Geometrical construction of Solids:
Concepts in the study and understanding of 3D geometric forms are based on ideas developed by Keith Critchlow in his book ‘Order in Space – A Design source book’, Thames and Hudson, (1969).
He argues that the primary idea of order and number is one of the ways of understanding our universe. He proposes that the fundamental element of this cosmos is space. Since its nature is emptiness and because it is empty it can contain and embrace everything. It is therefore necessary that to understand the tangible concepts such as '3-Dimensional form' is best understand it through the perspective of space.
For example:
if we were to regard the point as physically real, then it can be visualized and seen to occupy a position in space. By manipulation one can then study its behavior singly or collectively. Based on this presumption, concepts developed in this section attempt to bring understanding to the concepts in the configurations and construction of 3 Dimensional form.
Some Definitions:
• Point:
We visualize the point to be a minute version and see what volume it describes by tracing it systematically through space. Sphere is the most suitable form to give to the ‘point’ as it has complete rotational symmetry and is least biased. Point can be referred to as a ‘spherepoint’.
• Line:
If a point moves in an unchanging direction, from a starting position, a trace of its path is called a ‘line’.
• Plane:
Moving the line in any other direction than the first direction describes the planar trace.
• Solid:
The trace of the third change in the direction describes ‘solid’ .
The economic unfolding of the dimensions of space can be visualized based on the concept of spherepoint as shown below.
There are three fundamental ways in which the three moves can be made:
(1) Tetrahedron:
• The most economical four-faced pyramid (the solid Pyramid).
• Strongest of the solids, being most able to resist the external forces from all direction.
• It has the greatest surface area for volume of all polyhedra.
(2) Cube:
• Transitional phase between tetrahedron and sphere.
• Is a ‘sociable’ and close-packing unit.
(3) Sphere:
• This is formed by cyclic movement or rotation through each dimension.
• Has the least surface area for volume.
• Most suitable for restraining internal forces.
Platonic Solids: A Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex. Thus, all its edges are congruent, as are its vertices and angles.
There are five (and only Five), polyhedra that fall into the category of platonic solids. While an infinite number of polygons may be drawn on a plane surface, it is not possible to construct more than five regular polyhedral in three-dimensional space. These being the tetrahedron; the octahedron; the icosahedron, the cube and the dodecahedron.
Platonic Solids derive their name from Plato because of his efforts to relate them to the important entities of the universe. The tetrahedron represented molecule of fire, the octahedron the molecule of air, the icosahedron the molecule of water and the cube the molecule of earth, while the dodecahedron represented the all-containing ‘ether’ or the heavens
Entities of the Platonic Solids:
(1) Tetrahedron:
• Edges: 6
• Vertices: 4
• Faces: 4
• 4 Equilateral Triangles
• Symmetry: 2, 3, 3 - Fold
• Dual: Tetrahedron
(2) Cube:
• Edges: 12
• Vertices: 8
• Faces: 6
• 6 squares
• Symmetry: 2,3,4-fold
• Dual: Octahedron
(3) Octahedron:
• Edges: 12
• Vertices: 6
• Faces: 8
• 8 equilateral triangles
• Symmetry: 2.3.4-fold
• Dual: Cube
(4) Dodecahedron:
• Edges: 30
• Vertices: 20
• Faces: 12
• 12 pentagons
• Symmetry: 2,3,5-fold
• Dual: Icosahedron
(5) Icosahedron:
• Edges: 30
• Vertices: 12
• Faces: 20
• 20 equilateral triangles
• Symmetry: 2.3.5-fold
• Dual: Dodecahedron
**
Dual: The line joining the centre-point of the faces of one of the figures results in the other figure.
Evolution of the Basic Spherepoint Configurations:
A. Four spheres in tetrahedral configuration are the greatest number that can be in simultaneous contact.
B. The tetrahedron, outlined on its edges, with a second set of spheres introduced into the interstices; eight spheres in all.
C. The second set of spheres shows that the tetrahedron is its own dual – i.e. the lines joining the centre points of the faces repeat the original figure.
D. The next most regular grouping of spheres is six in octahedral configuration; each sphere touches four others.
E. The octahedral group outlined with edges, with eight additional spheres in the interstices.
F. When the edges of the second set of spheres are outlined, the cubeemerges as the dual of the octahedron; the lines joining the centre point of the faces of the octahedron results in a cube. Conversely, octahedron is the dual of the cube.
G. The closest packing of equal spheres around a nucleus of equal size gives the dymaxion or cuboctahedron. The nuclear sphere is surrounded by twelve spheres, each touching four neighbours in addition to the nucleus.
H. The grouping without the nucleus tends to close into the triangulation of the icosahedral grouping; twelve spheres are in closer configuration, each touching five others.
I. The icosahedron, with its edges outlined, shown on its 5-fold axis with a sphere introduced into each interstice – 32 spheres in all.
J. The added set of spheres, when outlined, shows that the regulardodecahedron is the dual of the icosahedron. This demonstrates a hierarchy of the five regular or Platonic solids by the criteria of numerical and structural economy.
From the arrangements of spherepoints , the following principles can be adduced:
• 4 equal spheres are the greatest number that can be in simultaneous contact- the first regular pattern;
• 6 equal spheres are the next regular pattern, with each sphere touching four neighbours;
• 12 equal spheres may surround and touch a nucleus of equal size.
Introducing additional spheres into the interstices of the three regular triangulated patterns generates the dual solid of each. In the first case, the tetrahedron is its own dual; in the second case, the cube is the dual of octahedron; and in the third case, the regular dodecahedron is the dual of icosahedron. This provides five regular solids from three triangulated close packing of equal spheres by the introduction of a second set of spheres in their interstices.
Yet another way of exploring the hierarchy of solids, adopting the same principles of economy is using the points of contact between the spherepoints rather than their centres,
In drawing 1 we see that if the six point of contact (A,B,C,D,E and F) between the first four spherepoints are joined (E and F being furtherest from and nearest to the eye respectively). The result is the octahedron, a figure composed of eight equilateral triangles, As the apices of theoctahedron are exactly half-way along the edges of the basic tetrahedron formed by the joining the centres of the spherepoints, we can regard the octahedron as the first ‘octave’ subdivision of tetrahedron.
In drawing 2 we see that if the octahedron is isolated and its apices are simultaneously expanded to become spherepoints in closed-packed relationship one to another, then the links linking the twelve points of contact of these spherepoints (a,b,c,d,e,f,g,h,I,j,k,l) form the figure called thecuboctahedron or dymaxion, which is made up of a total of 24 edges describing 8 equilateral triangles and 6 squares. The distance between its apices is identical to that from any apex to the centre of the configuration.
Drawing 3 shows that if the cuboctahedron is isolated and the apices expanded as the spherepoints as before, the resulting figure is seen to be stable on only eight faces-the triangular relationships- and unstable on only 8 faces-the square relationships. Triangulation is incomplete. The figure is in equilibrium, but it is unstable because it has no nuclear sphere.
Drawing 4 shows that without this nucleus, the spheres tend to close into a totally stable, triangulated position providing the figure called the icosahedron (a,b,c,d,e,f,g,,h,j,k,l) which is made up of 30 edges and 20 triangular faces. This is the third and final regular triangular close-packing pattern of equal spheres- regular meaning that all planar, linear, and angular relationships are equal.
It can be seen from the drawings of the transpositions that there are 3 sizes of spherepoints shown, each one half of the preceding it – 3 octave subdivisions. Hence the primary solids have been shown in yet another way to be related in a hierarchy of occurrence.
Exploring the relationships between the points of contact of the six spheres making up the octahedral pattern and the twelve spheres making the icosahedral pattern further, we find:
• The 12 points of contact of six equal spheres resulted in the cuboctahedron or dymaxion, made up from the eight triangular faces and six square faces. Regularly assembled, the six square faces make up the cube and the 8 triangular faces constitute the octahedron.
• The 30 points of contact of 12 equal spheres result in an icosidodecahedron, made up of 20 equilateral triangles and 12 pentagons which if regularly assembled would have formedicosahedron dodecahedron respectively.
Hierarchy between the solids could also be established by n-fold symmetry:
The tetrahedron is allocated the first sphere. The next two spheres are allocated to Octahedron and icosahedron as they are prime representatives of 2,3,4 and 2,3,5–fold symmetry respectively. 3 spheres containing the three triangulated inherently structural platonic figures are grouped together. If the two pairs of spheres, containing the prime and secondary representatives of the 2,3,4 and 2,3,5-fold symmetries, i.e. octahedron and cube with icosahedron and dodecahedron, are placed in close packing, the relationship between the four spheres provide the tetrahedron completing the cycle and establishing it again as the master or ‘over’ solid.
The prime and secondary representatives of the 2, 3, 4-fold symmetrical figures are duals. Similarly, the prime and secondary representatives of the 2, 3, 5-fold symmetrical figures are duals.
Only seven polygons or shapes singly or combination are needed to define the primary ‘orders’ of ‘surface’ or of solid space. These 3,4,5,6,8,10,12-sided polygons are shown together here in the large diagram ; they are all generated from two primary circles , with a common radius A, which is the edge length common to all the polygons. The first polygon is made up by linking A and B to the point of intersection of the primary circles, forming an equilateral triangle. This is the only polygon whose surface is totally enclosed within the common areas of the two primary circles (it is also the only inherently ‘stable’ structural shape). The subsequent polygons are generated by the progressive unfolding of the sides of the primary triangle.
There are thirteen possible semi-regular solids. Each edge of a semi-regular solid is the same length and is characterized by the centre angle subtended by an edge at the centre of the enclosing sphere commonly denoted theta.
The group of drawings shows the semi-regular solids and their surfaces arranged in two parallel columns as successive truncations of the
icosahedron, on the left, and, on the right of the
octahedron.
The surfaces or polygons indicated on the outer edges of the three rows of shapes are the shapes; they represent the amount of surface of the original solid left after truncation in each successive case. The column of surfaces shown vertically shaded is the new faces formed by truncation. The surfaces shown dotted are edges opposed to corner truncations. Only in three instances does this result in an additional shape.
The arrangement is in order of hierarchy from most parent face to least after each successive truncation. An alternative arrangement would ensure if the cube and dodecahedron were taken as parent solids.
A single nuclear sphere totally surrounded by equal and similar spheres has only twelve degrees of freedom for other spheres to touch it simultaneously. This arrangement of thirteen spheres in its most regular pattern is known as the
cuboctahedron or
dymaxion. On the two previous pages it has been shown that there are six truncations respectively of the regular octahedron and icosahedrons. These twelve figures together with the one possible truncation of the regular tetrahedron give the thirteen finite Archimedean solids. The definition of an Archimedean or semi-regular solids requires that all its vertices lie in the surface of a circumscribing sphere. Thus the thirteen circumspheres of the Archimedean solids can be grouped in regular cuboctahedron or dymaxion pattern that is the twelve spheres representing the possible truncations of the octahedron or icosahedron around the unique truncation of the tetrahedron. This truncation of tetrahedron has the opposite property of reflecting the twelve degrees of freedom in being the only semi-regular solid figure with twelve.
As with the platonic or regular solids, there are duals to each of the Archimedean or semi-regular. The center-point of each face of each of the Archimedean solids is the vertex of the dual. Each of the solids and its dual come into correspondence when the centre points of their respective edges (crossing at 90 degrees) lie in the surface of a common sphere, known as the intersphere. The full numerical value of the inspheres, interspheres and circumspheres for these figures.
Figure 2 shows the twelve spheres close packed around an equal and similar nuclear sphere.
Figure 3 shows the twelve spheres closed into icosahedral pattern after the removal of the nucleus.
In figure 4, the icosahedral grouping is shown expanded, each of the spheres containing the respective dual of the Archimedean solids represented in figure 1.
The six duals of the octahedral family are shown in the top half of the diagram, those of the icosahedral family in the bottom half.
In the configuration the two families of symmetry separate out into an above and below reflection of six.
