Introduction
The geometry of polyhedra has fascinated many from time immemorial. Greek Philosophers and Mathematicians from all over the world, over the centuries, have tried to unravel the mysteries of Polyhedra. As kids we played with 2-dimensional objects such as polygons or studied their properties in school geometry. If we go one step further and try to assemble these polygons in 3-dimensionsal space, we get what is called as polyhedra.
Platonic Solids
The blogpost describes the geometry of polyhedra and how it is used to construct geodesic domes. It was Plato the Greek philosopher who first described a class of polyhedra called Platonic Solids. These 3-dimensional object have unique properties characterised by faces that are all of the same shape and dimension. The same number of faces meet at each vertex and at the same angle. There are only five platonic solids as listed below and shown pictorially.
- Tetrahedron– 4 triangular faces.
- Cube– 6 square faces.
- Octahedron– 8 triangular faces.
- Dodecahedron– 12 pentagonal faces.
- Icosahedron– 20 triangular faces.
The fascinating property of these platonic solid is that their vertices are all equidistant from a central point. This means that the solids can be enclosed by a sphere called a circumsphere. The circumsphere will contain all the vertices of the platonic solid. This unique property is utilised to define the shape of a geodesic dome.
Archimedean Solids
Archimedes who came into the picture more than a century after Plato introduced another set of polyhedra which combine the platonic solids into more intricate shapes. The polyhedra allow more than one regular polygon as a face. However, regularity is introduced by congruent faces and vertices having the same ordering of polygons surrounding it. There are a total of 13 Archimedean solids, some of which are shown below.
Polyhedra-all things wise and wonderful
Over the years mathematician defined many types polyhedra such as (1) Kepler-Poinsot, (2) Johnson Solids, (3) Stewart Toroids and many more. To know more about polyhedra refer to [1].
For the purpose of this blogpost, I will only consider platonic solids. From a constructability point of view these are probably the easiest for constructing geodesic domes.
Frequency and Class of Platonic Solids
Before we go into the details of designing and constructing geodesic domes it would be worthwhile to understand some important terminology regarding subdivision of platonic solid faces, i.e. the Frequency and Class. These two properties influence the domes aesthetics, shape, complexity and strength.
Frequency of a Polyhedron
Frequency defines the number of subdivisions of the base polygon that forms the polyhedra. Each edge of the polygon is divided into smaller segments, creating additional polygons within the face. The frequency is denoted with a “V” preceded by a number (e.g., 3V, 4V), indicating how many times each side of the base edge is subdivided. The figure below shows a dodecahedron with subdivisions of1V, 2V and 3V.
The geodesic domes are obtained by projecting the vertices on to the circumsphere. The coordinates of the points of a Dodecahedron and Icosahedron are provided in the enclosed excel sheet (which is free for download). The points can be extrapolated depending on the radius of the circumsphere (of dome) that you want to design. Change the number highlighted in blue to obtain the coordinates of the dome of desired diameter. Great circle arcs can be plotted between the vertices. A structural analysis software package such as RFEM-6, SACS or SESAM may be used for plotting the arcs of the great circle. The figure below shows the arcs generated for a dodecahedron modeled in RFEM-6.
Class of the Polyhedron
The class defines the method and geometry of the subdivision. Class I subdivisions run parallel while Class II subdivisions are perpendicular to the edge. Class III subdivisions are between Class I and Class II.
Combination of Frequency and Class
The meaning of Frequency and Class of a Polyhedra are best explained with the illustrations given below. One triangular face of an Icosahedron is subdivided into 8 parts with orientation of the subdividing segments parallel (Class I), perpendicular (Class II) and at an angle (Class III) [2].
Design of a Geodesic Dome
A top-level design specification for a hemispherical dome should then consist of 5 important defining elements which are (1) The base platonic solid, (2) Dimension of the Circumsphere, (3) The frequency (4) The Class (I, II or III) and (5) Requirement to be Hemispherical or Truncated.
The higher the frequency the more curved the appearance of the dome. For even frequency domes the triangulation can be equally divided into 2 hemispheres. For odd frequency domes, the great circle does not lie on the edge of the base polygon. Hence a perfect hemisphere cannot be obtained. Truncation is provided based on the specified design height.
Example of a Hemispherical Dome
Let us say I would like to construct a perfectly hemispherical dome. Hypothetically, I could inform my fabricator of the 5 primary elements listed below.
- Base platonic solid shall be an Icosahedron.
- The radius of the circumsphere shall be 5m.
- The frequency shall be 6V.
- The triangulation shall be Class I
- The dome shall be a perfect hemisphere.
The picture below shows the illustration of the hemisphere based on the above requirements. In future blog posts I will demonstrate how to design this structure using glass panels and an aluminum frame.
Summary
In the above sections I have described the basic theory of Polyhedra and how they are used to generate the geometry of Geodesic Domes. The basic specifications for describing a dome are provided. In coming blogposts, I intend to provide a full design for a geodesic dome, made of glass panels and an aluminum frame. The illustrations were developed using RFEM-6 and Stella4D-Pro software packages.
Disclaimer
This blogpost is for information purposes. All terms and conditions stated in the disclaimer page shall apply.
References
[1] Euler’s Gem, The Polyhedron Formula and the Birth of Topology, David S. Richardson, Published in 2008 by Princeton University Press, ISBN: 978-0-691-12677-7.
[2] Divided Spheres, Geodesics & the Orderly Subdivision of the Sphere, Edward S. Popko with Christopher J.Kitrick, Second Edition 2022, published by CRC Press, ISBN: 9780367680039
Software Packages
[1] RFEM-6
[2] Stell4D-Pro
Free Downloads
[1] Excel Sheet to create Dodecahedrons and Icosahedrons. Coordinates of a Dodecahedron and Icosahedron