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Multiple View Geometry in Computer Vision
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PRACTICAL GEOMETRIC PATTERN DESIGN: DECAGONAL PATTERNS IN PERSIAN TRADITIONAL ART
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Sacred Geometry Journal: Patterns in Nature
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Geometric Methods and Applications: For Computer Science and Engineering (Texts in Applied Mathematics, 38)
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Shapes
Sacred Geometry Shapes Sacred Geometry is often referred to as the “architecture of the universe”, it is found throughout the natural world. It is all around us and is one of the very few subjects that satisfy both the left brain and right brain hemispheres simultaneously. It satisfies the left brains desire for logical, sequential and objective data. It also satisfies the right brains desire for random, intuitive and subjective data. It is simply not possible to cover all aspects of sacred geometry, although I will try to include as much as possible in this article. People have spent entire lifetimes studying a single aspect of sacred geometry; such as a rabbi studying the kabbalistic tree of life. The following content provides an overview on all Sacred Geometry symbols. The Graphic below shows all of the basic Sacred Geometry shapes. VESICA PISCIS Vesica Piscis is a mathematical shape formed by intersecting two identical circles so that the center of each circle intersects the perimeter of the other. The vesica piscis is commonly used for Venn Diagrams and in various art and in seals organizations. Various other symbols can be derived from the vesica piscis including the Ichthys, also known as the “Jesus Fish”, the triquetra symbol, and the Reuleaux triangle. It has also been a symbol used by Freemasons for many years. Historically, the vesica piscis has been used as a system of proportions for architecture, illustrated in the works of Vitruvius. SEED OF LIFE The Seed of Life is a universal symbol that also represents the seven days of creation. It is found in all of the major religions and can be seen in many sites around the world, such as churches, mosques, prayer books and ancient temples. The two dimensional projection of the seed of life contains seven overlapping circles. It has been used for centuries as a symbol of blessing and protection. EGG OF LIFE A symbol that has been used to represent new life, rebirth and fertility. The Egg of Life is the second iteration of a process that constructs the Flower of Life. Adding six additional circles to the seed of life you are able to produce the Egg of Life. This can also be formed by the eight spheres contained within the three dimensional Metatron’s Cube. FLOWER OF LIFE The Flower of Life was known around the world in ancient times and has been found in Ireland, Turkey, Egypt, Greece, China, Israel, Germany, England, Tibet, Japan and Sweden. The typical depiction Flower of Life always stops after 19 circles, the remaining circles are hidden outside the bounds of the outer circle. In the past this image and knowledge related to the Flower of Life was so sacred that they couldn’t allow it to become common knowledge. Out of the complete Flower of Life pattern a new shape emerges; the The Fruit of Life. The Flower of Life is said to be the actual blueprint of the universe continuing the basic design of every atomic and molecular structure, life form, etc; essentially everything in existence. Modern science and quantum physics are confirming the cosmic significance of these ancient sacred geometric forms which represent the building blocks of our physical universe. The ancients knew about the energetic nature of our existence. Other sacred patterns such as the Egg of Life, Seed of Life and the Tree of Life also exists within the Flower of Life pattern. GERM OF LIFE TREE OF LIFE The Tree of Life is another popular universal symbol it represents numerous systems across varying cultures and religions. Like all the other symbols in this list it is not identified by a single culture and it has been used around the world for centuries. So what does the Tree of Life represent? A connection to everything including the things we cannot see or the void beyond. It reminds you that you are not alone in the Universe but rather part of a vast interconnected net
work. The Tree of Life provides a hierarchical structure for all of the forces in the Universe. It can also be used as a road map of one’s soul. The 10 spheres are called the “Sephiroth” translating to eminiation. These are connected by different paths. The first Sephira on the Tree of Life is at the top. It represents cosmic consciousness. The Sephira at the base represents the material world. The rest of these Sephiroth represent different qualities of the Soul. These can be divided into three pillars; the Pillar of Severity, The pillar of Mildness and the Pillar of Mercy. FRUIT OF LIFE The 13 spheres represent an extremely powerful sacred geometric pattern known as the Fruit of Life, considered the Holy of Holies of Sacred Geometry because it’s hidden within the ancient and well-known Flower of Life pattern. The fruit of life is composed of 13 spheres; this is actually the 2d representation of a 3d structure Thirteen represents unity and transition between worlds / dimensions. Those of you who play an instrument know the chromatic scale of 12 notes the thirteenth note is actually repeating the first note but on a higher frequency; when you attain a higher octave you enter a higher dimension. If we connect all the Centers of the 13 circles the result is 78 lines which create a structure known as Metatron’s Cube. This structure contains all five platonic solids, these three-dimensional figures and their different combinations reveal every possible geometric law forming the physical reality we are experiencing. The spheres of the Fruit of Life represents the female movements or elements of creation and provide the field for the 78 ray like male movements of creation. The geometry of Metatron’s Cube is set to underlie what modern physics calls the quantum vacuum or void and is fundamental to the creation process at the quantum level. What we’re talking about here is what’s been called zero-point, life force or bio force energy. METATRON’S CUBE One of the most fascinating aspects of sacred geometry is the symbol known as Metatron’s Cube. So who is Metatron? If you investigate, Metatron was an archangel who was mentioned in Islamic, Judaic and Christian medieval mythologies. He was considered to be the scribe of God, this would make him the equivalent of the ancient Egyptian scribes of the gods known as Thoth. According to legend a scribe of God would understand the basic principles of creation which are almost always associated with sacred geometry. Many times the Archangel Metatron is pictured with or holding a cube. Metatron’s Cube 3D Model by pardesco on Sketchfab The graphic known as Metatron’s Cube didn’t show up until later when a Medieval Italian Mathematician Leonardo Pisano, discovered Metatron’s Cube. He read the history and legends of Metatron and was also familiar with sacred geometry and was probably an initiate in one of the schools that held those secrets. Metatron’s Cube traces back to nature’s first pattern, and the 13 circles that are found inside the fourth circle in Nature’s first pattern. When you connect the centers of all the 13 circles, you get the shape known as Metatron’s Cube. Leonardo Pisano also discovered that you could create all 5 of the platonic solids within Metatron’s Cube. Metatron’s Cube is a transcendental form and beautifully illustrates the mental universe through the unfolding of 2-dimensional realities into 3 dimensional realities The Hermetic axiom reads “all is mind, the universe is mental” and Metatron’s Cube is teaching us about this conceptual mental nature of the universe and the current experiments in quantum physics are also teaching us this exact same thing. Without first being a conceptual equivalent in consciousness nothing could exist. Sacred geometry is the creator’s conceptual architecture on which all things precipitate. GRID OF LIFE The star tetrahedron is the basis of the 64 grid tetrahedron also known as the Grid of Life. This is the masculine equivalent to the Flower of Life pattern. As the name suggests, this shape is made up of 64 tetrahedrons. Which as you know is one of the platonic solids. When the 64-tetrahedron is overlaid on the flower of life this represents a powerful concept; the spheres represent space and the lines lines represent where space and time come together. One of the most fascinating aspects of the 64 tetrahedron is the number 64 itself and how it appears throughout the natural world. Here are a few examples: There are 64 codons in human DNA 64 is the number of cells we have before the cells start to bifurcate (differentiate) after conception. 64 is the fundamental number in computer memory and bits of code 64 is encoded in the Tetragrammaton within the Hebrew Bible, which is the four letter theonym YHWH that means God in Hebrew. 64 is the number of generations from Adam until Jesus 64 manifestations of Lord Shiva in Hinduism 64 squares in chess and checker boards 64 hexagrams in the I Ching 64 tantras which is a form of Hinduism 64 is the maximum number of strokes in any Chinese character This metaphysical symbol can produce many trees of life which radiate inward, this is why the 64 tetrahedron is symbolic of linking up ones individual paths in harmony with our soul family. It is said that connecting with your soul / star family will amplify each individual’s energy to accomplish and manifest one’s goals. A true symbol of the mastermind principle of harmony with among a group of individuals. VECTOR EQUILIBRIUM 1 VECTOR EQUILIBRIUM 2 TORUS The shape known as a ‘Torus’ is a type of vortex. So what is a torus? The word “torus” comes from the Latin word meaning swelling. In sacred geometry it is the first shape to emerge from the Genesis Pattern. The torus is found in everything from atoms, to life forms and even in all cosmic bodies such as stars and galaxies. It is a primary shape in existence. For many years ‘seers’ have taught that the human aura appears as a series of nested torus formations around the body. Energy flows through the center of the body and loops around to connect the feet and head. The flow from the feet to the head is bi-directional.The energy is flowing through the surface of the torus and inside an in a spiraling pattern, this is why the sacred geometry shape is shown with curves lines throughout the middle, while also adding the illusion of a 3d perspective to the 2d depiction. The Torus is a 2d depiction of a 3 dimensional shape, like many other sacred geometry forms. The 3d shape is known as a horn torus. This can be mathematically described here. MERKABA The Merkaba symbol (also spelled Merkabah) translates to light, spirit, body. The symbol consists of 2 intersecting tetrahedrons that rotate in opposite directions, this creates an energy field that is commonly referred to as a light body. Merkavah mysticism is an early school of Judaism, practiced between 100 BCE – 1000 CE. The teachings were based on visions found in the Book of Ezekiel and hekhalot literature. These stories speak of ascents into the heavens and to the Throne of God. Everyone has access to their own personal Merkaba or vehicle of light with practice in meditation. It is said that once one has access to their Merkaba it can provide a powerful protection and transport your consciousness to higher dimensions. The Merkavah symbol can remind us of the potential power we hold when we unite our bodies energy to Source. This symbol combines opposing energies in perfect harmony and has allegedly been used by ascended masters to connect to higher realms. In modern esoteric teachings, the Merkaba is considered an interdimensional vehicle. It is also known as a star tetrahedron and can be imagined as a 3-dimensional Star of David. HOW TO DRAW SACRED GEOMETRY SHAPES All of the basic Sacred Geometry shapes can be easily derived from this cubic grid developed by Pardesco. How is Sacred Geometry Created? Let me show you how nature created its very first pattern. It is interesting to note that Genesis in the Bible essentially describes the same process I’m about to show you. However it should be known that this knowledge is most likel
y taken from the ancient Egyptian mystery schools and it is not original to the bible. In the beginning, there was an infinite void or complete emptiness. A space with zero dimensions and no time. The first creation started when nature, god, spirit or whichever term you prefer projected itself in all six directions, starting from a single point. In this depiction nothing material has been created, only imaginary lines that represent pure consciousness expanding out front, back, left, right, up and down. To define boundaries in space all that needs to be done is connect the end of each point to every other point. The beauty of this is that you cannot make a mistake, it is simply impossible. Once all the ends are connected we have what is called an octahedron, which if viewed from the correct perspective reveals a hexagram, also known as the Star of David. Now that boundaries have been created, relative movement within the void are possible; where as previously consciousness had nothing to move relative to. In the Bible, Eve was supposedly created from Adam’s Rib – when you compare this story to the next step in the process you can begin appreciating the idea that maybe the Bible is only an exoteric recreation of older esoteric wisdom. The octahedron is made up of perfectly straight lines. In sacred geometry straight lines are considered to be male and curved lines female. At this stage we have a purely male shape, this is why the freemasons consider the male to be the generating principle. The next step in the process involves spinning the octahedron in one full rotation around each of the axes defined previously. Again it is impossible to make a mistake in this step. The end result will always be that the parameters of a perfect sphere have been traced. Now we have a fully female shape. Eve has now been created from Adam. Consciousness now has a reference point, the spherical membrane, on which to repeat the process. Moving to the outer edge of the spherical membrane and creating another identical sphere results in a very special shape known as the Vesica Piscis. This shape can be found twice in the human body in the appearance of the human eye behind the eyelids and also the lens within the eye, which adjusts the focus of light. To project a third sphere we move to the newly created intersection between the two original spheres. After this third sphere is created, there is only one rule that is followed to complete the first process; always move to the innermost origin points before projecting another sphere. This results in the process being perfect and prevents a chaotic mess being created. Once seven of these spheres have been projected in a vortex like motion, what we have as a result is the Seed of Life. This basic underlying pattern lays the foundation for the creation of matter and the universe itself. It also gives rise to a shape known as tube Torus if the pattern is spun around its central point. The torus is the only shape in existence that can fold upon itself. It is also composed of seven equal segments which can be displayed visually using the seven colors of the visible light spectrum. The Seed of Life is not the complete pattern, there are several more steps to take each revealing another layer and new set of sacred geometry patterns. Following the rule of: always move to the innermost point before projecting, we add an additional six spheres on top of the existing seed of life pattern. This is referred to as the second vortex motion, if this is repeated once more it gives rise to the next complete pattern, known as the Flower of Life. This symbol is found all across the world today, but the earliest known depiction was found at the temple of Osiris in Egypt, etched into the granite using a process unknown to modern science. If we proceed through two more vortex motions, adding additional spheres to the Flower of Life we reach what is known as the Fruit of Life. This is now the complete template for all material things. The Tree of Life can be derived from the seed of life pattern. It is most commonly recognized as mystical concept within kabbalah as well as in various world philosophies and theologies. In kabbalah it is used as a tool to understand the nature of god and the manner in which the world was created out of nothing. The details of this are too complicated to go over in this guide. The next form is known as the Egg of Life, created from eight spheres. We are unable to see the eighth sphere which is located behind the central sphere. This pattern is the most balanced and optimized form for eight spheres existence. The reason why it is called the egg of life is because nature uses this form in the embryonic creation of every single living organism on the planet with no exceptions. We now proceed to one of the most important sets of extrapolations, which can demonstrate the beauty of sacred geometry both visually and mathematically. If we take the fruit of life and extract thirteen circles, what we get is known as Metatron’s Cube. In total there are sixteen spheres in the three dimensional version of Metatron’s Cube. It is interesting to note that all the parameters for a hypercube, also known as a tesseract can be easily extracted from Metatron’s Cube. It should also be noted that Metatron’s Cube is also deeply connected to the platonic solids. Below is a 2D projection of a tesseract: Many of the ancient traditions were fascinated by a science of simple 3 dimensional geometric shapes, that were originally discovered by Plato (who invented school). The Platonic Solids have more symmetry than any other shape you could possibly make. The most advanced hyper dimensional models, that scientists are trying to build multidimensional in the Universe you can see the Platonic Solids show up frequently, because they know that in order to build higher dimensions you need a lot of symmetry. THE PLATONIC SOLIDS In the original model Plato came up with, each shape represents an element. The Cube represents Earth, the Tetrahedron represents Fire, the Octahedron represents Air, the Icosahedron represents Water and the Dodecahedron represents Aether. Connecting the edges of four equilateral triangles creates the tetrahedron. Connecting the edges of size squares creates the hexahedron. Connecting the edges of eight equilateral triangles creates the octahedron. Connecting the edges of 12 pentagons creates the dodecahedron. Connecting the edges of 20 equilateral triangles creates the icosahedron. He wrote about this in their philosophical texts. There are five platonic solids: the tetrahedron, octahedron, hexahedron, the icosahedron and the dodecahedron. What makes these shapes so special is that the faces, angles and edges are all congruent and that each one can fit perfectly inside a sphere with all the corners touching. There are no other shapes in existence that can satisfy these parameters. The five platonic solids also form pairs, known as jewels; the hexahedron and octahedron make up pair and the icosahedron and dodecahedron make up the other. The tetrahedron is the odd one out, as it is a jewel with itself by connecting the midpoints of each face. Each platonic solid is also associated with an element. THE GOLDEN SPIRAL \u0026 FIBONNACI SEQUENCE Closely related to the golden ratio is the Fibonacci Sequence. It was first described by mathematicians in India in about 700 CE and introduced to the west by Leonardo of Pisa, also known as Fibonacci. In his book, Liber Abaci. The easiest place to find this sequence in nature is in plants. No matter what type of flower you look at you will see the number of pedals equals one of the numbers in the Fibonacci series. Rows of seeds in sunflowers and pinecones always add up to Fibonacci numbers. Learn more about the Fibonacci sequence and its relationship to Phi here. The Golden Ratio. What is Phi? The golden ratio appears in ancient Greek architecture and some famous artwork. It is found throughout nature in various forms and is likely to have been quantified after considering a regular pentagon with a side length of 1. The length of the diagonal connecting two adjacent side
s of a pentagon is the golden ratio. Closely related is a shape known as the pentagram, if the shortest of these lengths are 1, then the other lengths are powers of Phi. Ages ago the Renaissance thinker Luca Pacioli was fascinated by this ratio and wrote a book called “The Divine Proportion” which was illustrated by Leonardo da Vinci. The book studies various geometric solids and the ratio’s appearance in architecture and the human body. Many artists throughout time have embraced this ratio in there work, such as Salvador Dali painting The Sacrament of the Last Supper on a canvas in the shape of a golden rectangle. Another example is the ancient Greek sculptor Phidias who used the golden ratio to illustrate the idea of physical perfection. A golden rectangle can be progressively broken down into squares whose side lengths depend on Phi, this is often depicted with the Golden Spiral which connect the corners of each of these squares. Three golden rectangles can be created by connecting the twelve vertices of the icosahedron. You can read more about the mathematical properties of the golden ratio here. Check out our full collection of Sacred Geometry Art available on our website! Additional Resources: Sacred Geometry Wiki Trinity Knot Meaning Vesica Piscis Wiki Ichthys Symbol Explanation Metatron’s Cube Meaning Hypercube Wiki
Homeschool // kindergarten
What comes next? Assess your students’ understanding of patterns by having them identify and continue shape patterns.
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Flower Nature Patterns Sacred Geometry Contemporary Art Digital Art by Dean Marston
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Colorful drawings, rich forms, a captivating depth to look at …. these drawings fascinate me all my life: Mandalas! In Hinduism and Buddhism the mandala (Sanskrit: ‘circle’) acts as spiritual and ritual symbol. It represents the Universe. The basic form of most mandalas is a square with four gates containing a circle with a center point. But also C. G. Jung, a Swiss psychiatrist and psychotherapist, used it for his work. In his autobiography “Memories, Dreams, Reflections,” Jung wrote: “I sketched every morning in a notebook a small circular drawing,…which seemed to correspond to my inner situation at the time….Only gradually did I discover what the mandala really is:…the Self, the wholeness of the personality, which if all goes well is harmonious.” Jung recognized that the urge to make mandalas emerges during moments of intense personal growth. As Jungian analyst Marie Louise von Franz explains: “The mandala serves a conservative purpose—namely, to restore a previously existing order. But it also serves the creative purpose of giving expression and form to something that does not yet exist, something new and unique….The process is that of the ascending spiral, which grows upward while simultaneously returning again and again to the same point.” (C. G. Jung: …
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Looking for a geometric tattoo or adding geometric shapes to your new ink? Click here to learn about the hidden meanings of every shape.
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