four faces meet at
four vertexes,
with three
3-sided faces to a vertex
and six edges.
The Tetrahedron is the shape of the atomic geometry of Carbon—the backbone of organic molecules, and thus a primary geometry of biology's molecular architecture. Carbon can make four electron bonds of equal energy, so its electron orbits create the tetrahedral geometry implicit in all organic molecules, including sugars, amino acids and lipids. Hydrocarbon, carbohydrate and protein all incorporate the Tetrahedron's simplest geometry at their structural core.
The Octahedron is the next step up in size and complexity in 3-D geometry: four 3-sided faces arranged around a single vertex. A pyramid is half of an octahedron. Octahedron's
eight faces meet at
six vertexes,
with four
3-sided faces to a vertex
and 12 edges.
Place a point in the center of each triangular face of the octahedron, then connect the points together. The resulting figure is a familiar four-sided figure—the Cube.
The Hexahedron is the geometric complement to the Octahedron, in which faces of one become vertexes of the other, and vice versa. This is the Cube—the proverbial "box"—the common, everyday geometry of our man-made, rectilinear reality. Cube's
six faces meet at eight vertexes,
with three 4-sided faces to a vertex
and 12 edges.
These first three 3-D forms are rather simple and familiar, and not too challenging to imagine and manipulate. But now you must the step beyond 4-square reality and make the leap into 5-sided symmetry. In pentagonal reality, nothing seems to meet at normal, right angle edges anymore. In our four-square, right angle culture, these next two 5-sided shapes are unknown to most people, and too complex to contemplate.
The Icosahedron is the next step up in geometric size and complexity, with five triangles arranged around each vertex. This beautiful but confusing figure has
20 faces meet at
12 vertexes,
with five
3-sidedfaces to a vertex
and 20 edges.
The Dodecahedron is the fifth and final regular solid of 3-D geometry. The Dodecahedron is the geometric complement of the Icosahedron—faces of one become vertexes of the other, and vice versa. Dodecahedron's
12 faces meet at
20 vertexes,
with three
5-sided faces at each vertex,
and 20 edges.
In sports, a soccer ball is a truncated dodecahedron (a hexa-dodecahedron).
In geology, garnet crystals portray this Dodecahedron shape.
In biology, DNA's twin spiral staircase is formed by two 5-carbon sugars (ribose) of a racheting Dodecahedron. Thus, in a virus, both its protein shell and its DNA contents derive from PHI-ratio, 5-sided symmetry.
The triangles of the Tetrahedron and Octahedron create structure. But the 5-sided forms of Icoscahedron and Dodecahedron enclose, contain and embrace space.
If each shape has edges of equal length, the Dodecahedron encloses the largest space of the five. Yet, like a Cube, the Dodecahedron has no triangles to lock it into rigid form. Thus, the Dodecahedron is unstable, and easily collapses.
PHI-ratio recursion in 3-D creates fractal nests of alternating Dodecahedra and Icosahedra—impossible to depict in a 2-D diagram, or for ordinary mortals to imagine.
Four into Five
In 3-D
Even in this 3-D complexity, the "union of square and circle" still applies. Draw diagonals on the faces of a dodecahedron (white), and a cube (gold) appears nested inside. Each edge of the cube is in PHI ratio to the dodecahedron's edges. And thus, ordinary 3-D, 4-square reality can be enfolded and nested in fractal PHI-ratio, 5-sided symmetries.
Five different cubes can nest inside one dodecahedron. The nest of cube-in-dodecahedron reveals again PHI's link between our 4-square space and 5-sided living architectures. Each cube edge is a PHI-ratio to a pentagon side.
Icosahedron and dodecahedron of equal radii nest together to form a stable, rigid form composed of interlocking triangles.
Carbon atoms can pack together in flat sheets (graphite) and crystals (diamonds). Carbon also forms large, open structures with 60 atoms in this 5-sided, icosa-dodeca shape. The 1996 Nobel Prize for Physics was awarded to the discoverers of this third form of carbon: C60, named "Fullerene" (or "Bucky Balls") for Buckminster Fuller.
If this shape is rotated 180 degree and combined with the original, the edges of these two icosa-dodeca meet in a match to form a nest of full—or "great"—circles and 120 right triangles. This is the Universal Grid which Buckminster Fuller proposed to map the surface of a sphere.
In biology, this is the geometry of a cell membrane—the primary shape of biology.
In geology, this is the Earth Grid. Earth's major mountain ranges, mid-ocean ridges, continental faults, tectonic plates, and other geophysical features align along this global grid. So do sacred sites, holy places, principal cities, and transportation routes of human culture.
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In 3-D, the simplest shape we can make is the Tetrahedron. This regular shape is built with four equilateral triangles, arranged in three's around a single point—or "vertex." Tetrahedron's