Plato's Timaeus conjectures on the composition of the four elements which the ancient Greeks thought made up the universe: earth, water, air, and fire. Plato conjectured each of these elements to be made up of a certain Platonic solid: the element of earth would be a cube, of air an octahedron, of water an icosahedron, and of fire a tetrahedron. Each of these perfect polyhedra would be in turn composed of triangles. Only certain triangular shapes would be allowed, such as the 30-60-90 and the 45-45-90 triangles. Each element could be broken down into its component triangles, which could then be put back together to form the other elements. Thus, the elements would be interconvertible, so this idea was a precursor to alchemy. Plato's Timaeus posits the existence of a fifth element (corresponding to the fifth remaining Platonic solid, the dodecahedron) called quintessence, of which the cosmos itself is made. In three dimensional space, there are ONLY FIVE natural frequency modes for spherical EM standing wave, resulting in the formation of the five Platonic solids shown below. Each platonic would be perceived as the formation of a stable form of matter, anything in between will tend to be unstable, and will degrade to its nearest stable form, giving off its extra elements as EM energy, with radioactive elements being such an example.
|Cube/ Hexahedron||Earth||Solid||Molecules are limited to vibration about fixed position. Solids have a definitive volume and shape and high density. When energy is applied to a solid (eg heated) the solid becomes a liquid at its melting point. The solid phase is the lowest energy state of matter. See Hutchison effect. Speed of sound in steel is 5960m/s, for glass 5640m/s.|
|Icosahedron||Water||Liquid||Molecules free to move throughout the liquid but held by intermolecular forces, giving it a definitive volume but no definite shape and a lower density. When energy is applied, evaporation occurs and it becomes a gas at its boiling point. If energy is lowered it becomes a solid at its freezing point. Speed of sound in water is 1482m/s.|
|Octahedron||Air||Gas||In gas state, molecules are free to move in every direction, and a gas has no definite shape or volume and its density is lower than liquids. When energy is applied, electrons gain enough energy to leave the atom structure and a gas starts getting ionised. When fully ionised it becomes a plasma. See sonoluminescence. If energy is lowered a gas becomes a liquid. Speed of sound in air is approx 343m/s but dependant on pressure, temperature. For Helium it is 965m/s!|
|Tetrahedron||Fire||Plasma||When a gas is given energy, molecules are torn apart into their component atoms and individual electrons are pulled away. This highly energised mixture of electrons and ions forms the plasma. If energy is lowered, plasma becomes gas. If plasma is given further energy, the atom structure within it is broken into its constituent electromagnetic energy and can no longer be considered a state of matter. Indeed the Plasma phase is the highest energy state of matter.|
|Dodecahedron||Universe||Vacuum||The vacuum (or ether) has a structure as well. Vacuum is made up of pure electromagnetic energy, which can be re organised in any of the other platonic structures to be perceived as one of the other states of matter. Vacuum is a sea of electromagnetic energy which cannot be detected unless an imbalance is created (example: casimir plates).|
Euclid, 300 BC and the Ancient Greeks, in their inherited love for geometry, called the five solids shown below, the atoms of the Universe. In the same way that we today believe that all matter, is made up of combinations of atoms so the Ancient Greeks also believed that all physical matter is made up of the atoms of the Platonic Solids and that all matter also has a mystical side represented by their connection with earth, air, fire, water and aether. Similar to our conventional atom model which shows a nucleus surrounded by electrons in orbits creating spheres of energy, the Greeks felt that these Platonic solids also have a spherical property, where one Platonic Solid fits in a sphere, which alternately fits inside another Platonic Solid, again fitting in another sphere. It is fascinating to see how any one of these solids can fit inside one another. The concept of one sphere fitting inside another sphere is surprisingly frequently seen in different cultures. Indeed, the mechanism of platonic solids is so perfect, that perhaps as we are approaching in this study, their concept of platonics as being the building blocks of matter, might be more evolved than our present knowledge of the atom model. As shown in the photograph below, as in so many other aspects of their science and philosophy, the Greeks were not the originators of these concepts. The photograph is of a collection Neolithic stones, unmistakably showing the same basic "platonic" shapes. These (from the Ashmolean Museum, Oxford, UK) are at least 3,000 years old (>1000BC). Indeed, we know that from the Vedic times, around 3000 B.C. to 1000 B.C., Indians (Indo-Aryans) had classified the material world into the four elements; earth (Prithvi), fire (Agni), air (Maya) and water (Apa). To these four elements was added a fifth one; ether or Akasha. According to some scholars these five elements or Pancha Mahabhootas were also identified with the various human senses of perception; earth with smell, air with feeling, fire with vision, water with taste and ether with sound. Whatever the validity behind this interpretation, it is true that since very ancient times Indians had perceived the material world as comprising these 5 elements. The information one can get from these carved out shapes shows that a highly developed generation of human kind gave a lot of scientific importance to these shapes, and perhaps carving out these stones was one of their attempts to pass over their knowledge to others, including us!
What's so special about these geometric shapes? Here are the main rules for these geometric solids:
The platonic solids are those polyhedra whose faces are all regular polygons, which means they have congruent legs and angles. Leonhard Euler (1707-1783) who was a Swiss mathematician, noticed that no matter how one cuts a sphere into polygons, sometimes called a triangulation, there is a quantity which remains constant; in other words, there is a number related to the sphere independent of the triangulation. This number is now called the Euler characteristic. Each of the platonic solids is in fact a triangulation of the sphere into polygons.The Euler characteristic is given by F-E+V, where F is the number of polygonal faces, E is the number of edges, and V is the number of vertices in the triangulation. Euler showed that for any triangulation of the sphere, we get an Euler characteristic equal to 2, no matter which platonic solid is chosen. Euclid proved around 200 B.C. that there are exactly five regular solids in three dimensions. Ludwig Schlafli proved in 1901 that there are exactly six regular solids in four dimensions, and also proved that the only regular solids in dimensions greater than or equal to five are the generalized tetrahedron, cube, and octahedron.
|Each shape can be attached to a multiple number of the same shape or other platonic shape to generate a bigger platonic solid or even a non platonic one, as happens during generation of crystals. In a way, one may regard a crystal lattice structure as a picture of the mechanism within the atom itself. So as you see, this theory works well at quantum level as well as at molecular level, which makes it unique.|
Similar to the two-dimensional case of the Chladni plate, the Platonic Solids are simply representations of waveforms in three dimensions. Each tip or vertex of the Platonic Solids touches the surface of a sphere in an area where the vibrations have canceled out to form a node. Thus, what we are seeing is a three-dimensional geometric image of vibration / pulsation within a sphere.
This explains why an atom does not necessarily look spherical. It does not however indicate that an atom is restricted to any particular size, and this means that an atom mechanism can be 'grown' as much as its spherical boundary is set. We know, from the art of growing crystals, that a crystal tends to use up similar atoms to grow up, retaining its original structure. Sonoluminescence, described earlier, we see how a mechanism in all respects similar to an atom can be setup to work in the size of a small bubble, many times greater than any known atom. As we can see, we no longer have to restrain atoms to a certain size; they are capable of existing on various scales and maintaining the same properties. Once we fully understand what is going on in the vibrating sphere, we can design materials that are extremely hard, extremely light, or extremely unstable at our wish.
As we know, most physics parameters cannot fit in a 3 dimensional space, but in addition to space, require a further dimension we call time. So, although a 3D platonic may give us a good picture of what an elementary particle looks like, it will not give us any indication about its movement in time. As we will see later on, a moving 3D shape can be integrated over time and be fully described by a stationery 4D shape. Thus in order to understand the motion of 3D platonics we need to consider platonics of a higher dimension. In four dimensions, the five Platonic Solids have six analogues. Interestingly enough higher dimensions have only three platonic solids, so the 4th dimension is the special case with the largest variety. In 4D, Polyhedra are called polytopes. The Simplex and the Hypercube are relatively easy to understand, and illustrated with projections, as analogues of the Tetrahedron and the Cube.
|3 Dimensional Platonics|
|4 Dimensional "Platonic" Polytopes|
|1. 5-cell, Pentatope or Simplex||tetrahedra||5||10||10||self-dual|
|2. 8-cell, Tesseract or Hypercube||cubes||16||32||24||16-cell|
|n-Dimensional "Platonic" Polytopes, n > 4|
|Polytope||number of (n-1) D cells||vertices||duals||3-d analogue|
|1. (n + 1) cell||n + 1 n-cells||n + 1||self-dual||Tetrahedron|
|2. 2n-cell||2n (2n-2)-cells||2n||2n-cell||Cube|
|3. 2n-cell||2n n-cells||2n||2n-cell||Octahedron|
Very interesting is the fact that, in ALL dimensions greater than four, there are exactly three analogues to the Platonic Solids. Also these 3 analogues: the Tetrahedron, cube and octahedron, exist in all dimensions. This is, curiously, exactly half the forms we find in 4 dimensions. Also, note that the 3D platonics (or their duals) are found in the cells making up the 4D polytopes. In a way, we can say that the 4 dimensional state, has the highest structural entropy of all, and that is where we live in!. In 1908, a Russian physicist, Minkovsky gave a new concept of space-time continuum, which may be regarded as the geometrical interpretation of the Special Relativity Theory. Minkovsky considered that space and time, being relative, describe a fourth dimension. The space-time is composed of individual events each of which is described by four complex numbers, three space coordinates x, y and z, and one time coordinate t. How does our brain react to 4D space? We tend to see the universe around us as a 3D space, changing in time. What actually our brain is doing, is to take one of the 4D axis as reference (=time) and differenciate (or photograph) the other 3 dimensions with respect to it. This results in a sequence or 3D images over time, but the reference dimension (time) is arbitrarily taken as reference only in our perspective, whilst in reality it is a space dimension in its own right.
|Tetra <-> Tetra||Hexa <-> Octa||Dodeca <-> Icosa|
|Edge length to circumscribed sphere radius for tetrahedron= 163.3%||Edge length to circumsribed sphere radius for hexahedron (cube)= 115.47% & octahedron = 141.42%||Edge length to circumscribed sphere radius for icosahedron = 105.15% and dodecahedron = 71.364%|
|Inscribed to Circumscribed sphere radius ratio for tetrahedron= 33.33%||Inscribed to Circumscribed sphere radius ratio for BOTH hexahedron (cube) & octahedron = 57.735%||Inscribed to Circumscribed sphere radius ratio for BOTH icosahedron and dodecahedron = 92.624%|
|Inscribed to Circumscribed sphere volume ratio for tetrahedron= 3.7%||Inscribed to Circumscribed sphere volume ratio for BOTH hexahedron (cube) & octahedron = 19.245%||Inscribed to Circumscribed sphere volume ratio for BOTH icosahedron and dodecahedron = 79.465%|
|Inscribed Planck's spherical volume for tetrahedron= 1.8793E-107 m3||Inscribed Planck's spherical volume for hexahedron (cube) = 2.762E-106m3 and for octahedron= 1.503E-106m3||Inscribed Planck's spherical volume for icosahedron= 9.538E-106m3and dodecahedron = 3.05E-105m3|
A very interesting characteristic of these five platonic solids, is the so called DUALITY. The dual of a platonic is the shape formed having its vertices at the centre of each face of the parent platonic. The importance of duality is re-confirmed in the 1000 BC old stones shown above, by the presence of white dots, that show the vertices of the dual platonic within each stone. As shown above, you can see that the tetrahedron is the dual of itself, whilst an octahedron is the dual of a hexahedron/cube (and vice versa), and a dodecahedron is the dual of the icosahedron (and vice versa). Thus each platonic can have nested platonics within it of diminishing sizes down to an infinetely small side lengths, and yet every nested structure will still have all characteristics of a platonic solid. In the case of the tetrahedron, where the number of faces is equal to the number of vertices, its dual will be the same shape of its parent platonic shape. From the above calculations, it is shown that the ratios between both radius and volume of any circumcribed sphere to its inscribed sphere is a constant, not only for the case tetra-tetra, but also to the other two dual platonics, even if the platonic shape of the duals is not the same.The limiting edge size of any platonic is equal to half Planck's length (1.616E-35m), since each side of the platonic is vibrating at its fundamental frequency, where node to node distance is equal to half a wavelength.
This length is the lower limit at which the classical description of gravity ceases to be valid, and below which 'length', and time to travel it, have no meaning. At this value of length, the theories of quantum mechanics and general relativity become incompatible, and so it seems reasonable that it should be at this value that our platonic standing wave should interact with gravity, otherwise it will be, at best, only as good as the present theories. There is a corresponding Planck time associated with the Planck length which is the time required for an EM wave or photon to travel the elementary Planck length at the speed of light, which equates to 5.39E-44 seconds. In the Duals table above, a value named Planck's spherical volume has been worked out for each platonic shape, representing the volume inside the inscribed sphere for the particular platonic shape with edge length equal to half Planck's length. This will later on be shown to be the matter-antimatter interface volume, known in Superstring theories as the light cone.Platonic & Archimedian solids - by Daud Sutton