Welcome to the Icosahedral Server
Have you ever constructed an icosahedron from a hexagonal paper template? Creating such a model is very beneficial
in understanding viral symmetries, and the core differences between T numbers. For this reason we have created
the icosahedral server to aid you in the construction of your own icosahedron. A
T=3
icosahedron is shown to the left, built with the hexagonal templates found here. The hexamers (the polygons with
6 sides) denoted by the blue lines are left asis on the hexagon sheet. The green lines are found within the
units representing pentamers, and are generated by folding and cutting the paper template. Icosahedrons with
any T numbers (T=h**2+h*k+k**2, see paradigm section) can be generated using this server. The output of the program
found here can be either an ODL file, so that the 3D model can be viewed by O, or a postscript file, so that
a projection of the model (as shown here) can be printed out. These files can be downloaded directly from this
site. By the time you have finished browsing this server, you will have learned a great deal about the quasiequivalences
of viruses.
Icosahedron construction
Geometric principles for generating icosahedral quasiequivalent surface lattices. These constructions show the relation
between icosahedral symmetry axes and quasiequivalent symmetry axes. The latter are symmetry elements that hold
only in a local environment.
Hexamers are initially considered planar (an array of hexamers forms a flat sheet as shown in a) and pentamers are
considered convex, introducing curvature in the sheet of hexamers when they are inserted. The closed icosahedral
shell, composed of hexamers and pentamers, is generated by inserting 12 pentamers at appropriate positions in the
hexamer net. To construct a model of a particular quasiequivalent lattice, one face of an icosahedron is generated
in the hexagon net. The origin is replaced with a pentamer and the (h,k) hexamers is replaced by a pentamer. The
third replaced hexamer is identified by 3fold symmetry (i.e. complete the equilateral triangle of the face). The
icosahedral face for a T=3 surface lattice is defined by the triangle with blue edges (h=1, k=1). Two possible
T=7 lattice choices are also marked with green and yellow lines (h=2, k=1 or h=1, k=2, these being mirror images
of each other), and require knowledge of the arrangement of hexamers and pentamers and the enantiomorph of the
lattice for a complete lattice definition.
In figure b), seven hexamer units (bold outlines in a) defined by the T=3 lattice choice are shown and the T=3 icosahedral
face defined in (a) has been shaded. A three dimensional model of the lattice can be generated by arranging 20
identical faces of the icosahedron as shown, and folded into a quasiequivalent icosahedronIn figure b), seven
hexamer units (bold outlines in a) defined by the T=3 lattice choice are shown and the T=3 icosahedral face defined
in (a) has been shaded. A three dimensional model of the lattice can be generated by arranging 20 identical faces
of the icosahedron as shown, and folded into a quasiequivalent icosahedron.
References:
Johnson, J.E. & Speir, J.A. (1997) Quasiequivalent Viruses: A Paradigm for Protein Assemblies JMB, 269, 665675
Caspar DLD, Klug A. Physical principles in the construction of regular viruses. Cold Spring Harbor Symp Quant
Biol 1962;27:1.
Paper Model Templates



Paper 
Hexagonal template sheet 

T=1 
(h,k) = (1,0) 

T=3 
(h,k) = (1,1) 

T=4 
(h,k) = (2,0) 

T=7 
(h,k) = (2,1) 

T=7 
(h,k) = (1,2) 

T=9 
(h,k) = (3,0) 

T=12 
(h,k) = (2,2) 

T=13 
(h,k) = (3,1) 

T=13 
(h,k) = (1,3) 

T=16 
(h,k) = (4,0) 

T=19 
(h,k) = (3,2) 

T=19 
(h,k) = (2,3) 

T=21 
(h,k) = (4,1) 

T=21 
(h,k) = (1,4) 

T=25 
(h,k) = (5,0) 

T=27 
(h,k) = (3,3) 

T=28 
(h,k) = (4,2) 

T=28 
(h,k) = (2,4) 

T=31 
(h,k) = (5,1) 

T=31 
(h,k) = (1,5) 

T=36 
(h,k) = (6,0) 

T=37 
(h,k) = (4,3) 

T=37 
(h,k) = (3,4) 

T=39 
(h,k) = (5,2) 

T=39 
(h,k) = (2,5) 

T=43 
(h,k) = (6,1) 

T=43 
(h,k) = (1,6) 

T=48 
(h,k) = (4,4) 

T=49 
(h,k) = (5,3) 

T=49 
(h,k) = (3,5) 

T=49 
(h,k) = (7,0) 

T=52 
(h,k) = (6,2) 

T=52 
(h,k) = (2,6) 

T=57 
(h,k) = (7,1) 

T=57 
(h,k) = (1,7) 

T=61 
(h,k) = (5,4) 

T=61 
(h,k) = (4,5) 

T=63 
(h,k) = (6,3) 

T=63 
(h,k) = (3,6) 

T=64 
(h,k) = (8,0) 

T=67 
(h,k) = (7,2) 

T=67 
(h,k) = (2,7) 

T=73 
(h,k) = (8,1) 

T=73 
(h,k) = (1,8) 

T=75 
(h,k) = (5,5) 

T=76 
(h,k) = (6,4) 

T=76 
(h,k) = (4,6) 

T=79 
(h,k) = (7,3) 

T=79 
(h,k) = (3,7) 

T=81 
(h,k) = (9,0) 

T=84 
(h,k) = (8,2) 

T=84 
(h,k) = (2,8) 

T=91 
(h,k) = (6,5) 

T=91 
(h,k) = (5,6) 

T=91 
(h,k) = (9,1) 

T=91 
(h,k) = (1,9) 

T=93 
(h,k) = (7,4) 

T=93 
(h,k) = (4,7) 

T=97 
(h,k) = (8,3) 

T=97 
(h,k) = (3,8) 

T=100 
(h,k) = (10,0) 

Icosahedron gallery




OMAC FILE 

T=1 
(h,k) = (1,0) 

T=3 
(h,k) = (1,1) 

T=4 
(h,k) = (2,0) 

T=7 
(h,k) = (2,1) 

T=7 
(h,k) = (1,2) 

T=9 
(h,k) = (3,0) 

T=12 
(h,k) = (2,2) 

T=13 
(h,k) = (3,1) 

T=13 
(h,k) = (1,3) 

T=16 
(h,k) = (4,0) 

T=19 
(h,k) = (3,2) 

T=19 
(h,k) = (2,3) 

T=21 
(h,k) = (4,1) 

T=21 
(h,k) = (1,4) 

T=25 
(h,k) = (5,0) 

T=27 
(h,k) = (3,3) 

T=28 
(h,k) = (4,2) 

T=28 
(h,k) = (2,4) 

T=31 
(h,k) = (5,1) 

T=31 
(h,k) = (1,5) 

T=36 
(h,k) = (6,0) 

T=37 
(h,k) = (4,3) 

T=37 
(h,k) = (3,4) 

T=39 
(h,k) = (5,2) 

T=39 
(h,k) = (2,5) 

T=43 
(h,k) = (6,1) 

T=43 
(h,k) = (1,6) 

T=48 
(h,k) = (4,4) 

T=49 
(h,k) = (5,3) 

T=49 
(h,k) = (3,5) 

T=49 
(h,k) = (7,0) 

T=52 
(h,k) = (6,2) 

T=52 
(h,k) = (2,6) 

T=57 
(h,k) = (7,1) 

T=57 
(h,k) = (1,7) 

T=61 
(h,k) = (5,4) 

T=61 
(h,k) = (4,5) 

T=63 
(h,k) = (6,3) 

T=63 
(h,k) = (3,6) 

T=64 
(h,k) = (8,0) 

T=67 
(h,k) = (7,2) 

T=67 
(h,k) = (2,7) 

T=73 
(h,k) = (8,1) 

T=73 
(h,k) = (1,8) 

T=75 
(h,k) = (5,5) 

T=76 
(h,k) = (6,4) 

T=76 
(h,k) = (4,6) 

T=79 
(h,k) = (7,3) 

T=79 
(h,k) = (3,7) 

T=81 
(h,k) = (9,0) 

T=84 
(h,k) = (8,2) 

T=84 
(h,k) = (2,8) 

T=91 
(h,k) = (6,5) 

T=91 
(h,k) = (5,6) 

T=91 
(h,k) = (9,1) 

T=91 
(h,k) = (1,9) 

T=93 
(h,k) = (7,4) 

T=93 
(h,k) = (4,7) 

T=97 
(h,k) = (8,3) 

T=97 
(h,k) = (3,8) 

T=100 
(h,k) = (10,0) 
