# Icosadeltahedral Geometry of Geodesic Domes, Fullerenes and Viruses: A Tutorial on the T-Number

## Abstract

**:**

## 1. Introduction

## 2. Icosa(delta)hedral Geodesic Domes

## 3. Icosahedral Fullerenes

## 4. Caspar–Klug Classification of Viruses: The T-Number

#### 4.1. Viruses as Overlapping Dome and Fullerene Designs

#### 4.2. Possible Ambiguities of the CK Designs

#### 4.3. The T-Number and the Capsid Size and Shape

## 5. Some Modifications and Variations of the Icosadeltahedral Geometry in Viruses

## 6. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The icosahedron with subdivided faces (left) and the geodesic dome (right), with vertices all lying on a sphere. (

**b**) The unwrapped and flattened icosahedron with subdivided faces, i.e., the net of the polyhedron. The small red dots denote the twelve pentavalent vertices. (

**c**) The Coxeter construction with $(m,n)=$ (2,1), (3,3), (4,0).

**Figure 2.**Gallery of icosadeltahedral geodesic domes for $m>n$ and $m<5$. For the sake of clarity, the back sides of geodesic domes are not shown, i.e., only half of the dome and four of twelve icosahedral vertices can be seen. The upper-right corner of the figure contains comparison between chiral $(3,2)$ and $(2,3)$ domes. Note that they are mirror images. The spiky shape is a $(3,2)$ icosadeltahedron in which triangular faces are very nearly equal and equilateral—note that such a requirement produces a very aspherical polyhedron.

**Figure 3.**Gallery of icosahedral fullerenes for $m>n$ and $m<5$. For the sake of clearer representation, the back sides of fullerenes are not shown. The average carbon–carbon bond length in all fullerenes is about 0.142 nm [30], and this can be used to estimate their size. The upper-right corner of the figure contains a comparison between $(1,1)$ dome and $(1,1)$ icosahedral fullerene (buckminsterfullerene, not to scale with other depicted fullerenes). Note that these polyhedra are dual to each other.

**Figure 4.**Cut-and-fold construction of $(2,1)$ icosahedral fullerene. The vectors ${\mathbf{a}}_{1}$, ${\mathbf{a}}_{2}$, and $\mathbf{E}$ discussed in the text are denoted. The triangular faces (Coxeter constructions) of $(1,1)$, $(2,0)$, $(3,1)$, and $(4,2)$ fullerene-like icosahedra are shown in the bottom of the figure. The figure also illustrates the concept of ‘’jumping” over pentagons and hexagons in order to determine the t-number of the structure.

**Figure 5.**Panels (

**a**,

**b**) represent polyhedral models of T = 3 viruses. These polyhedra can be termed as omnicapped truncated icosahedra or omnicapped Buckminsterfullerenes. The “pentamers” are colored in a darker tone and borders (contacts) between “capsomers” are represented by thicker lines. The polyhedron in panel (

**b**) is quite similar to turnip yellow mosaic virus [40]. Panel (

**c**) represents a model T = 1 (pT3) virus in which the building block is a “protein trimer” outlined by dashed lines.

**Figure 6.**The overlapping of hexagonal and triangular meshes (left) and CK (1,1) shape and (3,0) geodesic dome (right). The base vectors of the triangular (${\mathbf{a}}_{1},{\mathbf{a}}_{2}$) and hexagonal (${\mathbf{A}}_{1},{\mathbf{A}}_{2}$) meshes are denoted.

**Figure 7.**(

**a**) Three different arrangements of three proteins on the faces of the geodesic dome. When the three proteins are near the dome vertices (left), one first recognizes hexamers and pentamers. When they are near the center of the dome faces (middle), one first recognizes trimers. When they are near the centers of the dome edges (right), dimers can be most easily observed. (

**b**,

**c**) The protein density in a virus can often be thought of both as consisting of trimers or dimers (left) and of pentamers and hexamers (right). This is illustrated on a simplified geometrical model of protein density on a T = 3 virus.

**Figure 8.**The mean radius of the capsids plotted against their T-number. The vertical gray lines were drawn between the minimum and the maximum value of the mean radius in each T-class. The thick line shows a square-root dependence of mean radius, as discussed in the text, $\overline{R}={r}_{0}\sqrt{\mathrm{T}}$, with ${r}_{0}=$ 8 nm. Adapted from [50].

**Figure 9.**An alternative view of (2,0), (2,1), (2,2), and (4,1) geodesic domes. The dome triangles have been united or replaced by other polygons, so that (2,0) and (2,1) domes consist of pentagons and triangles, (2,2) domes of pentagons, rectangles, and triangles, and (4,1) domes of pentagons, hexagons, and triangles.

**Figure 10.**The prolate (top) and oblate (middle) icosadeltahedral designs obtained by elongating and shortening the isometric icosadeltahedral shape along the five-fold axis of symmetry by $b=\left|\mathbf{b}\right|$. The influence of elongation vector $\mathbf{b}$ on shapes is illustrated by the three shapes shown at the bottom of the figure. Adapted from [30].

**Figure 11.**The two nets for the two halves of the capsid of geminiviruses and the folded shapes, which may be thought of both as.

**Figure 12.**A possible way to construct a polyhedron similar to a capsid of the HIV virus. Note that the top pyramid-like shape is smaller than the bottom pyramid-like shape, which can be seen by the different sizes of the triangles in the polyhedron net (adapted from [65]). The folded polyhedron is also shown in two different views. Such constructions can be obviously varied in many different ways. One of the representative structures of an HIV capsid is also shown (adapted from [21]).

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**MDPI and ACS Style**

Šiber, A.
Icosadeltahedral Geometry of Geodesic Domes, Fullerenes and Viruses: A Tutorial on the T-Number. *Symmetry* **2020**, *12*, 556.
https://doi.org/10.3390/sym12040556

**AMA Style**

Šiber A.
Icosadeltahedral Geometry of Geodesic Domes, Fullerenes and Viruses: A Tutorial on the T-Number. *Symmetry*. 2020; 12(4):556.
https://doi.org/10.3390/sym12040556

**Chicago/Turabian Style**

Šiber, Antonio.
2020. "Icosadeltahedral Geometry of Geodesic Domes, Fullerenes and Viruses: A Tutorial on the T-Number" *Symmetry* 12, no. 4: 556.
https://doi.org/10.3390/sym12040556