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3 Hyper-dimensional Geometry
Mar 20, '12
27

Chapter Three

HYPER-DIMENSIONAL GEOMETRY

The answer to the very big lies in understanding the very small; the answer is, therefore, within. Now with the advent of super-computers we are able to model and map very complex surfaces and to show other people what they look like. This was not possible for people 100 years ago and before. Gaston Julia the famous French mathematician could see the mathematical equations of chaos and visualise them in his own head, but he was unable to externalise them in order to make them meaningful to other people. Eventually he did so in a series of cartoons, but this lacked the impact of the actual visual images of the Julia sets named after him.

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Enter a Polish refugee living in France, Benoit Mandelbrot. Mandelbrot was born in Poland in 1924 into a family with a very academic tradition. His father, however, made his living buying and selling clothes while his mother was a doctor. As a young boy, Mandelbrot was introduced to mathematics by his two uncles.

In 1945 Mandelbrot's uncle had introduced him to Julia's important 1918 paper, claiming that it was a masterpiece and a potential source of interesting problems, but Mandelbrot did not like it. Indeed he reacted rather badly against suggestions posed by his Uncle Sice, he felt that his whole attitude to mathematics was so different from that of his uncle. Instead Mandelbrot chose his own very different course which, however, brought him back to Julia's paper in the 1970's after a path through many different sciences, which some characterise as highly individualistic or nomadic. In fact the decision by Mandelbrot to make contributions to many different branches of science was a very deliberate one taken at a young age. It is remarkable how he was able to fulfil this ambition with such remarkable success in so many areas.

With the aid of computer graphics, Mandelbrot who then worked at IBM's Watson Research Center, was able to show how Julia's work is a source of some of the most beautiful fractals known today. To do this he had to develop not only new mathematical ideas, but also he had to develop some of the first computer programs to print graphics.

His work was first elaborated in his book Les objets fractals, forn, hasard et dimension (1975) and then more fully in The fractal geometry of nature in 1982. But it wasn't until 1985 that Benoit Mandelbrot announced to the world on the front cover of Scientific American the discovery of the ultimate fractal set that contains all others, now named after him - the Mandelbrot set. The study of complex numbers involving real and imaginary number components real started to open the way for a paradigm shift. Complexity was revealed to be underpinned by chaos, but just the right amount of chaos was needed to give rise to spontaneous order. An awareness of the hither to obscured higher dimensional world that creates our reality was suddenly on the high street.

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The Mandelbrot set - a universal image of dynamic complexity

Is 3-D all there is? The key to understanding future science is to understand the multi dimensional nature of the universe. With the advent of the mathematics of Ron Pearson’s theory we can understand that there can be several, if not an infinite, number of dimensions and that these are merely separated by frequency of vibration. These can therefore exist side by side, very much as TV stations. People living in the invisible universe with etheric bodies will experience their reality exactly the same as we experience our physical reality in the 3-dimensional plane.

In the end we are only processing information and, with the advent of computer games, simulation and virtual reality, experiences obtained through virtual information are almost as good as the experiences obtained through real information. The processors, our minds and brains, merely process information and the data inputs are our sensors. 85% of our information enters through our eyes, from photons that stimulate the retina. 10% of information comes from our hearing and this is why pupils often drift off to sleep when the teacher drones on for too long - it is much better to be more visual. The other 5% are reserved for physical touch, taste and smell, but despite being a relatively small proportion of the total input of signals to the brain, the experience of, for example, a particular smell or scent can be extremely memorable and stimulate a lasting experience.

I often use the analogy of a woman’s perfume, which is extremely evocative and memorable. Even 20 or 30 years later a particular perfume will trigger an emotional response and memory. This is quite a powerful memory mechanism but, again, our own senses are extremely limited. The bandwidth of frequency and wavelength that we can observe in visible light is extremely small. Insects use a far higher range of frequencies and wavelengths. They can see on average at least four bandwidths in ultra-violet. Many animals use infra-red, such as snakes, which will target in on infra-red hot subjects, especially when they move, in order to inject their venom. Bats, using ultra-sonic hearing, to detect the small wing beats of moths. Thus a whole, invisible universe was only opened up with the advent of the discovery of radio and television waves. Interestingly some of the early pioneers of these media were of the opinion that they had discovered the answer to humanities spiritual conundrum and that within these wave constructs would be found the realm of the afterlife.

Most notable of these scientific frontiersmen were the first sender of a radio message Sir Oliver Lodge, an esteemed physicist Sir William Crookes and the inventor of television John Logie Baird. For voicing their opinions in open contradiction to the organised authorities of church and state their work was censored and their lives put on hold indefinitely. It is only now that they are receiving the true recognition that they deserve for their advances in Physics. They reported many of their observations, but unfortunately, because of the lack of mathematical equations to substantiate their observations, they were much ridiculed by the popular media.

As we now have the mathematics for the survival of physical mortality and associated phenomena, this should no longer be a barrier. We now know that it is a multi-dimensional universe and that this has been a close guarded secret for many, many years by secret occult groups.

The development of the atomic bomb relies, for example, on the knowledge of higher dimensional geometry. For the final explosion to occur a geometric process is necessary, which relies on very fine tuning to do with the position on the Earth’s surface, the position of the sun and the position of the moon finally to unlock the atomic power through fission. We can therefore see that scientists have been aware of this for at least 50 years. It is perhaps one of the best-kept secrets in the scientific community and the world?

In my early scientific career, I went through a phase of believing, as do most people, that everything we knew was freely on display and that if one read the literature in the public domain one would know all the answers and all the possible research. But I was very wrong. I discovered in 1989, to my own incredulity, that I had had a completely naive perception of the workings of the world.

There are two camps of science; white science, which is openly on view, with squeaky-clean peer reviewed public domain literature and then there is the deliberately occulted black science. This is the science of the X-Files and it was only with the advent of the X-Files television programme that the public became more aware of this dark side of science. Quite simply, if you were to produce a wonderful flying machine that was far more advanced than anybody else's then you would not patent it because the people at the Patent Office would immediately siphon off your ideas to the military. You would not put it directly into the public domain, because you would immediately lose your economic or strategic advantage, you would deliberately keep it a secret!

This then is the basis of the black world. Black Science exists and is very real and I have made an intensive study of many aspects of Black Science in order to uncover information relevant to my quest for understanding how the universe works. Now, with the Internet, information can be exchanged much more freely and access to this Forbidden Science can be readily obtained by any individual with the knowledge and perception to know what they are looking for. Therefore, the powers that be now rely on ignorance to enshroud their secrecy. The public have been officially dumbed-down through state imposed educational curricula exploiting the mass heard instinct to restrict individual flair and thinking. Interestingly I have noted that most entrepreneurs tend to come from a public (private - US equivalent) school background where enterprise, excellence and individuality have been encouraged to flourish.

The control of information and disinformation is a very powerful tool in manipulating the electorate. We live in a society obsessed with spin and the Government spin doctors manipulate the popular crowd just as they have done from time immemorial, until they are persuaded to make the correct decision, which was the one the politicians wanted all along. This has been policy from the earliest of times and is part of what I term the Roman control mentality: bread and circuses to keep the crowd happy; give them what they want, distract them by any means possible and get them to vote for what you want to do!

It is very much this case with hyper dimensional geometry, it was considered an abstract esoteric subject of little practical worth to the sensibilities of the practical Roman business mind set. It was back of the envelope stuff, it didn’t have any real monetary meaning and therefore was not worth learning. Mathematical lessons at school did not include geometry of this nature. Geometry itself was considered to be relatively obscure and of limited value. We therefore find a regime of ignorance whereby the average person is put off knowing about these things. Yet to our ancient forefathers it was considered to encompass everything, the philosophy of a higher learning and to be of infinite value.

Mathematics tends sadly to be a subject that most people dislike and almost regard as a punishment, but it is not really their fault. Mathematics can be divided into three sections. First there is secular mathematics, which is the mathematics of money and how things work in real life, quantities and volumes. Then there is symbolic mathematics, which becomes more complex in the form of algebra and solving many complex problems through using symbols and letters. And finally, there is sacred mathematics, which is perhaps the highest form of mathematics and is to do with how the universe works.

Sacred geometry part of that latter component, for a long time was held in obscurity, known only to a few occult groups and religious organisations that knew its true value and practised its tenets. Chief among these in modern terms are the Knights Templar, whose obsession with the Golden mean ratio and the secrets of star shaped pentagram were self-evident from their literature, diagrammatic representations, cryptic cartography and architecture. But we can trace their roots back further to the Pythagorean Mystery Schools and back beyond those to Egypt, from whence the Pythagorean acolytes gained most of their knowledge. The Egyptians themselves based much of their culture on Sumeria, and the Sumerian's could trace their past back to an antediluvian culture that existed before the Great Flood, when a much higher civilisation existed in isolated pockets on the Earth.

The understanding of geometry is, therefore, the understanding of the universe. The classical Greek mathematicians knew the value of this study and of all their sciences and arts, geometry was considered the highest. The Platonic School had the motto above its doorway ‘Let none enter who do not understand geometry’. The Pythagoreans were obsessed with the pentagram and the dodecahedron as being the mathematical mechanism by which reincarnation was thought to occur. One particular student who divulged the inner secrets of the dodecahedron to the public was executed for this transgression such was the seriousness of this subject taken. Later on this was to transfer into the Knights Templar of which Leonardo da Vinci was a grand master! Among the other reputable grand masters were Sir Isaac Newton and Sir Robert Boyle, another great British scientist of his day. Is it any wonder, then, that these people had the knowledge they did? They had access to information that the average person would not have had access to and crucially the genius to make use it.

Now, come along with me on a journey most strange and venture into the mystic realms of other spatial dimensions! Geometry is the key and, as we will see with a few simple explanations, everybody can understand how this works. Nature is very elegant, very minimal, it creates the most out of the least and it is above all ruthlessly efficient. Therefore, in nature the shortest distance between two points is a straight line. This is the basis of geometric construction. The forces of attraction and repulsion will take the line of least resistance, which is usually the shortest. Therefore, if we begin to construct our number theory and figures we start with the numbers 1-10 in the linear sense and expand them into two spatial dimensions - 2D. We can see that the first shape we can make is the circle, for all things are resolved within the circle, it is both the beginning and the end, alpha and omega, the one, the whole and a hole! It encloses the maximum space for the minimum boundary, it is the first shape to manifest by spinning from the void, the point, the origin; all is joined by the circle. Through division the circle becomes two, joined circumference to centre. Thus is born the duality of our physical universe and the birth passage for all higher geometric shapes. The most sacred of shapes produced by this mathematical union is the ellipse or vesica piscis (Latin for fish's bladder). From within and through this sacred shape is born the first stable form in two dimensions, the triangle. With its three points and three sides of equal length in two dimensions it is the return to unity expressed in cosmic principles. It is the first stable, regular shape in our geometric tool kit. Any engineer knows that you use triangles to build bridges and structures, because it is the most efficient and strongest shape. Taking three straws of equal length join them simply with a piece of sellotape at each corner and you will create a stable shape. Similarly, three pieces of metal hinged at their ends will form a rigid triangle. Expressed in iconic terms only God was depicted with a triangular halo, a visual manifestation of the principle of trinity much displayed throughout the Christian faith.

Now we extend this two-dimensional shape into a three-dimensional shape called a tetrahedron. The tetrahedron is the basis of modelling for everything in three dimensions. Taking the triangle, which we have formed with just three straws, we erect three more straws upon its base and join them to a single point at the top. We now have a triangular pyramid. It is called a tetrahedron because tetra - means ‘four’ in Greek and –hedron means ‘faces’. So the object has four faces. If the object is made with triangles of card or similar material then we have a solid tetrahedron. If it is made with straws we have a skeletal tetrahedron, which has four corners or vertices, and is composed of six straws. If we then take two tetrahedra and interlock them into a star shape, we create the star tetrahedron, which is also known in two-dimensions as the Star of David as displayed on the flag of Israel. We see that if we put this on its side it becomes a cube, so a cube contains two tetrahedrons interlocking within it and is made by merely connecting the extra points with lines.

Now we come to understanding the fourth dimension. We follow the same principles in mathematics and simply join each corner or vertex to a fourth line. If we place our finger on the straw triangle we can see that just two lines or straws connect each corner. If we place our finger on a corner or vertex of the tetrahedron we see that each corner is connected with three lines. So to make a fourth-dimensional shape we simply add another point in the middle of the tetrahedron and then join four more lines or straws to the four internal corners of the tetrahedron. This now is then called the hyper-tetrahedron or simplex.

This is the first fourth-dimensional shape. My attention was drawn to it by the epic pictogram found in the wheat at Barbury Castle in 1991. This was a giant theosophical mandala composed of what appeared at first to be a triangle on the ground, but when analysed was found to be a two-dimensional depiction of a four dimensional tetrahedron! Thus my attention was drawn most dramatically for the first time to the subject of hyper dimensional geometry. The crop glyph had performed its function perfectly, for it had raised my own personal conscious awareness to this wonderful and exciting new frontier of reality. And so crop circles started to play an integral part in my own personal development and as an anomalous phenomenon they attracted my increased curiosity to the point that I felt compelled to go into the fields each summer to experience them first hand. They were vital in unlocking my own spiritual evolution and development and in helping me to grasp the significance of hyper-dimensional geometry. Therefore the fourth dimension is to be found inside, and around every 3-D point of matter is a four-dimensional energy filled space creating it. As I came to understand much later, the four-dimensional space is the real thing. The three-dimensional point is, but an illusion created by positive and negative forces distributed in space. This is most important to grasp as we take the next hyper step in comprehension, for our 3-D reality is but an illusion created by four dimensional space.

To expand this point within the tetrahedron so that we can actually live in an organised space, we have to create a hyper cube. The hyper-cube has been the subject of fascination for such artists as Salvador Dali. He used the tesseract of the hyper-cube to depict famously a hyper-dimensional crucifixion of Christ. Dali was fascinated and obviously aware of multiple dimensions. In the hyper-cube, four lines join all 16 vertices or corners at each individual point. This is not as complex as it sounds! If we take a cube made of eight corners or vertices and twelve straws, we then simply join each corner to a larger cube, which encompasses the smaller cube. We now have a cube within a cube. This is called the hyper-cube and is one form of this shape.

The link between the two cubes can be so small that we would never notice the one cube inside the other cube. Therefore we would never notice the other dimension around us. We can see quite easily then that our own physical body could be the inside cube and that our ethric or electrical energy body could be the fourth dimensional outside cube or vice versa. We can then start to understand how it is possible for spirit to inhabit matter.

If we focus on a picture of the hyper-cube it will tend to start moving. This is a process whereby our consciousness and our sub-consciousness is registering the shape and trying to make sense of it. The natural function for this is to whirl. This power of using perception and consciousness has been used to great effect in Eastern cultures, notably in India, with the use of yantra, which are sacred diagrams used for meditation. The sacred shapes and diagrams tend to be triangles and the most famous one of all, the Sri yantra, is composed of a series of interlocking triangles producing a very hypnotic shape. This shape represents the primal vibration or ‘OM’. Recently it was found that with an electroscope and a microphone, when the word ‘OM’ was sounded, the vibrationary pattern on the electroscope was exactly the same as the Sri yantra pictures of visualisation. One has to ask - how did these people know that this is the electronic equivalent of what they were sounding? They were supposed not to have electronics. Similarly in modern times, Hans Jenny a Swiss mathematician found that many complex shapes could be induced by frequencies of vibration acting on various droplets of media such as water and oil. Did the ancients have access to this knowledge? I would say most certainly!

We have looked at cubes but, really, everything in the universe is made of spheres. The physical universe is dominated by spherical form. The reason is mathematical and very, very simple. The sphere operates using the maximum volume for the minimum surface area. One can ask the question in two dimensions, why is a pizza round? A pizza is round in shape, because you get more topping on the surface for a minimum amount of crust. It is quite simply, economy of circumference for a given surface area.

The four-dimensional equivalent of the sphere is the hypersphere and the hypersphere in three dimensions as a shadow projection appears as a toroidal doughnut - this is the doughnut with the hole in - and this turns out to be a primal shape for how things are made in three dimensions. A wound toroidal doughnut becomes a hypersphere and really is a sphere within a sphere. A toroidal doughnut can be used to form a neutral macro-particle. Charles Cagle, in 1997, copyrighted a diagram showing a macro-particle in the shape of a toroidal doughnut.

With spin-rotated stabilisation the toroidal doughnut is the perfect shape for maintaining structure. By simply observing smoke rings we can see this in action. When one blows smoke rings using a cigar or a smoke box apparatus to produce smoke rings, the rings prove to be remarkably persistent and exhibit many unexpected properties. To try this take a cardboard or plastic box and cut a circular hole into one end. Remove the wall on the opposite end and attached a membrane of rubber, made from a toy balloon and secure with duct tape or similar. The box is then filled with smoke and when the rubber is tweaked and snapped a doughnut shape is expelled in the form of a smoke ring from the annulus or circular hole. These smoke rings are remarkably stable and can extinguish a candle flame at quite a considerable range. The rings will move across a room and maintain their shape. They are self-sustaining, able to bounce off of each other and behave as if they are solid objects.

With this observation we start to understand how atoms can be an illusion yet also appear to be a solid object by simply rotating electrical charges about a nucleus. Finally, when the smoke ring breaks down it dissipates and disappears into millions of small particles, but whilst the toroidal doughnut is functioning the object has stability. A larger electronic version of this has been turned into a sonic or acoustic cannon, whereby a three-foot diameter toroidal doughnut of sound was produced and projected across a football stadium with the capacity of knocking over a person. One can see that this could be used as quite an effective piece of technology in various non-lethal force situations.

It has also been found that the magnetic fields surrounding UFO craft appear from video analysis to be toroidal doughnut in shape. The nesting toroidal doughnuts of electro-magnetic energy would allow the craft to move effortlessly and silently through space and particularly within our atmosphere. There would be no sound or shock wave, as the air would be sucked in, creating a vacuum, which would allow the craft to slip into the space created. Also when craft were hovering in formation the toroidal doughnut structure of the magnetic fields would interlock with other craft to produce a stable formation capable of manoeuvre as one object. The use of toroidal doughnut magnetic fields is used with some success in atomic fusion generators and a magnetic bottle or magnetic toroid field is used to contain the extremely high temperatures in excess of 100,000ºC such that all matter would melt instantaneously upon touching that heat. Therefore the only thing that can contain it is a virtual magnetic bottle created of toroidal doughnuts. Doughnuts are strangely the key to many things!

As with everything, nature has got there first. We only have to look at our own blood cells, especially the red ones, to see that the toroidal doughnut shape is favoured in transport systems. It is the perfect shape for containing an extremely large surface area and the doughnut-shaped blood cells, although they are filled in the middle, circulate around the body without too many snarl-ups endlessly throughout one’s lifetime. Rubber rings mimic this. If you go to one of the big water parks such as Blizzard Beach in Disney World you will see the rubber rings going around the rapidly flowing watercourse with people merrily enjoying themselves. Again, it is the toroidal doughnut shape that is perfect for going around an irregular course of fluid and bumping off the sides and not getting stuck by obstructions. So we learn, that from Disney to red blood cells, toroidal doughnuts are the way to go.

Now we come to nesting polytopes. Polytopic landscapes are higher dimensional landscapes created by high dimensional geometry and there are several very good interactive CD-ROM's for the PC, which can be found in the bibliography and are well worth looking at. With the advent of computers we can all view the complexity of these landscapes in our own homes. Before we had to rely on seeing the odd video or television programme, but now we can all grab hold of a CD-ROM and view these magical shows ourselves. Polytopic landscapes found particularly on the two computer CD programs Polytopia 1 and Polytopia 2, produce higher dimensional geometric shapes, which then give shadow projections into two dimensions. These beautiful shadow projections fall onto a two-dimensional area and display many complex and attractive forms. It suddenly struck me as I saw these for the first time, that this is exactly what crop circles are, higher dimensional energy forms hitting a two-dimensional landscape and producing a flattened crop. Many of the patterns of complexity in our crop fields each summer are therefore alluding to higher-dimensional mathematical shapes. Yet, all this is thanks to our infamous Doug and Dave, who claim to have invented most of them over a 10 year period from the mid eighties to the mid nineties. Not bad for two old age pensioners who allegedly came out of the pub most summer nights and decided to do hyper-dimensional geometry on a whim!

Life is a never ending continuous set of revelations and the next revelation came when I saw a video by Dan Winter, a rebel esoteric mathematician, describing among other things the golden mean Phi-ratio heart muscle contraction. He also went on to describe the Earth energy grid and the fact that nesting polytopes were responsible for how energy interlocks to produce energy fields. Up until then I hadn’t realised how the platonic solids, which are the five regular platonic solids used in classical Greece, interlocked and interacted as energy fields. It just goes to show how bad my education was as a child, in that nobody had ever told me this. We had studied platonic solids once in passing, but we had not been told that they interlocked. It was probably a case at the time of, ‘Well you don’t need to know this and well we don’t know it either, so let forget it!’ But upon examining mathematical geometry books myself I then found that this was indeed the case and that classical mathematics books do in fact contain this information.

It just goes to show that there are no secrets, it is just that the information is so vast that one does not really bump into it unless somebody points you in the right direction. So in life it is the case of meeting the right person or teacher who will point you in the right direction. As Carl Sagan once said, ‘The Earth is like a giant library containing billions of books, but the trick is to read the right books.’

If we take the five platonic solids, they are in order of number of faces; the tetrahedron, which we have already discussed, having four faces; the cube, with six regular faces; the octahedron, with eight regular faces; the dodecahedron, with twelve regular faces - do means 2 and deca means 10, so 2 plus 10 is 12 faces and the icosahedron, which has 20 regular faces. All of these shapes can be made quite easily with straws and/or just simple geometric nets created out of card and it is well worth playing with these shapes to get some idea of how they work. For it is only when you start using straws that you realise, as with the star tetrahedron that it is really a cube on its side, so playing with straws is highly recommended! For the more advanced experimenter I chanced upon a marvellous geometric construction set called Zometool, this is easily found via the internet and well worth buying. A true understanding of higher mathematical shapes is acquired by construction of the many projects included in the boxed set booklet, this leads very rapidly to self experimentation and discovery with new shapes in sacred and hyper-dimensional math.

If we now look at nesting, we come across the concept of dual platonic solids. Two dual platonic solids are simply ones that fit neatly inside each other so that the point, corner or vertex of one solid immediately sits in the centre of the face of the next higher solid. These duals include the cube nested with the octahedron; but the highest one and the most important one, is the dodecahedron nested within the icosahedron. The 12-faced, pentagonal, 12 patch leather ball, as Plato and Pythagoras called it, fits exactly inside the icosahedron such that all the corners touch the faces of the higher geometric shape and vice versa. Nesting can continue alternately dodec/icosa/dodec/icosa indefinitely so that complex interlocking energy fields can project outwards from a given object.

What has this got to do with anything? Well, quite simply, this is how energy fields nest and interact to produce our reality. Once one realises this we can see the relevance of studying this form of geometry. The discovery of the Earth energy grid was one of the things that attracted my mind to this geometry. The Russians discovered very early on in the early 60’s that the basic energy grid of the Earth was like a 12-patch leather football, a dodecahedron, and they then started to produce tentative maps of the Earth's energy fields in publications

Dan Winter demonstrates rather nicely how this nesting may work to produce our world, in the fact that if you take a dodecahedron shaped quartz crystal and hold it at an angle you can see a holographic cube in the middle. This holographic cube very neatly describes our reality. A higher dimensional energy field in the shape of the dodecahedron is creating our 3-D reality as the holographic cube in the middle and as, of course, you twist the energy field so the cube disappears. This corresponded neatly with the description given by a lady of how a UFO disappeared in front of her. She held her hands one above the other vertically, palms facing, and described it as a movement with her hands, being able to describe it in words only as a collapsing, twisting and folding. This consisted of compressing the hands towards each other whilst twisting.

Now, for a long time this description puzzled me, but upon seeing Buckminster Fuller’s favourite toy, the dynamaxion or bend'a'flexor, as he called it, which is a cube octahedron with flexible rubber joints, the exact same movement is used to twist from the cube octahedron to the icosahedron and this, I think, is a demonstration of how it is possible to change dimensions? As the energy field is collapsed, the object that looks so real in 3-D then twists and collapses into the next dimension, whereupon it becomes invisible! This is entirely consistent with the witness’s statement.

Nesting polytopes and nesting energy fields are therefore essential study in order to understand how matter interacts inter-dimensionally and in how energy fields interlock to create our holographic reality. The leading workers in this field that I can recommend for study are Carl Munck, an American researcher, who has done much to elucidate the early pioneering work on the grid, along with Bruce Cathie, the New Zealand airline pilot who has written several books on the Earth energy grid.

Nineteen point five is perhaps the key number for understanding how energy can cascade between dimensions from within a sphere, from 4-D into 3-D and as the Earth is a sphere we should consider it's application together with the planets in our solar system and indeed the Sun itself. This was brought to public attention by Richard Hoagland, who was fronting a loosely based group of scientists who all contributed their own mathematical and special expertise to a study of the Mars surface anomaly known as the Face, especially Carl Munck from the defence-mapping sector and Mark Carlotto. His associates had been attracted by the apparent discovery of a large rock structured Face on Mars amongst the Viking data sent back to Earth in the mid 1970's. They noticed that there was also associated geometry relating to this strange value repeatedly described amongst the ruins surrounding the face. Despite much scepticism by people initially, a real advance in thought was inspired by this chance find.

For if we look at the geometry of what was being discussed by this group called the Mars Mission, the geometry put forward was valid. It is checkable; we can all go away and check this ourselves so it is a real piece of information that can be used and tested despite the controversy. Again, scientific observation comes into play from this mathematics in that we can observe that the major energy events on all the planets of the solar system occur at either 19.5 degrees North or South of the equator. The Sun with its enigmatic sunspots, that indicate major energy disturbances on its surface, can be found to have these dramatic events precisely distributed in bands at both 19.5 degrees North and South. But, why 19.5 degrees? The group discovered that if you placed a tetrahedron inside a sphere so that the corners or vertices touched the circumscribing sphere in the equatorial plane and the fourth point the pole, then the three corners of the base of the tetrahedron touched at exactly 19.47° (19.5° when rounded up).

This geometry was both elegant and fascinating at the same time. Just what could it all be about? Well, this is what we are going to discuss in the next chapter as I think really the whole 19.5 phenomena deserves a chapter of its own.

More fun with dimensions: to understand how it is possible to have a one-sided, or one-dimensional, piece of paper when introducing the idea of polytopic landscapes, one can use the Möbius Strip. This is quite simply a length of paper cut into a long ribbon which is then given one twist before joining the two loose ends with a piece of sellotape. You are then in possession of a one sided piece of paper. To prove this to the audience or student, simply place a pencil onto the paper and start to revolve the paper beneath the pencil. You will find to your utter amazement that the pencil circumnavigates the piece of paper twice and, like Robinson Crusoe discovering his own footsteps, you come back to the same position without the pencil ever leaving the side of the paper it is touching - we therefore now have a one-sided piece of paper! This can be taken further into three dimensions with Klein bottles - worth investigating!

M C Escher, the famous Dutch mathematical artist, used this to great effect in one of his famous prints, with ants walking around the Möbius Strip to give a living emphasis to it. He was a great artist and able to use spatial concepts, which he saw in his head and derive fantastic pictures of illogical landscapes based upon these false isometric drawings, that play on our optical senses in the fact that optical illusions are easy to perform on the human eye. The human eye will always look for a straight line and where there is no straight line it will connect the points to form an imaginary straight line. This can lead to lots of fun and optical illusions, as exploited by the great artist and his paintings.

Isometric drawings are a great exercise in understanding how things should look in two dimensions when you are describing a three-dimensional object and great fun can be had with dotted paper, drawing both possible and impossible shapes as we explore that concept. Really, mathematics does not have to be boring and this exercise often gets people interested in spatial mathematics.

It can also be further extended into examining complex shapes that form just one dimension, such as Klein bottles, which are a fun part of maths in looking at 3-D shapes that have only one surface. As Professor Ian Stewart said, ‘the trouble with us is that we know too much about 3-D and we always assume when we look at something that has an outside there must be an inside. But in mathematics, especially higher mathematics, this is not necessarily true because you can have a one sided piece of paper, just as you can have a one-sided piece of space, in which case there is no middle bit.

It appears, therefore, that we may live in this sort of universe, as creatures existing on the surface of a complex membrane or sheet of space-time and that there really is no inside to our 3-D physical universe - now just how weird is that?

Mar 20, '12
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