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Post by ♥Starlene on Sept 3, 2007 21:28:49 GMT -5
Here is a very cool article about the 5 platonic solids... I have been reading about sacred geometry and learning about this stuff and thought I would share!
((hugs))
The Five Platonic Solids
The Platonic solids, also known as the regular polyhedrons, are the three-dimensional bodies whose surfaces consist of identical, regular polygons which meet in equal angles at the corners. There are five such polyhedrons, the Tetrahedron, the Octahedron, the Cube, the Icosahedron and the Dodecahedron. The first three have apparently been known since ancient times. The others was definitely known by the Pythagoreans, since one of them, Timaeus of Locri, invented the "Platonic" correspondence between them and the elements. Plato later publicised their results, which is the reason they bear his name. Here is a table with their properties:
Faces Edges Vertices Schfli Dual Plato symbol Tetrahedron 4 6 4 {3,3} Tetrahedron Fire Octahedron 8 12 6 {3,4} Cube Air Cube 6 12 8 {4,3} Octahedron Earth Icosahedron 20 30 12 {3,5} Dodecahedron Spirit Dodecahedron 12 30 20 {5,3} Icosahedron
These solids naturally fall into three groups, based on their symmetries and duals. The Octahedron and Cube, which are duals of each other, form one group, while the Dodecahedron and Icosahedron form another. The Tetrahedron form a third group with only itself as a member since it is its own dual. Note that the five elements are similarly divided: the spiritual elements are duals to the material elements (and a similar duality holds for actives and passives), and the fifth is left out or its own opposite (one is reminded of the concept of positive and negative aethyr in [CL]). Thus, from my mathemagickal standpoint, Quintessence belongs more naturally to the Tetrahedron, the Cube and Octahedron corresponds as normal to Earth and Air while Fire and Water correspond to the Dodecahedron and Icosahedron respectively.
The Tetrahedron
The Tetrahedron classically represents Fire, and each fa ce is also the alchemical triangle of fire. The Golden Dawn called it the Pyramid of Fire, and used it as the admission badge for the path of Shin. The three upper triangles represents Solar Fire, Volcanic Fire and Astral Fire, while the bottom triangle, often hidden from view is the latent heat. The upper triangles are also linked to the three fire-signs Aries, Saggittaurius and Leo.
Note that each face and each vertex can be put into a one-to-one correspondence with an element. Each element touches the others, showing that the superficial divisions of Fire and Water, Air and Earth are really unities. No element is superior to any other, and they all balance each other into a very stable structure (Buckminster Fuller designed his entire mathematics and architecture on this simple fact). This represents is in my view the state before the divisions between the elements, and thus resonant with the Quinta Essentia, from which the element were formed.
Its worth noting that the tetrahedron is its own dual. At the same time it belongs to the 4.3.2 symmetry group, the same as the octahedron and the cube belongs to. In a way this reflects "Keter is in Malkuth, and Malkuth is in Keter", the material world subtly reflects the spiritual world and vice versa.
The Octahedron
The Octahedron corresponds classically to Air. It has 8 faces (corresponding to Hod and mental activity?), 6 vertices and 12 edges. The edges naturally correspond to the zodiac. They can be arranged in such a manner that the four triplicities border a triangular face each without overlap. These faces cover half the surface, leaving 4 incomplete faces with signs from three elements along each edge (this may signify an absence of the left-out element. The octahedron thus consists of both the abundance of each element and its absence). At each corner two elements meet (creating the same planetary correspondences as in the tetrahedron, with the sun at the centre as usual). In this arrangement, each square "equator" corresponds to one quadruplicity.
Another common use of the octahedral symmetry is used in banishing rituals (mainly the LBRP and the Rose-Cross Rite). The sphere encircled by three orthogonal circles is the natural projection of the octahedron onto the surface of a sphere. In most rituals the horizontal equator corresponds to the cherubic signs. This also corresponds to the six directions of the Yetziratic Sealing Rite [DK], see below for the discussion of the symmetric group.
The octahedron fits air very well, since the various symmetries and correspondences are so clear and easily viewed. As we will see in the case of the cube, many of these symmetries are hidden or hard to discern in the case of Earth, perhaps signifying that the intellect allows us to see the structure of the world more easily than our physical senses, which are parts of the system we try to study.
The Cube (Hexahedron)
The Cube (hexahedron) naturally corresponds to Earth. It is stable, the basis of western architecture and salt crystallises into cubes. It has six faces, making some groups attribute it to Tiphareth. The six faces naturally fit the sephira, and can of course be linked to the planets except for the sun, which is placed in the centre. Another natural link is the folded out cube, which forms a cross. The eight corners of the cube neatly corresponds to three complementary dualities. When two dualities interact, the four elements are created. Now the four elements are dualized again, and we get eight corners representing the relative absence and abundance the each element. This is naturally dual to the faces of the octahedron. In the same way the six faces correspond to the six vertices of the octahedron (i.e. meetings between two quadruplicities). It is however not possible to arrange the three quadruplicities along the edges to enclose whole faces without overlaps. Does this signify the imperfections and limitations of the material world?
Its an interesting fact that the cube isn't stable. If a model is made using toothpicks and peas, it can easily be shown that it tends to distort or collapse. It is however possible to inscribe a tetrahedron inside a cube so that its vertices meet four corners of the cube and its edges lie in the faces of the cube. This will stabilise it completely (spirit stabilises and orders matter). If two tetrahedrons are inscribed using different sets of vertices, they intersect and form a geometric body known as the "Stella Octangula" (which is an octahedron with pyramids added on its faces). This is a very neat representation of the complementarity between positive and negative forces, which seems to underlie much of the structure of the cube.
It is worth noting that the duality of the cube and octahedron fits the duality between Air and Earth. Both belong to the same symmetry family (called 4.3.2), to which all normal minerals and crystals belong (only the so-called quasicrystals belong to the icosahedral symmetry family). It is also an interesting fact that of the platonic polyhedrons, only the cube can fill space completely, without interstices or overlaps. Thus we see that despite that the only way to create a completely consistent universe out of one element is to use matter. The other elements are not able to bind together in the right way to form a stable world, but will either move around or form imperfect patterns.
The Icosahedron
This polyhedron traditionally corresponds to water, possibly because it rolls quite easily. Its 20 faces could correspond to the sephiroth and qlippoth, but I have so far not found any significant arrangement. While the octahedron and cube, belonging to 4.3.2 have many symmetries involving the four elements, trinities and dualities, the icosahedron and doedecahedron, belonging to 5.3.2 have links to the five elements and the trinities and dualities. Thus they correspond closer to the whole system than the more material elements, which deal with just the four elements.
In nature these symmetries are rare, and are usually found in viruses and radiolaria. One reason for the rarity of these symmetries may be that they don't interconnect as well as the 4.3.2 group. In crystals, molecules and viruses with 5.3.2 symmetries organize according to the 4.3.2 group instead, subjugating their own symmetries. The higher elements decay into the lower in order to form the world.
These symmetries are harder to discern, since traditionally we humans have a tendency to avoid high-order groups, especially odd symmetries (its worth noting that the number five is sacred to the Discordians since it is the smallest number of factors the human mind is unable to handle at once).
The 12 vertices can of course be viewed as the zodiac. In this case each sign is linked to five other signs along the edges which corresponds to the five elements, a quite interesting set of corresponences (this is of course reflected in the faces and edges of the dodecahedron in a similar way). This seems to imply a network between the signs, where each sign is transformed into five others by the actions of the five elements.
One obvious way of arranging the elements in such a pattern is the following: choose two edges opposite to each other and assign them to an element. Then there are four edges along the "equator" if the two edges are regarded as the poles which can be assigned the same element. These edges are orthogonal to the first, and each pair of opposite edges are orthogonal to all others. In fact, if the opposite edges are joined with lines through the interior, a very neat structure of interlocking rectangles result, where each rectangle locks the other rectangles without touching them. Each pair of rectangles doesn't interlock, but together they form a synergetic whole. In this way each element can be assigned to its own edges in a proper way. It is interesting to note that the pattern inside each element belongs to the 4.3.2 symmetry group.
The icosahedron can be inscribed in the octahedron if its vertices are placed on the octahedron-edges in the golden ratio. In this case eight faces of the icosahedron lie in the plane of the faces of the octahedron, and the rest lie in the interior. As a general rule, the golden ratio is intimately linked to the 5.3.2 family of solids. This construction is symbolic of how the creativity and feeling of Water is needed to form the rational thought of Air.
The icosahedron corresponds to water. It seems to tie together things in complex, apparently random ways and encompass them without necessarily elucidate their interrelationships. As one can see, the complexity of the icosahedron and dodecahedron "liquiefies" the various correspondences. The number of possible arrangement is much larger than for the relatively simple cubes and octahedrons.
The Dodecahedron
This solid is classically attributed to spirit, probably because it was the last discovered and because of the pentagonal faces. Its twelve faces has naturally been attributed to the zodiac, and there have even been dodecahedral calendars. The symmetries discussed above exist in a dual form here too.
The dodecahedron can be seen as the union of five intersecting cubes, whose corners touch the vertices of the dodecahedron (this is a rather complex structure and hard to visualize). At each vertex three different cubes meet. Along each side of the dodecahedron an edge from a cube runs, creating a rather neat system of correspondences between the five elements and the edges like the system mentioned above for the icosahedron.
Another way of placing polyhedrons in the dodecahedron is to use five intersecting tetrahedrons, whose corners touch the vertices. This is a most elegant configuration where the tetrahedrons seem to twist around each other. It exists in two different forms, essentially corresponding to clockwise and counterclockwise rotation. The space occupied by all five tetrahedons is a smaller icosahedron, another nice example of the power of duals. It could perhaps be seen as a "construction drawing" of Fire, where the Quinta Essentia takes on its various elemental properties, and combines them in an eternally rotating and twisting form.
The evolution of the Quinta Essentia into the four elements may thus be described as follows: The original form of the Tetrahedron is created out of the primordial chaos by being the simplest and most stable form. It combines in various ways with itself, either by moving and mixing, forming the Dodecahedron and Fire, or by linking together and building the Icosahedron and primordial Water. However, while both polyhedrons are close to being perfect spheres, they don't fit together. These imperfect interactions betwen the growing numbers of polyhedrons force them to order themselves according to cubical symmetries, and Earth and Air are formed. As we will see, this fits with some results within group theory.
((hugs))
The Five Platonic Solids
The Platonic solids, also known as the regular polyhedrons, are the three-dimensional bodies whose surfaces consist of identical, regular polygons which meet in equal angles at the corners. There are five such polyhedrons, the Tetrahedron, the Octahedron, the Cube, the Icosahedron and the Dodecahedron. The first three have apparently been known since ancient times. The others was definitely known by the Pythagoreans, since one of them, Timaeus of Locri, invented the "Platonic" correspondence between them and the elements. Plato later publicised their results, which is the reason they bear his name. Here is a table with their properties:
Faces Edges Vertices Schfli Dual Plato symbol Tetrahedron 4 6 4 {3,3} Tetrahedron Fire Octahedron 8 12 6 {3,4} Cube Air Cube 6 12 8 {4,3} Octahedron Earth Icosahedron 20 30 12 {3,5} Dodecahedron Spirit Dodecahedron 12 30 20 {5,3} Icosahedron
These solids naturally fall into three groups, based on their symmetries and duals. The Octahedron and Cube, which are duals of each other, form one group, while the Dodecahedron and Icosahedron form another. The Tetrahedron form a third group with only itself as a member since it is its own dual. Note that the five elements are similarly divided: the spiritual elements are duals to the material elements (and a similar duality holds for actives and passives), and the fifth is left out or its own opposite (one is reminded of the concept of positive and negative aethyr in [CL]). Thus, from my mathemagickal standpoint, Quintessence belongs more naturally to the Tetrahedron, the Cube and Octahedron corresponds as normal to Earth and Air while Fire and Water correspond to the Dodecahedron and Icosahedron respectively.
The Tetrahedron
The Tetrahedron classically represents Fire, and each fa ce is also the alchemical triangle of fire. The Golden Dawn called it the Pyramid of Fire, and used it as the admission badge for the path of Shin. The three upper triangles represents Solar Fire, Volcanic Fire and Astral Fire, while the bottom triangle, often hidden from view is the latent heat. The upper triangles are also linked to the three fire-signs Aries, Saggittaurius and Leo.
Note that each face and each vertex can be put into a one-to-one correspondence with an element. Each element touches the others, showing that the superficial divisions of Fire and Water, Air and Earth are really unities. No element is superior to any other, and they all balance each other into a very stable structure (Buckminster Fuller designed his entire mathematics and architecture on this simple fact). This represents is in my view the state before the divisions between the elements, and thus resonant with the Quinta Essentia, from which the element were formed.
Its worth noting that the tetrahedron is its own dual. At the same time it belongs to the 4.3.2 symmetry group, the same as the octahedron and the cube belongs to. In a way this reflects "Keter is in Malkuth, and Malkuth is in Keter", the material world subtly reflects the spiritual world and vice versa.
The Octahedron
The Octahedron corresponds classically to Air. It has 8 faces (corresponding to Hod and mental activity?), 6 vertices and 12 edges. The edges naturally correspond to the zodiac. They can be arranged in such a manner that the four triplicities border a triangular face each without overlap. These faces cover half the surface, leaving 4 incomplete faces with signs from three elements along each edge (this may signify an absence of the left-out element. The octahedron thus consists of both the abundance of each element and its absence). At each corner two elements meet (creating the same planetary correspondences as in the tetrahedron, with the sun at the centre as usual). In this arrangement, each square "equator" corresponds to one quadruplicity.
Another common use of the octahedral symmetry is used in banishing rituals (mainly the LBRP and the Rose-Cross Rite). The sphere encircled by three orthogonal circles is the natural projection of the octahedron onto the surface of a sphere. In most rituals the horizontal equator corresponds to the cherubic signs. This also corresponds to the six directions of the Yetziratic Sealing Rite [DK], see below for the discussion of the symmetric group.
The octahedron fits air very well, since the various symmetries and correspondences are so clear and easily viewed. As we will see in the case of the cube, many of these symmetries are hidden or hard to discern in the case of Earth, perhaps signifying that the intellect allows us to see the structure of the world more easily than our physical senses, which are parts of the system we try to study.
The Cube (Hexahedron)
The Cube (hexahedron) naturally corresponds to Earth. It is stable, the basis of western architecture and salt crystallises into cubes. It has six faces, making some groups attribute it to Tiphareth. The six faces naturally fit the sephira, and can of course be linked to the planets except for the sun, which is placed in the centre. Another natural link is the folded out cube, which forms a cross. The eight corners of the cube neatly corresponds to three complementary dualities. When two dualities interact, the four elements are created. Now the four elements are dualized again, and we get eight corners representing the relative absence and abundance the each element. This is naturally dual to the faces of the octahedron. In the same way the six faces correspond to the six vertices of the octahedron (i.e. meetings between two quadruplicities). It is however not possible to arrange the three quadruplicities along the edges to enclose whole faces without overlaps. Does this signify the imperfections and limitations of the material world?
Its an interesting fact that the cube isn't stable. If a model is made using toothpicks and peas, it can easily be shown that it tends to distort or collapse. It is however possible to inscribe a tetrahedron inside a cube so that its vertices meet four corners of the cube and its edges lie in the faces of the cube. This will stabilise it completely (spirit stabilises and orders matter). If two tetrahedrons are inscribed using different sets of vertices, they intersect and form a geometric body known as the "Stella Octangula" (which is an octahedron with pyramids added on its faces). This is a very neat representation of the complementarity between positive and negative forces, which seems to underlie much of the structure of the cube.
It is worth noting that the duality of the cube and octahedron fits the duality between Air and Earth. Both belong to the same symmetry family (called 4.3.2), to which all normal minerals and crystals belong (only the so-called quasicrystals belong to the icosahedral symmetry family). It is also an interesting fact that of the platonic polyhedrons, only the cube can fill space completely, without interstices or overlaps. Thus we see that despite that the only way to create a completely consistent universe out of one element is to use matter. The other elements are not able to bind together in the right way to form a stable world, but will either move around or form imperfect patterns.
The Icosahedron
This polyhedron traditionally corresponds to water, possibly because it rolls quite easily. Its 20 faces could correspond to the sephiroth and qlippoth, but I have so far not found any significant arrangement. While the octahedron and cube, belonging to 4.3.2 have many symmetries involving the four elements, trinities and dualities, the icosahedron and doedecahedron, belonging to 5.3.2 have links to the five elements and the trinities and dualities. Thus they correspond closer to the whole system than the more material elements, which deal with just the four elements.
In nature these symmetries are rare, and are usually found in viruses and radiolaria. One reason for the rarity of these symmetries may be that they don't interconnect as well as the 4.3.2 group. In crystals, molecules and viruses with 5.3.2 symmetries organize according to the 4.3.2 group instead, subjugating their own symmetries. The higher elements decay into the lower in order to form the world.
These symmetries are harder to discern, since traditionally we humans have a tendency to avoid high-order groups, especially odd symmetries (its worth noting that the number five is sacred to the Discordians since it is the smallest number of factors the human mind is unable to handle at once).
The 12 vertices can of course be viewed as the zodiac. In this case each sign is linked to five other signs along the edges which corresponds to the five elements, a quite interesting set of corresponences (this is of course reflected in the faces and edges of the dodecahedron in a similar way). This seems to imply a network between the signs, where each sign is transformed into five others by the actions of the five elements.
One obvious way of arranging the elements in such a pattern is the following: choose two edges opposite to each other and assign them to an element. Then there are four edges along the "equator" if the two edges are regarded as the poles which can be assigned the same element. These edges are orthogonal to the first, and each pair of opposite edges are orthogonal to all others. In fact, if the opposite edges are joined with lines through the interior, a very neat structure of interlocking rectangles result, where each rectangle locks the other rectangles without touching them. Each pair of rectangles doesn't interlock, but together they form a synergetic whole. In this way each element can be assigned to its own edges in a proper way. It is interesting to note that the pattern inside each element belongs to the 4.3.2 symmetry group.
The icosahedron can be inscribed in the octahedron if its vertices are placed on the octahedron-edges in the golden ratio. In this case eight faces of the icosahedron lie in the plane of the faces of the octahedron, and the rest lie in the interior. As a general rule, the golden ratio is intimately linked to the 5.3.2 family of solids. This construction is symbolic of how the creativity and feeling of Water is needed to form the rational thought of Air.
The icosahedron corresponds to water. It seems to tie together things in complex, apparently random ways and encompass them without necessarily elucidate their interrelationships. As one can see, the complexity of the icosahedron and dodecahedron "liquiefies" the various correspondences. The number of possible arrangement is much larger than for the relatively simple cubes and octahedrons.
The Dodecahedron
This solid is classically attributed to spirit, probably because it was the last discovered and because of the pentagonal faces. Its twelve faces has naturally been attributed to the zodiac, and there have even been dodecahedral calendars. The symmetries discussed above exist in a dual form here too.
The dodecahedron can be seen as the union of five intersecting cubes, whose corners touch the vertices of the dodecahedron (this is a rather complex structure and hard to visualize). At each vertex three different cubes meet. Along each side of the dodecahedron an edge from a cube runs, creating a rather neat system of correspondences between the five elements and the edges like the system mentioned above for the icosahedron.
Another way of placing polyhedrons in the dodecahedron is to use five intersecting tetrahedrons, whose corners touch the vertices. This is a most elegant configuration where the tetrahedrons seem to twist around each other. It exists in two different forms, essentially corresponding to clockwise and counterclockwise rotation. The space occupied by all five tetrahedons is a smaller icosahedron, another nice example of the power of duals. It could perhaps be seen as a "construction drawing" of Fire, where the Quinta Essentia takes on its various elemental properties, and combines them in an eternally rotating and twisting form.
The evolution of the Quinta Essentia into the four elements may thus be described as follows: The original form of the Tetrahedron is created out of the primordial chaos by being the simplest and most stable form. It combines in various ways with itself, either by moving and mixing, forming the Dodecahedron and Fire, or by linking together and building the Icosahedron and primordial Water. However, while both polyhedrons are close to being perfect spheres, they don't fit together. These imperfect interactions betwen the growing numbers of polyhedrons force them to order themselves according to cubical symmetries, and Earth and Air are formed. As we will see, this fits with some results within group theory.
Hugs and Stuff Girly <3
~love hugs and stuff~ I'm just very selective about the reality I accept." - Calvin and Hobbs{=}{/=}{=}{/=}