Icosahedron

Greg Frederick

The regular icosahedron has an irrational volume in both cubes and tetrahedra. Fuller accounted for this irrationality by noting that the icosahedron exists only as discrete shells and lattices; it does not close pack, either with itself or any other regular polyhedron to fill all space. Nature, therefore, had no need to assign it a volume commensurate with any of the other regular polyhedra.

Its isolation is further reinforced by its axes of spin and the 31 great circles they describe. No great circle unique to the icosahedron passes through any point of contact with spheres radially close-packed in the isotropic vector matrix. Six of its great circles pass through no points of contact, including those between spheres arranged into icosahedral shells and lattices. This isolation suggests the icosahedron to be a model for potential energy, charge, and other holding patterns of energy.

The icosahedron of edge length a has the following properties:

  • φ=golden ratio, (1+√5)/2)
  • Face angles: all 60°
  • Dihedral angle: acos(-√5/3) or 180°-asin(2/3) ≈ 138.189685°;
  • Central angle of edge: atan(2) ≈ 63.434949°
  • Volume (in tetrahedra): a³×5φ²√2 ≈ 18.512296
  • Cubic volume: a³×5φ²/6 ≈ 2.18169499
  • A and B quanta modules: n/a
  • Surface area (in equilateral triangles): 20a²
  • Surface area (in squares): 5a²√3
  • In-sphere radius: a×φ²√3/6 ≈ 0.755761
  • Mid-sphere radius: a×φ/2 ≈ 0.809017
  • Circum-sphere radius: a×(√(φ+2))/2 ≈ 0.9510565
The regular icosahedron of unit edge length showing a dihedral angle is 2 times arctan(1+phi), about 138.189685 degrees, a central angle of arctan(2), about 65.434949 degrees, and an edge-to-opposite-edge span that is the golden ratio, phi.
Dihedral and central angles of the regular icosahedron

If the nucleus is removed from a radially close-packed cluster of 13 unit radius spheres (see Vector Equilibrium and the “VE”), the remaining 12 spheres naturally rearrange themselves into the stable configuration of the icosahedron.

A radially close-packed cluster of 12 spheres around a nuclear sphere describe the vector equilibrium. When the nuclear sphere is removed, the 12 spheres rearrange to describe the regular icosahedron.
Twelve spheres close packed around a central nuclear sphere naturally rearrange themselves into an icosahedron when the central sphere is absent.

The icosahedron is famously derived from the golden ratio (φ), i.e., by connecting the vertices of three golden rectangles arranged orthogonally around a common center.

The regular icosahedron constructed from three golden rectangles.
The regular icosahedron constructed by connecting the vertices of three golden rectangles intersecting at 90° around a common center.

The following illustration shows how the dihedral angles of the regular icosahedron relate to the golden ratio.

A large rectangle, a+2b by a+b, divided into three smaller rectangles: b by a+b; a by a+b, and b by a+b. The diagonals of the two smaller rectangles cross at an angle of about 138.189685°.
The icosahedron’s dihedral angle, about 138.189685°, is equal to 2×arctan(1+φ); its supplementary angle (approximately 41.810315°) is therefore equal to 2×arctan(1/(1+φ)), and that angle’s complementary angle (approximately 69.094846°) is therefore equal to arctan(1+φ).

The icosahedron’s circumsphere and mid-sphere radii are the cosines of 18°, (√φ+2)/2, and 36°, φ/2, or the sines of 72° and 54°. The angles 72° and 36° define the golden triangle.

The golden triangle (36°, 72°, 72°,) with a unit base. The two remaining sides each have a length of the golden ratio, phi.
The golden triangle

The two angles of the golden triangle are formed in the icosahedron by a lines drawn perpendicular to the circumsphere or mid-sphere radii and intersecting with a sphere of unit radius, disclosing the relationship between the icosahedron, the unit-radius sphere, and the vector equilibrium (VE).

Two regular icosahedra of unit edge length. A line drawn from the center to a vertex (left) forms the circumsphere radius of √(φ+1)/2. A line drawn from the vertex perpendicular to the radius intersects the unit sphere at an angle of 18 degrees from the radius. A line drawn from the center to mid-edge (right) forms the mid-sphere radius of φ/2. A line drawn from mid-edge perpendicular to the radius intersects the unit sphere at an angle of 36 degrees from the radius.
Perpendiculars drawn from the circumsphere radius (left) and mid-sphere radius (right) of the regular icosahedron intersect with the unit sphere (radius = 1) at angles of 18° and 36° with the radius.

The 36° and 72° angles also show up in the face angles of the polyhedron that complements the icosahedron in the jitterbug transformation, i.e., when the VEs have contracted into the shape of the regular icosahedron, the octahedra have expanded into the shape of its complement, and vice versa. (See: Icosahedron Phases of the Jitterbug.)

Two edge-bonded regular icosahedra viewed from the front (left) and the sides (right) leave an angle of 108 degrees (left) and 36 degrees (right) between them, angles which match angles of the concave and convex faces respectively of the icosahedron's matrix-filling complement which nestles between them.
The regular icosahedron (gray) and its complement (pink) emerge together in the jitterbug, with the expanding or contracting VEs forming one, and the expanding or contacting octahedra forming the other.

The concave forms of the regular icosahedron and its space-filling complement have, despite their different shapes, identical face angles: eight equilateral triangles (60°, 60°, 60°) and twelve isosceles triangles of 36°, 36°, 54°.

A line connecting the opposing vertices of two adjacent faces of the regular icosahedron (left) forms a concavity of two isosceles triangles identical to those formed by the icosahedron's complement: 36°, 108°, and 36°.
The concave forms of the regular icosahedron and its space filling complement have identical face angles.

We can close up the cavities to create six irregular tetrahedra arranged orthogonally around their common centers. Curiously, the tetrahedral cavities of both the regular icosahedron and its complement have identical, rational cubic volumes of a³/12, or a combined cubic volume of a³/2, where a is the edge length.

Six irregular tetrahedra occupy the concavities of the concave regular icosahedron (left) and its concave complement (right).
Six irregular tetrahedra occupy the cavities of the concave regular icosahedron (left), and its concave complement (right). The combined cubic volume for both is exactly a³/2, where a is the edge length.

The in-sphere radius makes an angle with the mid-sphere radius of atan(2-φ), or about 20.905157°. Some tedious trigonometry works out its length to be φ²√3/6 or about 0.75576131.

The regular icosahedron constructed of three golden rectangles with the in-sphere radius drawn from the center to mid-face. The radius crosses the long edge of each golden rectangle exactly one unit length (a) in from the rectangle's far short edge, and length b in from the near short edge. The golden ratio states that a/b = (a+b)/a
In-sphere radius of the icosahedron

The central angle, i.e. the angle between two adjacent vertices of the icosahedron and its center of volume, is the same as the angle from a square’s mid-edge to one of its opposite corners, atan(2), about 63.43894882°.

A square of edge length 2a superimposed on a regular icosahedron whose circum-sphere diameter form one edge of the square. A line drawn from center of the icosahedron to a far corner of the square passes through a vertex and forms an angle identical to the icosahedron's central angle, arctan(2), or approximately 63.438949°.
The angle between two adjacent vertices of the icosahedron and its center of volume is the diagonal of a double square.

Fuller sometimes referred to the six-strut tensegrity sphere as the “tensegrity icosahedron.” Mathematicians are now calling it the Jessen Orthogonal icosahedron. See: Jessen Orthogonal Icosahedron and Tensor Equilibrium, and The Jessen Orthogonal Icosahedron in the Isotropic Vector Matrix. It would be perhaps more accurate to say that it approximates the shape of an octahedron (eight triangles, with six vertices meeting at mid-strut). But I prefer to call it “the tetrahedron in its tensor equilibrium phase.” It is also the shape that the VE and octahedron describe precisely midway through the jitterbug transformation of the isotropic vector matrix. See: Tensegrity; Tensegrity Equilibrium and Vector Equilibrium; Jitterbug, and; Icosahedron Phases of the Jitterbug.

The six-strut tensegrity sphere transforming into its polyhedron state, the tensegrity tetrahedron, by pulling its four triangular vertex loops in tight.
The six-strut tensegrity sphere, sometimes called the “tensegrity icosahedron” or “Jessen Othogonal Icosahedron”, is really a tetrahedron in its tensor equilibrium phase.

The regular icosahedron in its tensor equilibrium phase (left in the figure below) approximates the shape of the icosidodecahedron.

The 30-strut tensegrity in its spherical state (left), and in its polyhedron state, the tensegrity icosahedron (right).
The icosahedron in its tensor equilibrium phase approximates the shape of the icosidodecahedron (left).
The 30-strut tensegrity sphere transforming into its polyhedron state, the tensegrity icosahedron, by pulling its 12 pentagonal vertex loops in tight.
All structure is fundamentally reducible to tensegrity, and this applies to the regular icosahedron, shown here transforming between its tensor-equilibrium and polyhedral phases.

The icosahedron has an irrational volume and so is incommensurate with the A and B quanta modules.

The regular icosahedron can be constructed from a single paper strip of twenty equilateral triangles.

Illustration of the strip, divided into 20 equilateral triangles, from which the regular icosahedron may be constructed.

The construction is accomplished by 19 sequential folds of atan(2√5/5).

Animation of a regular icosahedron constructed by the sequential folding of a single paper strip.
The regular icosahedron may be constructed from a single paper strip.

The Raleigh Edition of the Dymaxion Airocean World Map (copyright 1954) is a variation of the cartographic projection method Fuller patented in 1946. Earth’s surface is mapped onto the planar facets of the regular icosahedron by a method Fuller called “triangular geodesics transformational projection.” It is one of the most distortion-free maps of the earth’s surface every devised. Great circles approximate straight lines, and its modules can be rearranged to reveal relationships and orientations not otherwise apparent on more conventional maps.

Buckminster Fuller's Dymaxion Airocean World Map
Fuller’s Dymaxion Airocean World Map, copyright 1954.

The map folds neatly into a regular icosahedron.

Buckminster Fuller's Dymaxion Airocean World Map folded into a regular icosahedron.
Fuller’s Dymaxion Airocean World Map folded into a regular icosahedron.

All the structural (i.e. triangulated) polyhedra may be constructed from an even number triangular helices, or what Fuller called one half of a structural quantum. The tetrahedron and icosahedron (but not the octahedron) require an equal number of clockwise and counter-clockwise helices.

The two helices each occupy their own hemisphere, with exclusively clockwise helices in one hemisphere, and exclusively counter-clockwise helices in the other. Each radiate from diametrically opposed vertices, in effect polarizing the icosahedron on one preferred axis of spin. See also: Isotropic Vector Matrix as Transverse Waves.)

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