Unfortunately, we live in a grander complexityÑvastly and far beyond these simple 2D, one plane, flatland figures. Space has not only length and width, but heightÑand depth. In three dimensions, there are no lines, nor circlesÑonly spheres and spirals.
Since few modern minds know much about 3D geometry, let's review the basics. It is very simple. In 3D, only five regular shapes can be constructed, with equal sides, equal angles and equal faces.
In 2D, the simplest shape is a Triangle. This 3sided form is the strongest, most stable and rigid, and so is the key element to build all other forms. All the regular shapes of 3D geometry are built from equilateral ("equalsided") triangles, with 60 degree interior angles.
The math of 2D is Trigonometry, based on ratios of the sides of right triangles. The simplest right triangle is a
3Ñ4Ñ5, which reveals the fundamental Theorem of Pythagorus, the ancient Greek mathematician who studied Sacred Geometry in Egypt's Mystery Schools:
four faces meet at four vertexes,
with three 3sided faces to a vertex
and six edges.
Tetrahedron is the shape of the atomic geometry of CarbonÑthe 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.
Octahedron is the next step up in size and complexity in 3D geometry: four 3sided faces arranged around a single vertex. A pyramid is half of an octahedron. Octahedron's
eight faces meet at six vertexes,
with four 3sided 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 result is a familiar foursided figure.
Hexahedron is betterknown as the common CubeÑthe proverbial "box"Ñthe everyday geometry of manmade, rectilinear reality. This everyday image of orderly geometry has
six faces meet at eight vertexes,
with three 4sided faces to a vertex
and 12 edges.
Hexahedron is the geometric complement to the Octahedron, in which faces of one become vertexes of the other, and vice versa.
These first three 3D forms are rather simple and somewhat familiar, and are not too challenging to imagine and manipulate. But now you must take the uncertain step beyond 4square reality, and make the leap into 5sided symmetry. After a lifetime in our foursquare, right angle culture, these next two 5sided shapes are unknown to most people, and too complex to contemplate. In pentagonal reality, nothing seems to meet at normal, right angle edges anymore.
Icosahedron is the next step up in geometric size and complexity, with five triangles arranged around each vertex. This beautiful, but complex and confusing figure has
20 faces meet at
12 vertexes,
with five
3sidedfaces to a vertex
and 20 edges.
Dodecahedron is the fifth and final regular solid of 3D geometry. This elegant enclosure is a shape that consists of
12 faces meet at 20 vertexes,
with three 5sided faces at each vertex,
and 20 edges.
Dodecahedron
is the geometric complement of the
Icosahedron
Ñfaces of one become vertexes of the other,
and vice versa.

In sports, a soccer ball is a truncated dodecahedron (a hexadodecahedron).
In geology, garnet crystals portray this dodecahedron shape.
In biology, DNA's twin spiral staircase is formed by two 5carbon sugars (ribose)Ñopposite faces of a racheting dodecahedron. Thus, in a virus, both its protein shell and its DNA contents derive from PHIratio, 5sided symmetry.
Triangles of a tetrahedron and octahedron create structure.
But 5sided shapes 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.
PHIratio recursion in 3D creates fractal nests of alternating dodecahedra and icosahedraÑimpossible to depict in a 2D diagram, or for ordinary mortals to imagine.
Four into Five
In 3D
Even in this 3D 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 3D, 4square reality can be enfolded and nested in fractal PHIratio, 5sided symmetries.
Five different cubes can nest inside one dodecahedron. The nest of cubeindodecahedron reveals again PHI's link between our 4square space and 5sided living architectures. Each cube edge is a PHIratio 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 5sided, icosadodeca shape. The 1996 Nobel Prize for Physics was awarded to the discoverers of this third form of carbon: C_{60}, 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 icosadodeca meet in a match to form a nest of fullÑor "great"Ñcircles 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 membraneÑthe primary shape of biology.
In geology, this is the Earth Grid. Earth's major mountain ranges, midocean 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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