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We will now move onto the final Platonic solid, the Dodecahedron.

In this article we will also cover the associated Archimedean and Catalan solids of the dodecahedron.  These include the:

  • Truncated Dodecahedron & Dual: Triakis Icosahedron
  • Snub Dodecahedron & Dual: Pentagonal Hexecontahedron
  • Icosidodecahedron & Dual: Rhombic Tricontahedron
  • Truncated Icosidodecahedron & Dual: Disdyakis Triacontahedron
  • Rhombicosadodecahedron & Dual: Deltoidial Hexecontahedron

 

 

The Dodecahedron – 6480°

The dodecahedron is the most elusive Platonic solid.

It has:

  • 12 regular pentagonal faces
  • 30 edges
  • 20 corners

 

There are 160 diagonals of the dodecahedron.  60 of these are face diagonals.  100 are space diagonals (a line connecting two vertices that are not on the same face).

The dodecahedron expresses the principle of equality in all directions (that arises from the Monad, or sphere), as does all the Platonic solids.

It is associated with Aether, Spirit, Consciousness, Quintessence, or the Heavens.

Plato said of the dodecahedron, “There remained a fifth construction which God used for embroidering the constellations on the whole heaven.”

Translated another way it reads, “There remained one further construct, the fifth; the god decorated it all over and used it for the whole.”

 

As Michael Schneider writes, “It isn’t above the four elements, the configured states of matter, but encompasses them, infusing the force of life and excellence throughout their structure.”

 

It was called “the sphere of the twelve pentagons” by Plato.

“Like a pomegranate ready to burst forth its seeds, the dodecahedron represents the archetype of life and fecundity made visible.  It’s a three-dimensional pentagonal web on which life expresses its fullness.”1

The dual of the dodecahedron is the icosahedron.

If the dodecahedron has edge length = 1, the associated dual icosahedron will have edge length = phi.

The Sum of the dodecahedron’s angles = 6480º.

 

 

Projections of the Dodecahedron

 

 

Spherical Close-Packing

A Dodecahedron can be built from 32 Spheres

The Dodecahedron – 32 spheres – The dodecahedron is the dual of the icosahedron.  The dodecahedron can be seen here.

It began with 12 spheres of the Icosahedron.

20 more spheres are introduced into the interstices to create the dodecahedron.

 

The Dodecahedron in the Fruit of Life and Metatron’s Cube:

 

The Spherical Dodecahedron:

 

The Stereographic Projection of the Dodecahedron:

 

Volume and Surface Area of the Dodecahedron

Volume = √5/2 φ4s3                or         V = ¼ (15 + 7√5)s3                 s = side length

 

Surface Area = 3(√25+10√5s2)                                                         s = side length

 

Note, if all 5 Platonic solids are built with the same volume, the dodecahedron will have the shortest edge lengths.

 

A dodecahedron sitting on a horizontal surface has vertices lying in four horizontal planes which cut the solid into 3 parts.

Each of the three parts are equal in volume.  Each is 1/3 of the total.

 

When set in the same sphere, the surface area of the icosahedron and dodecahedron are in the same ratio as their volumes – their inspheres are also identical.

 

Here is the dodecahedron and its insphere:

 

Here is the dodecahedron and its circumsphere:

 

 

The Net of the Dodecahedron

The Dodecahedron has 43,380 nets.  Here is one of the most common:

 

 

Phi in the Dodecahedron

Credit: Rafael Araujo

 

12 of its 20 vertices are defined by 3 perpendicular φ2 rectangles.

The remaining 8 vertices are found by adding a cube of edge length φ.

See page 152 in the Quadrivium.

 

“In addition, the center of each face of the regular dodecahedron form three intersecting golden rectangles.”2

 

Rectangles drawn inside a dodecahedron are φ2:1 (or 1:φ2)

Credit: page 341 in Designa.

 

Below is an image with the 3 golden rectangles in the dodecahedron and icosahedron.

 

 

Angles of the Dodecahedron

Sum of its Angles = 6480°

1 Pentagon = 540°;   540 x 12 = 6480°

720 x 9 = 6480 = sum of the angles of a dodecahedron

 

Tetrahedron (720) + Octahedron (1440) = Dodecahedron (6480)

 

 

The Dodecahedron in Chemistry and DNA

The dodecahedron contains symmetry of perfect self-embedding & self-similarity and is used for these purposes.  This is specifically seen in DNA.  All the following topics are covered extensively in the Science section of Cosmic Core.

 

In Dr. Robert Moon’s work the dodecahedron is associated with the element palladium.

The dodecahedron: (26 + 20) 46 corners = 46 protons.  Element = palladium.

Uranium (92) is also associated with the dodecahedron but would be two dodecahedra side-by-side.

 

 

The dodecahedron is also found in the structure of quasicrystals, such as the Ho-Mg-Zn quasicrystal.

 

 

The dodecahedron is also found in the structure of garnet and diamond, though this is usually rhombic dodecahedral.

Dodecahedral Garnet

Rhombic dodecahedral Boracite

 

 

The dodecahedron shows up all over the place in DNA.

The braided thread of the DNA helix is a ratcheted dodecahedron.  The helix is a formed from ten regular pentagons oriented about a decagon.

A complete turn of the helix is formed by progressive rotation and extension of the ten regular prismatic pentagons.

Credit: Dr. Mark White

Furthermore, Dr. Mark White discovered that the 20 edges of a dodecahedron can be used to represent the 20 standard amino acids in DNA and show the complex relationship of how they all fit together.

 

 

The Dodecahedron in Nature

All the following topics are covered extensively in the Science section of Cosmic Core.

The dodecahedron shows up as the structure in several diatoms and radiolaria.

Braarudosphaera pentagonica

Braarudosphaera Bigelowii

 

 

The phi ratios inherent in the dodecahedron are seen all throughout nature from plants, to insects, to animals, to the human body.

 

 

The dodecahedron is also seen in the global grid as a complement to the icosahedron.

 

 

The rhombic dodecahedron is seen in the large-scale structure of galactic clustering, and some believe the universe itself is shaped as a dodecahedron.

Credit: Conrad Ranzan

 

 

Dodecahedral Shape of the Universe

An article in National Geographic from October, 8, 2003 asks if the universe is shaped as a dodecahedron.3

This study of astronomical data (density fluctuations in cosmic background radiation) hints that the universe is finite and bears a rough resemblance to a dodecahedron, a 12-sided volume bounded by pentagons.

 

The paper in Nature by Jean-Pierre Luminet, Jeffrey Weeks, Alain Riazuelo, Roland Lehoucq, Jean-Phillip Uzan states in its abstract:4

“The current ‘standard model’ of cosmology posits an infinite flat universe forever expanding under the pressure of dark energy. First-year data from the Wilkinson Microwave Anisotropy Probe (WMAP) confirm this model to spectacular precision on all but the largest scales1, 2. Temperature correlations across the microwave sky match expectations on angular scales narrower than 60° but, contrary to predictions, vanish on scales wider than 60°. Several explanations have been proposed3, 4. One natural approach questions the underlying geometry of space—namely, its curvature5 and topology6. In an infinite flat space, waves from the Big Bang would fill the universe on all length scales. The observed lack of temperature correlations on scales beyond 60° means that the broadest waves are missing, perhaps because space itself is not big enough to support them. Here we present a simple geometrical model of a finite space—the Poincaré dodecahedral space—which accounts for WMAP’s observations with no fine-tuning required. The predicted density is Ω0 ≈ 1.013 > 1, and the model also predicts temperature correlations in matching circles on the sky.”

Fluctuations of the Cosmic Microwave Background as observed by the WMAP Satellite

 

The Dodecahedron structure inside the WMAP sky found by F.Roukema et all. Image from: The optimal phase of generalized Poincar’e dodecahedral space hypothesis implied by the spatial cross-correlation function of the WMAP sky maps.  

 

 

Please note the link to this article has sadly been removed.  You can find it on the Way Back Machine here: https://web.archive.org/web/20031010171053/http://news.nationalgeographic.com/news/2003/10/1008_031008_finiteuniverse.html

 

 

Stellations of the Dodecahedron

There are 3 stellations of the dodecahedron.  These are the:

  • Small Stellated Dodecahedron
  • Great Dodecahedron
  • Great Stellated Dodecahedron

These stellations are three of the four Kepler-Poinsot Polyhedra.

 

 

The Small Stellated Dodecahedron

The small stellated dodecahedron has 12 pentagrammic faces and 125 faces by sides.

It has the same topoolgy as the pentakis dodecahedron but with much taller isosceles triangle faces.

Net of the Small Stellated Dodecahedron

 

The shape can be constructed by using 12 of the following shapes, folding them into 5-sided pyramids, and connecting them together as you would a regular dodecahedron.

Here is the truncation sequence from the small stellated dodecahedron to its dual, the great dodecahedron.

 

 

The Great Dodecahedron

The Great Dodecahedron is the dual of the small stellated dodecahedron.  It is composed of 12 pentagonal faces (six pairs of parallel pentagons), with five pentagons meeting at each vertex.

Net of the Great Dodecahedron

 

It can be constructed by folding together 20 of the following shapes and arranging them like the faces of an icosahedron.

The Great dodecahedron has the same topology as the triakis icosahedron, with concave pyramids instead of convex ones.

Triakis Icosahedron

 

 

The Great Stellated Dodecahedron

The great stellated dodecahedron is composed of 12 pentagrammic faces with three pentagrams meeting at each vertex.

It shares its vertex arrangement with the regular dodecahedron, and it is a stellation of a smaller dodecahedron.

It is related to the triakis icosahedron, but with much taller isosceles triangle faces.

It’s dual is the Great Icosahedron, one of the 58 stellations of the icosahedron.

The Great Icosahedron

 

To build the great stellated dodecahedron fold 20 of the following shapes into points and arrange them like the faces of an icosahedron.

 

 

The Dodecahedron and the Tetrahedron

A tetrahedron nests within the dodecahedron.  See below how the corners of the tetrahedron fall at vertex points of the dodecahedron.

Five full tetrahedra will fit inside a dodecahedron in this way.  This is called the Compound of Five Tetrahedra.

The dodecahedron has 20 points that it shares with 5 tetrahedra. Note each tetrahedron shares one point with the top pentagon, one with the bottom pentagon, and two with the zigzag of ten points around the middle of the dodecahedron.

If you connect the outer vertexes you will draw the outline of a regular dodecahedron.

To see this, look at the top five points on this shape.  They are each of a different color.  Connecting those five points creates a pentagon.  You can then see how the dodecahedron forms around this compound.

Could a dodecahedron be a single spinning tetrahedron that pauses in 5 positions?

 

 

Compound of Ten Tetrahedra

This compound can be seen as a different stellation of the icosahedron.

Like the Five Tetrahedra compound, it also can be seen as a faceting of a regular dodecahedron.

Each of the ten tetrahedra here are seen in a different color.  It is interesting to see how they naturally interweave.

It also shares the vertex arrangement of a regular dodecahedron, as seen above.

 

 

The Cube and the Dodecahedron

A cube fits within a dodecahedron.

A cube has 12 edges.  The cube will have one edge along each of the twelve faces of the dodecahedron, on which the edge is a diagonal.

Each of the colored lines below represents a different cube found in the dodecahedron.  There are 5 total: red, yellow, green, blue, and black.

The compound of five cubes that forms a dodecahedron is discussed below.

 

dodecahedron side = 1

cube side length = phi

 

You can also see how one can build a dodecahedron by adding 6 ‘roof’ shapes to the 6 faces of a cube.

An outward expansion is required, where each face of the cube sprouts a slanting “rooftop” made of five equidistant lines in order to turn into the dodecahedron.

 

 

The Chamfered Cube and the Dodecahedron

Chamfering a cube is the same as cantellation.  The edges are truncated instead of the corners being truncated.

The cube transforms into the dodecahedron if the edges of the cube are pushed into planes.

Credit: Frank Chester

 

 

A Dodecahedron folds inside out to form a Cube

See here for an animated visualization: https://www.youtube.com/watch?v=PyXyOOR63Ms

 

 

Compound of Five Cubes

The compound of five cubes is composed of 5 cubes as its name suggests.  It is dual to the compound of five octahedra.

Compound of Five Cubes

 

Compound of Five Octahedra

 

It can also be seen as a faceting of a regular dodecahedron.

Each of the 5 cubes can be seen below in a different color.

It has:

  • 30 squares
  • 60 edges
  • 20 vertices

 

 

The Icosahedron & Dodecahedron

The Icosahedron and dodecahedron are duals.  Connecting the centers of the faces creates the dual.

Or, the dodecahedron’s points can be truncated to yield the icosahedron.

They both nest perfectly inside the other as well.

 

 

The Icosahedron & Dodecahedron: Purusha & Prakriti

“Purusha and Prakriti are the eternal creative dichotomy in Hindu mythology.  Purusha is the anthropocosmic, paradigmatic Man or Seed that projects Prakriti, the eternally enchanting Feminine, in order that her womb may give birth to his own embodiment in the world of form.”5

 

The Icosahedron as Purusha

The Hindu tradition associates the icosahedron with Purusha – the seed-image of Brahma, the supreme creator.

Purusha is synonymous with the Monad.  It is the unmanifest Source of Cosmic Consciousness.

Purusha is envisioned as unmanifest and untouched by creation – just as in the following drawings, the icosahedron is untouched by the other forms.

Reference Construction Lesson #41: The Genesis of Platonic Solids.

 

As we said, Purusha is thought of as the ‘Monad’ – the unpotentiated source from which all arises.

The icosahedron is a structure of triangles – three being a dynamic ‘male’ number.

All other volumes arise naturally out of the icosahedron, making it one obvious choice for the first form.

The key is to know the method of how to find the vertices of the first icosahedron – given to us on the radius of a circle and its division by φ.

 

 

The Dodecahedron as Prakriti

Prakriti is the feminine power of creation and manifestation.

Prakriti is thought of as the ‘Dyad’ – the womb of creation, the action that creates and gives form to consciousness.  Within this womb of creation, all shapes and forms are present in potentiation.

It is the Universal Mother – the quintessence of the natural universe.

Prakriti touches all forms with her silent, observing partner.

The dodecahedron has a structure of 5 (3 male + 2 female) = giver of life.

The star born within its pentagon is the configuration of Cosmic Man, the perfecter of life, the Golden Proportion.

 

 

Phi – the Seed; Fire of Spirit

Both Prakriti (dodecahedron) and Purusha (icosahedron) have phi proportions.

“The outer progression, extending into vaster and vaster realms, demarcates the same progression, the same genesis:  icosahedron, the Purusha, generating the dodecahedron, the Prakriti, and within Prakriti the full play of manifested existence.  The whole coagulation is begun by the secret seed which contracts the circle, the infinite, undifferentiated spirit, into the icosahedron.  The seed is phi, the fire of spirit.”6

 

 

The Square Root of 2 – The Way Phi Acts in Nature

The golden spiral is shown in black.  The sequence of root 2 ratios is shown in red.  Phi = transcendent & Square Root 2 = Earthly

 

 

The Star Tetrahedron as Yin/Yang

The star tetrahedron, or stellated octahedron, is seen as the yin and the yang due to its upward pointing tetrahedron and downward pointing tetrahedron.

The tetrahedron is a volume of threeness – a primary symbol of function accompanied by its reciprocal.

 

 

The Octahedron at the Heart of the Star Tetrahedron

Recall that there is an octahedron at the center of the star tetrahedron.

The octahedron symbolizes the crystallization, the static perfection of matter.

It is the diamond at the heart of the cosmic solids.

It is the transformed and clarified lens of light – the double pyramid.

The result of the harmonic interaction of the star tetrahedron gives birth to the cube.

 

 

The Cube as Material Existence

There are four states of matter: earth, air, fire, water.

The cube and star tetrahedron touch (and fit within) the dodecahedron.

 

 

 

Associated Archimedean and Catalan Solids

 

  1. Truncated Dodecahedron – 20880º & Dual: Triakis Icosahedron – 10800º

 

The Truncated Dodecahedron – 20880º

The truncated dodecahedron is an Archimedean solid.

It has:

  • 32 Faces (12 regular decagonal; 20 regular triangular)
  • 60 Vertices
  • 90 Edges

It is formed by truncating the corners off the regular dodecahedron.

 

Projections of the Truncated Dodecahedron

 

The Spherical Truncated Dodecahedron:

 

The Net of the Truncated Dodecahedron:

 

 

The Triakis Icosahedron – 10800

The triakis icosahedron is the dual of the truncated dodecahedron.  It is a Catalan solid.

It has:

  • 60 Faces (isosceles triangle)
  • 90 Edges
  • 32 Vertices

Its faces look as follows:

 

The Net of the Triakis Icosahedron:

 

Projections of the Triakis Icosahedron

The triakis icosahedron is a Kleetope of the icosahedron.  This means that it is an icosahedron with triangular pyramids added to each face.

 

The Spherical Triakis Icosahedron:

 

 

 

  1. Snub Dodecahedron – 20880º & Dual: Pentagonal Hexecontahedron – 12960º

 

The Snub Dodecahedron – 20880º

The Snub Dodecahedron is an Archimedean solid that has two distinct forms that are mirror images (enantiomorphs) of one another.

They have:

  • 92 Faces (12 Pentagons; 80 equilateral triangles)
  • 150 Edges
  • 60 Vertices

The Snub Dodecahedron has the highest sphericity (0.982) of all Archimedean solids.

 

 

Projections of the Snub Dodecahedron

 

“The snub dodecahedron can be generated by taking the twelve pentagonal faces of the dodecahedron and pulling them outward so they no longer touch. At a proper distance this can create the rhombicosidodecahedron by filling in square faces between the divided edges and triangle faces between the divided vertices. But for the snub form, only add the triangle faces and leave the square gaps empty. Then apply an equal rotation to the centers of the pentagons and triangles, continuing the rotation until the gaps can be filled by two equilateral triangles.”7

 

Here is the transition from the rhombicosidodecahedron to the snub dodecahedron:

 

Net of the Snub Dodecahedron:

 

 

Pentagonal Hexecontahedron – 12960º

The Pentagonal Hexecontahedron is the dual of the Snub Dodecahedron.  It is a Catalan solid.

Like the snub dodecahedron, there are two mirror image forms (enantiomorphs).

They have:

  • 60 Faces (irregular pentagons)
  • 150 Edges
  • 92 Vertices

 

Here is the shape of the faces:

The ratio of edge lengths is about 1:1.749…

 

The Pentagonal Hexecontahedron can be constructed by adding pentagonal pyramids to the 12 pentagonal faces of the snub dodecahedron, then adding triangular pyramids to the 20 triangular faces that do not share an edge with a pentagon.

 

Net of the Pentagonal Hexecontahedron:

Projections of the Pentagonal Hexecontahedron

 

The Spherical Pentagonal Hexecontahedron:

 

 

 

  1. Icosidodecahedron – 10080º & Dual: Rhombic Tricontahedron – 10800º

 

Icosidodecahedron – 10080º

The Icosidodecahedron is an Archimedean solid that combines the 12 pentagonal faces of the dodecahedron with the 20 triangular faces of the icosahedron.

It has:

  • 32 faces (20 triangular; 12 Pentagonal)
  • 60 edges
  • 30 vertices

Its edges form 6 equatorial decagons that give a radial projection of 6 great circles.

The Golden Section is embodied by the ratio of the icosidodecahedron’s edge to its circumradius.

 

Defining the icosidodecahedron with 30 equal spheres leaves space for a central sphere that is √5 times as big as the others.

 

The icosidodecahedron is the rectification of both the dodecahedron and icosahedron.  It is the full-edge truncation between both of these solids, just as the cuboctahedron is between the octahedron and the cube.

 

Projections of the Icosidodecahedron

 

The Spherical Icosidodecahedron:

 

The Net of the Icosidodecahedron:

 

 

The Rhombic Triacontahedron – 10800º

The rhombic triacontahedron is the dual to the Icosidodecahedron.  It is a Catalan solid.

It is the most common 30-faced polyhedron.

It has:

  • 30 rhombic Faces
  • 60 Edges
  • 32 Vertices

The ratio of the long diagonal to the short diagonal of each face equals the golden ratio.  Thus the face is called a golden rhombus.

 

The Net of the Rhombic Triacontahedron:

“The rhombic triacontahedron is also interesting in that its vertices include the arrangement of four Platonic solids. It contains ten tetrahedra, five cubes, an icosahedron and a dodecahedron. The centers of the faces contain five octahedra.”8

The cube can transform into a rhombic triacontahedron by dividing its square faces into 4 squares and splitting middle edges into new rhombic faces.  This is seen below:

 

Projections of the Rhombic Triacontahedron

The rhombic triacontahedron has over 227 stellations.

 

The Spherical Rhombic Triacontahedron:

 

 

 

  1. Truncated Icosidodecahedron – 42480º & Dual: Disdyakis Triacontahedron – 21600º

 

The Truncated Icosidodecahedron – 42480º

The truncated icosidodecahedron is also known as the Great Rhombicosidodecahedron.  It is an Archimedean solid.

It has:

  • 62 Faces (30 square; 20 regular hexagon/ 12 regular decagon)
  • 180 Edges
  • 120 Vertices

“The name truncated icosidodecahedron, originally given by Johannes Kepler, is somewhat misleading. If one truncates an icosidodecahedron by cutting the corners off, one does not get this uniform figure: instead of squares the truncation has golden rectangles.”9

 

 

Projections of the Great Rhombicosidodecahedron

 

The Spherical Great Rhombicosidodecahedron:

 

The Net of the Great Rhombicosidodecahedron:

 

 

 

Disdyakis Triacontahedron – 21600º

The Disdyakis Triacontahedron is the dual of the truncated icosidodecahedron.  It is a Catalan solid.

It has:

  • 120 Faces (scalene triangle)
  • 180 Edges
  • 62 Vertices

It has the most faces of any Platonic, Archimedean or Catalan solid.

The faces look as follows:

 

The Net of the Disdyakis Triacontahedron:

 

Projections of the Disdyakis Triacontahedron

Projected onto a sphere, the disdyakis triacontahedron forms 15 great circles.

 

The Spherical Disdyakis Triacontahedron:

This is the same spherical projection as the compound of five octahedra.

 

Here is the Disdyakis Triacontahedron on a dodecahedron:

 

 

 

  1. Small Rhombicosidodecahedron – 20880 & Dual: Deltoidal Hexecontahedron – 21600º

 

The Small Rhombicosadodecahedron – 20880º

The Rhombicosidodecahedron is also called the small rhombicosidodecahedron.  It is an Archimedean solid.

It has:

  • 52 Faces (20 regular triangular; 30 square; 12 regular pentagonal)
  • 60 Vertices
  • 120 Edges

 

The rhombicosidodecahedron is an expanded or cantellated dodecahedron and icosahedron.  This means if you expand the icosahedron by moving the faces away from the origin the right amount without changing size or orientation of the faces, and you do the same to the dodecahedron, then patch the square holes you will get the rhombicosidodecahedron.

This is seen below:

 

Projections of the Small Rhombicosadodecahedron

 

The Spherical Small Rhombicosidodecahedron:

 

The Net of the Small Rhombicosidodecahedron:

 

 

The Deltoidal Hexecontahedron – 21600º

The Deltoidal Hexecontahedron is the dual to the Rhombicosidodecahedron.  It is a Catalan solid.

It is sometimes called a trapezoidal hexecontahedron.

It has:

  • 60 Faces (kites)
  • 120 Edges
  • 62 Vertices

 

Its faces look as follows:

 

The Net of the Deltoidal Hexecontahedron:

 

Projections of the Deltoidal Hexecontahedron

 

The Spherical Deltoidal Hexecontahedron:

 

  1. Schneider, Michael, A Beginner’s Guide to Constructing the Universe, Harper Perennial, 1994
  2. https://en.wikipedia.org/wiki/Regular_dodecahedron
  3. Universe is Finite, “Soccer Ball” Shaped, Study Hints – National Geographic, 8 October 2003, http://news.nationalgeographic.com/news/2003/10/1008_031008_finiteuniverse.html
  4. Luminet, Weeks, Riazuelo, Lehoucq, and Uzan, Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background, 9 October 2003, http://www.nature.com/nature/journal/v425/n6958/full/nature01944.html
  5. Lawlor, Robert, Sacred Geometry: Philosophy & Practice, Thames & Hudson, 1982
  6. ibid.
  7. https://en.wikipedia.org/wiki/Snub_dodecahedron
  8. https://en.wikipedia.org/wiki/Rhombic_triacontahedron
  9. https://en.wikipedia.org/wiki/Truncated_icosidodecahedron

 

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