Sándor Kabai, Szaniszló Bérczi, Lajos Szilassi

In this article we explore possible clusters of rhombic triacontahedra (RTs), usually by connecting them face to face, which happens when they are placed at the vertices of certain polyhedra. The edge length of such polyhedra is set to be twice the distance of a face of an RT from the origin (about 2.7527). The clusters thus produced can be used to build further clusters using an RT and a rhombic hexecontahedron (RH), the logo of Wolfram|Alpha. We briefly look at other kinds of connections and produce new clusters from old by using matching polyhedra instead of RTs.

Rhombic Triacontahedra (RT) and Rhombic Hexecontahedron (RH)

Here are the RT and RH.

To find the possible candidate polyhedra, let us summarize the angles of face centers relative to the center of a single face of RT, as seen from the origin.

face radians degrees; 1 0 0 degrees; 1 pi 180 degrees; 4 pi/2 90 degrees; 4 pi/5 36 degrees; 4 2pi/5 72 degrees; 4 3pi/5 108 degrees; 4 4pi/5 154 degrees; 4 pi/3 60 degrees; 4 2pi/3 120 degrees

The dimensions applicable in the clusters can be determined on the basis of the relationship of the cube and RT. For instance, the cube edge is equal to the longer diagonal of the face of the RT (a golden rhombus), . The diagonal of the cube equals the distance between opposite threefold vertices of the RT.

Additional information for finding possible candidate polyhedra comes from a chart of truncations prepared by Szaniszló Bérczi. Figure 1 shows regular (Platonic) solids projected on a sphere. Archimedean solids are deduced from the regular solids by the truncation operation. A Platonic or Archimedean solid can be identified by its vertex configuration, because it is uniform; this is given by the Steiner symbol, which lists the faces that meet at a vertex. For example, is the Steiner symbol for the cube, because three squares (4-sided faces) meet at each vertex. The RT-related structures should be arranged according to the third row of the table: , , , , , .

Figure 1. The periodic table of Platonic and Archimedean solids and tessellations supplemented by the sequence of one of the infinite numbers of two-dimensional hyperbolic tessellations. In order to emerge, the regular solids are given in their projected-onto-sphere form.

For more help, we can consider the relationship of RT to cube and to RH, on the basis of which all necessary dimensions can be calculated.

This is the length of the cube edge.

This is the length of the cube diagonal, which is equal to the length of the threefold axis diagonal of the RT.

This is the face distance of the RT.

This is the length of the fivefold axis diagonal of the RT.

The relationship of the RT, cube, and a golden rectangle can be used to determine dimensions.

The relationship of the RT, RH, and a plane perpendicular to the threefold axis that cuts the RT and RH in half can be used to determine the angles between the RT faces.

Here are the definitions used in the constructions.

Icosahedron (ICO)

The icosahedron (ICO) is one of the five Platonic solids.

Twelve RTs placed at the vertices of the icosahedron enclose an RH. Such RT clusters appear in photos of certain quasicrystals.

Here is a cluster of 12 RHs without transparency.

When dodecahedra are used instead of RTs, they are attached to each other along their edges.

Truncated Icosahedron (TICO) (5, 6, 6)

The truncated icosahedron (TICO) is the shape most widely used for a soccer ball. It is also the overall structure of the C60 molecule, Buckminsterfullerene.

Here ICOs replace RTs; the ICOs meet along their edges.

Icosidodecahedron (ID) (3, 5, 3, 5)

The icosidodecahedron (ID) can be constructed as a truncation of either an icosahedron or a dodecahedron.

Here is the corresponding cluster.

Let us add 12 RTs at the vertices of the ICO.

Now add 12 RHs at the vertices of the DOD.

Add some more RTs at the vertices of an ID. This construction can be continued by adding more RTs and RHs.

Truncated Dodecahedron (TD) (3, 10, 10)

This is a special case of dodecahedron truncation, when the edge length is uniform.

This cluster of 60 RTs can be interpreted as being assembled from 20 sets of three RTs.

Dodecahedron (DOD) (5, 5, 5)

The usual pentagonal dodecahedron is one of the five Platonic solids.

Here is its associated cluster.

This investigates how the basic polyhedron relates to its associated cluster.

The cluster of 20 RTs can be fitted with 12 RHs placed at the vertices of an ICO.

Great Rhombicosidodecahedron (GRID) (4, 6, 10)

This is a beautiful shape. Somewhat similar shapes can be seen in some photographs of quasicrystals.

Here is its associated cluster.

Small Rhombicosidodecahedron (SRID) (3, 4, 5, 4)

An SRID has 60 vertices. In the related cluster, each RT is connected to four adjacent RTs, as opposed to the TICO-based cluster, where each RT is connected to only three adjacent RTs.

Here is its associated cluster.

This cluster can also be extended by adding RHs, for example at the vertices of an ICO.

Summary of Truncations and Their Clusters

Here is a summary of the clusters that correspond to the various truncations.

Cube (4, 4, 4)

The cube is one of the five Platonic solids.

Here is its associated cluster.

Two Johnson Solids

A number of Johnson solids can be considered as parts of the above polyhedra. The two shown here could be used as structures.

Here are their clusters.

Connections with Vertices

Interesting clusters can also be produced when the RTs meet at their vertices. For instance, RTs at RT vertices meet with their fivefold tips.

RTs at the vertices of a rhombic dodecahedron (RD) meet with their threefold tips. Adjacent pairs of RTs overlap in a flat golden rhombohedron, creating another closed structure.


[1] Sz. Bérczi, “From the Periodic System of Platonic and Archimedean Solids and Tessellations to the 4D Regular Polyhedra and Tessellations (with Extensions to Some 5D Polytopes),” Symmetry: Culture and Science, 11(1–4), 2003 pp. 125–137.
[2] S. Kabai. “21609 Clusters of Polyhedra” from the Wolfram Demonstrations Project—A Wolfram Web Resource. www.demonstrations.wolfram.com/21609ClustersOfPolyhedra.
[3] G. Gévay, “Icosahedral Morphology,” Fivefold Symmetry (I. Hargitai, ed.), Singapore: World Scientific, 1992.

S. Kabai, Sz. Bérczi, and L. Szilassi, “Some Clusters Produced by Placing Rhombic Triacontahedra at the Vertices of Polyhedra,” The Mathematica Journal, 2012. dx.doi.org/doi:10.3888/tmj.14-14.

Image Reference:

1. Sz. Bérczi, “A szabályos és féligszabályos (platoni és archimedészi) testek és mozaikok periódusos rendszere,” Középiskolai Matematikai Lapok, 59(5), 1979 pp. 193–199.

About the Authors

Sándor Kabai is a retired engineer who received his B.Sc. in manufacturing technology from Bánki College in Hungary in 1970. He has written many Demonstrations for the Wolfram Demonstrations Project; see demonstrations. He consults in space research-related education in Hungary.

Szaniszló Bérczi received his M.Sc. in physics and astronomy at Eötvös Loránd University in 1974 and his Ph.D. in geology (planetary science) at the Hungarian Academy of Science in 1994. He has achievements in several fields of science. He made a new synthesis of the evolution of matter according to the material hierarchy versus great structure building periods. This model is a part of his Lecture Note Series at Eötvös Loránd University. He also organized a research group on the evolution of matter within the Geonomy Scientific Committee of the Hungarian Academy of Science (with Béla Lukács). He wrote the first book in Hungary about planetary science, From Crystals to Planetary Bodies (in Hungarian). At Eötvös Loránd University, he initiated and built (with colleagues) the Hungarian University Surveyor (Hunveyor) experimental space probe model and the Husar rover (Hungarian University Surface Analysis Rover) for training teachers, and developed new technologies for measurement. He wrote the first lecture note book in Hungarian on symmetry and structure building, which uses symmetry principles in forming cellular automata models. He organized the loan of NASA lunar samples from Houston JSC and NIPR Antarctic meteorites from the Tokyo Antarctic Meteorite Center to Eötvös Loránd University, and initiated planetary materials studies and comparisons on industrial materials and technologies. His booklet series on ancient Eurasian art surveys both figurative and ornamental arts of great Eurasian regions. His studies on ethnomathematics cover ancient Eurasian cultures.

Lajos Szilassi received an M.Sc. in constructive geometry from the University of Szeged, Hungary in 1966, and a Ph.D. in 2006. He is now an emeritus professor at the University of Szeged. His main teaching areas include geometry, elementary mathematics, and computer applications. His research fields are in computer-aided solutions of mathematical and geometrical problems, with visualization of the solutions. In 1977 he found a way to construct a toroidal heptahedron, which is known now as the Szilassi polyhedron.

Sándor Kabai
1203 Budapest
Határ 9


Szaniszló Bérczi
Eötvös Loránd University, Faculty of Science
Institute of Physics, Dept. Materials Physics
Pázmány Péter sétány 1/a. H-1117
Budapest, Hungary


Lajos Szilassi