View Full Version : Platonic Solids and the Tree of Life
As discussion of both the Platonic Solids and various renditions of the Tree of Life have been made, I thought I'd add this rather personal exploration (I have a two-metre 'restoration' version I constructed out of a wooden frame I made several years ago, hanging on a wall).
Firstly, the 'pre-fall' version (which is attached).
It one considers the three platonic solids which have triangular faces, they give a total of 12+6+4=22 vertices (the icosahedron has 12, the octahedron 6, and the tetrahedron 4).
In placing these in a connected way one above the other, the visual depiction they give, of their top layer only, is as attached, also with, incredibly, 22 visible edges ('paths').
Note that this version I call the pre-Fall version: the Tree of Knowledge has not as yet been consumed, and thus remains. In fact, the whole Tree has, though connected, three clearly distinct parts, corresponding to the Spirit, Soul and Body, the body sustained by other Spiritual beings at work deep at its core.
Following the fall, the Tree has a different structure, which I'll draw and post later, if anyone is interested.
Here, however, is an original 3-dimensional version of the Tree, consistent with traditional Kabalistic considerations:
JMD Wrote: "...if anybody is interested..."
And I raise my hand and say, YES, definitely!
Hello JMD,
I think your tree before the fall is a fascinating structure. It seems almost like a temple viewed from directly above.
In 3 dimensions it hides 24 edges, which would only come into view if the vantage spot were moved. I would very much like to see the model for after the fall.
In 3 dimensions, I have always had affection for the tetrahedron. One of the main things I find qabalistically pleasing is that a tetrahedron can be folded out of an equilateral triangle.
Seeing kether as point, [c]hockmah as line, and binah as plane, the resulting 3 pointed triangle can be folded into a solid, with for faces [all faces still equilateral].
5 connected tetrahedrons, would also make the same bird eye view as the icosahedron does, in you’re before the fall tree. [Still 22 visible edges, leaving 9 hidden in total, the structure would however have 13 vertices] (13 vetices in total with two of them hidden, makes me think about an old lingering wonder of mine about two hidden sephiroth)
I think 20 tetrahedrons makes an icosahedron.
Macavity
03-01-2003, 11:09
Finally found an HTML version of the "Anatomy of the body of God" by Frater Achad. Chapter 7 may interest people who like 3D structures and projections :)
http://w3.one.net/~browe/achad/anatomy/anatomy7.htm
There are other versions of the whole book in PDF around too...
Best, Mac
Thanks for the comments and the link.
I must admit that I have never studied Fr Achad's work, though I do know some who have, and find it quite severe (in the best of ways). I look forward to the opportunity of at least perusing it.
AmounrA, though five connected (non-see-through) tetrahedra would give a similar bird's eye appearance to the top section of an icosahedron, the connection of twenty tetrahedra to give a visual similarity to an icosahedron would just be that: a similar appearance. For that matter, one can always substitute any triangular face for a tetrahedron whose fourth vertex is hidden behind - I wonder, however, what this yields in meditation (I obviously haven't done this, and I probably prefer the 'purity' of the full platonic solids - though I will consider such wonderful considerations).
The Tree which follows is a quick rendition of one amongst other possible results. Here, the top section also forms a crucifix through which Gnosis becomes transformed and permits restoration, at a higher level, to occur.
With the restoration, the top section becomes truly a representation of spiritually conscious humanity, holding its own Da'at within itself.
Maybe further discussions may require of me clearer expressions of thoughts...
In the meantime, here is my restored Tree:
AmounrA has produced two wonderful 3-d renditions of my version of the Tree before the Fall. Here is its 'solid' rendition:
... and, in terms of my previously drawn version, more significantly this rendition:
JMD,
The tree before the fall is a very interesting structure. The Icosahedron relates to spirit, the Octahedron to the soul and the tetrahedron to body. I imagine that when we die, the tetrahedron falls away into dust; do you think we are the Octahedron? which I would image relates to the sub-conscious, and collective unconscious ‘levels’.
Another little thought is, what is the space these solids are in? I imagine them floating in darkness, almost tempted to use the word abyss. If these 3 shapes lined up as shown, but did not actually connect, with the Octahedron getting Hod, netzach, tiphareth, geburah & chessed, could these gaps be viewed as ‘ the abyss’?
Its just a thought, The tetrahedron would get malkuth and tiphareth & the Icosahedron would get [daath?] binah, [c]hockmah and Kether. The constitution of man.
The structure for after the fall opens up the middle path route to/from kether, the system connects? Can I ask for more information about what shape, or what kind of morphic movement has taken place with the Icosahedron?
With the restoration, it looks like the Icosahedron has ‘flowered’. It looks like the five sephiroth encasing Daath are shining ‘light’ onto it, the edges of the rays making the pentagram. [Perhaps revealing the hidden star].
Please, JMD tell more about these truly beautiful model! :-)
I'm glad it has caught your imagination too, AmounrA.
Here is a somewhat brief description of some of my work:
Prior to incarnation, the Tree is folded upon itself, but in the following manner: the Tetrahedron (which I consider of pure Spirit) contains the octahedron and icosahedron (in the manner described in another thread: the mid-points of the edges of the tetraheron coinciding witht the vertices of the octahedron, and the golden mean of the edges of the octahedron coinciding with the vertices of the icosahedron - an image of the latter is attached). With incarnation, an inversion occurs: the Tetrahedron is given over to higher spiritual beings for them to form the body (with Malkut & Yesod - which I presumed you meant, by the way, rather than Tifaret - also with aspects of Hod and Netzah). The Octahedron allows for the development of various Soul forces, and the Icosahedron remains very much our own higher being.
At death, a re-foldment occurs, whereby, firstly, the physical framework operating through the Tetrahedron is removed (thus permitting the body the decompose) - here the tetrahedron literally folds within the inner parts of the octahedron. This latter itself folds within the inner parts of the icosahedron, enabling memories, feelings and the like to dissipate.
As you have mentioned, the three solids can also be viewed somewhat separately, thus giving imaginative access to both the veil (Paroket) and the abyss.
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Following the Fall, what I see as occuring is the 'destruction' of Da'at: its supports have allowed the sphere to fall within the icosahedron. The redemptive act is representated by especially the mystery of Golgatha, leading to the formation of the Rose and its sacred Gnosis (the pentagramme of the 'restored' phase).
In the 'After the Fall' depiction, I see the 'edge' which once connected Keter to Da'at extending towards a sure foundation (Yesod), whilst the edges from Hockmah and Binah connecting. Those from Hesed and Geburah also extending and, respectively, connecting to Binah and Hockmah.
The 'restoration' occurs when the swirlings of the effects of the sacred sacrifice upon the cross begins to emerge into a self-sustaining new Rose-Da'at, acting upon the projection from Keter and 'splitting' the beam into two, thus linking Keter to both Hesed and Geburah. Da'at really being, in this final case, and as you have expressed it, a 'flowering'.
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I hope this has been reasonably clearly expressed...
The images of the Tree, however, like the Tarot, will reveal other hidden mysteries I am too only in the process of being made aware.
JMD, it has certainly captured my imagination, here are some thoughts, [thinking out loud in a way], before I go away and think about for winterJ
An equilateral triangle can be seen to contain the, Tetrahedron, Octahedron and Icosahedron in that these three solids all have equilateral triangle surface and line dimensions. It’s an interesting point I think, that there can only be 5 regular polyhedra, 3 of them can be made with an equilateral tri matrix. The two left out are the cube and the dodecahedron.
The problem with this is that the equilateral triangle only has 2 dimensions, and building with them, would form a hollow structure…a vessel?
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My take on what you said-
Almost in a manner of cell division, an equilateral triangle could be seen to grow, divide 3 new cells, which fold into the tetrahedron. Within/along side (?), the tetrahedron (womb?), a separate entity, one again growing from equilateral triangles could grow, the Octahedron… and again within the Octahedron comes the Icosahedron. Within/out, this cocoon, the tetrahedron could be seen as the first cell, higher spiritual beings being the motivators. Within the Octahedron, at the same time, forces developing the soul, and within the Icosahedron, our essence, our higher being. This is birth of physical life.
After the fall, daath sinks within the higher being, at this moment a solid connection runs through the system , the ray in a straight line hits tiphareth, then continues straight until yesod.(The Octahedron which resembles a pyramid and its refection), the it takes the tetrahedrons edge down to malkuth, uniting it into one, and a new life form, being starts to develop from the sacrifice of Daath, and the immediate rebirth of Daath [restoration]. A new rose is born.
At death, the tetrahedron folds into the Octahedron [soul], which then itself folds [assimilates] itself into Icosahedron [higher being]. The system then repeats.
Ravenswing
13-01-2003, 09:58
[QUOTE]Originally posted by jmd
[B]I'm glad it has caught your imagination too, AmounrA.
Here is a somewhat brief description of some of my work:
Prior to incarnation, the Tree is folded upon itself, but in the following manner: the Tetrahedron (which I consider of pure Spirit) contains the octahedron and icosahedron
*** from the mathematical discoveries of r buckminster fuller:
there is a figure called the cuboctahedron. this figure can be thought of as a cube with the corners cut off. it is also represented by a sphere surrounded by other close packed spheres. 12 spheres about 1. when the center sphere is removed, it looses stability and starts to collapse. it will then stablize as one of the triangular solids (a process fuller calls 'jitterbugging') the cuboctahedron might be considered what you call your folded tree.
fuller calls the cuboctahedron 'the face of god'. i consider figure to represent the wand level-- pure potential.
as the face collapses it can form two different figures.
if it spins on the way 'down' it forms the tetrahedron-- the basic unit of energy. this is the world where creation begins. i consider this the cup level.
if there is no spin, it collapses into the the octahedron. a funny thing here-- this is the sword level, the level of form. if you connect the midpoints of the octahedron, the smaller figure is the cuboctahedron; but now it is frozen in place. so the octahedron 'captures' the face of god. form is the result.
to release the face, the octahedron is 'blown up'. in it's expansion, it reaches the icosohedron phase. i see this as the world we know of... ****
ravenswing
Thanks for that input Ravenswing - I have only read one of B. Fuller's books, and that was many years ago. I'll have to carefully consider this 'cuboctahedron' - though it isn't what I was describing in the folded process, as these remain the perfect (or 'platonic') solids.
AmounrA, I have been away for a few days, and should have made a response to your post earlier - but these things are at times put off, and then semi-forgotten.
I agree with you that taking the (equilateral) triangle as a basic unit can lead to a very informative growth process. Instead of a duplication in the growth, however, a natural tetrahedron results as each vertex (ie, its furthest points) seek to unite - they thereby fold upon themselves and create this most incredible of forms, whose simplicity, with which I can only presume both you and Ravenswing would also agree, belies its hidden strength and potential.
With regards to a cube (otherwise also called an hexahedron) and its corners (ie, vertices) chopped, this, if continued deeper, results in the cube's transformation into an octahedron. In its earlier stages, it is wonderfully represented in Dürer's Melancholi I - a woodcut worthy, in my opinion, of as many reflections as words written upon it.
Thanks for the various commentaries and reflections - it is the kind of thread which demands even more careful thought than when working with more widely accepted and written about views.
""Instead of a duplication in the growth, however, a natural tetrahedron results as each vertex (ie, its furthest points) seek to unite - they thereby fold upon themselves and create this most incredible of forms, whose simplicity, with which I can only presume both you and Ravenswing would also agree, belies its hidden strength and potential.""
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Yes, folding would make more sense, the growth from point to plane in the 3 steps from kether to binah. Then the folding of the plane, into a tetrahedron.
At each point on the plane, the direction of the line takes a 60 degree turn. I wonder why?
If I had 3 spheres of equal size, all with the same gravity, they would attract each other . Drawing a line from each centre would form an equalateral triangle. {Two spheres would create a line from centre to centre}
Perhaps without the solid spheres , but rather 3 points of equaly attracting energy, they would form the plane, filling the centre with an energy film.. however, trying to visualise this folding is not so easy. It is perhaps easier to see another point of energy born out of the centre 'energy film'?
Your reflections remind me of some meditative visualisations I did a number of years ago with regards to what it would take to construct the simplest perpetual motion 'machine' (I would rather call it 'thought machine', for there are always other constrainst in the physical world).
My reflections lead me to consider placing repelling forces (like little 'magnetic' balls) within the entrails of a hollow sphere, and see what would happen.
If one is placed, its only cause for motion would be an external force - which I precluded in this thought experiment.
If two balls are therein placed, they would maintain their distance along a diametric line within the hollow sphere.
With three, an equilateral triangle would form (with, therefore, internal angles of 60 degrees - but other reasons come to mind for the 60 degrees you mention above).
With four balls, a tetrahedron would form.
With five, we have the first instance of non-equilibrium (interesting, I thought, given how the fives of the minor arcana are generally viewed) - and hence, incidentally, a highly dynamic perpetual motion ensues.
From here on, only four other numbers generate static equilibrial forms, corresponding to the platonic solids (six generates the vertices of an octahedron, eight of a cube, twelve of an icosahedron, and twenty of a dodecahedron). Other static shapes are also generated, but with internal anomalous forces - such as with 60 points, producing the shape of combined hexagons and pentagons reminiscent of a soccer ball (football, for Europeans)... I must admit that I didn't carry, in visualised form, the thought experiment much past this number.
Anyhow - this probably moves away from the thread, but I couldn't help remember such musings reading AmounrA's post.
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With regards to the 60 degrees, if two arms of equal length, being a distance apart of their length, swing out, their extremities but meet at sixty degrees on the plane.
I think this brings the Archimedean solids into bat. A football is a truncated Icosahedron. Do you think the early formation of structures from point dot would splinter into such solids?
I wonder what the effect would be if the magnetic balls where different sizes?
The relation to the 5 disrupting the balance is most apt to the tarot :-). 5 is a facinating number in the sense of the golden geometry with Phi contained within not just the pentagram, but each arm. This type of magick contained within 'flat' geometry, does make you think about the fractal nature that may lie within 'solids'.