a given inflation factor.

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Symmetry preserving operations preserve the group – that is: don’t destroy symmetries – but can also increase it (that is: create new symmetries): ambo Can that also happen for polyhedra that are not self-dual and when the operation can not be written as a product of

• perations involving ambo?

For the torus that is possible (inflation factor 9). . . . . . for polyhedra we don’t know such an example.

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Symmetric polyhedra can be seen as redundant: all you need is the fundamental domain and the group tells you how to distribute the content of the fundamental domain. A decorated polyhedron is also redundant: all you need is the decoration and the polyhedron tells you how to distribute it. group ↔ base polyhedron fundamental domain ↔ decoration

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Algorithmic question:

How do you efficiently decide whether a polyhedron is a decoration of a smaller

• ne?

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• Thanks for your attention!
• Are there questions, proposals, or even

answers?

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