Sacred geometry is the study of geometric patterns and sacred numbers that appear in sacred spaces and nature. Some key concepts discussed in the document include fractals that appear throughout nature, the golden spiral and ratio, platonic solids, the flower of life, tree of life, Pythagorean symbols like the vesica piscis and pentagram, crop circles, astrological aspects, mandalas, the tesseract, sangaku puzzles, ley lines, and influences of sacred geometry and Fibonacci sequences in music. The document provides examples and descriptions of these geometric patterns found in architecture, art, nature, and other religious and cultural contexts.

1. Sacred Geometry viraj joshi (2011)

2. What is Sacred Geometry? Plato stated that God geometricised the universe while creating it, and he still geometricises it continuously. Sacred Geometry is the arrangement, configuration, composition and planning of forms used for the architectural plans of sacred buildings, such as churches, mosques, temples, altars, and other religious monuments, and is also employed in religious art. People, since ancient ages have been giving sacred meanings to certain geometric proportions and forms. The forms have importance in religious scriptures of many cultures. The source of these forms has many a times been from the nature, or the discovery of such forms, perfect in proportions, forms, or number has always left man curious. 2 viraj joshi (2011)

3. Fractals A rough or fragmented geometric shape that can be split into parts is at least approximately a reduced-size copy of the whole is called a fractal. Examples of fractals are abundantly found in nature. 3 viraj joshi (2011) A peacock’s feathers Frost crystals formed on cold glass Aloe Plant

4. Fractals 4 viraj joshi (2011) Human Lungs Shoreline Ferns

5. Fractals 5 viraj joshi (2011) Lightning Mountains Branching of a tree

6. Golden Spiral In geometry, a golden spiral is a logarithmic spiral whose growth factor is related to φ, the golden ratio. φ has the value 1.613 and is the average of the ratio of each consecutive 2 numbers in the fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...) 6 viraj joshi (2011)

7. Golden Spiral Occurence in nature: 7 viraj joshi (2011)

8. Golden Spiral Photographs, buildings and paintings with the Golden Ratio tend to have very good composition. 8 viraj joshi (2011)

9. Symmetry in Nature 9 viraj joshi (2011) Bilateral Symmetry Vitruvian Man

10. Symmetry in Nature 10 viraj joshi (2011) Radial Symmetry Helical symmetry

11. Platonic Solids 11 viraj joshi (2011) In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. The Metatron's Cube is a figure in the Euclidean plane created from 13 equal circles with lines from their centers extending out to the other 12 circles. Six circles are placed in a hexagonal pattern around a central circle, with six more extending out along the same radial lines. This pattern shares 2 dimensional resonance with the Flower of Life and the first three Platonic solids, although if additional lines are drawn the other two platonic solidscan be found as well.It is a Sacred Geometry figure. Its name makes reference to Metatron, though there has been no evidence shown relating the stories of Metatron to this design.

12. Platonic Solids 12 viraj joshi (2011) Platonic Solids Metatron’s Cube

13. Flower of Life 13 viraj joshi (2011) The Flower of Life is the modern name given to a geometrical figure composed of multiple evenly-spaced, overlapping circles. They are arranged to form a flower-like pattern with a sixfold symmetry, similar to a hexagon. The center of each circle is on the circumference of six surrounding circles of the same diameter. It is considered by some to be a symbol of sacred geometry, said to contain ancient, religious value depicting the fundamental forms of space and time.

14. Flower of Life 14 viraj joshi (2011) Seed of Life Flower of Life

15. Flower of Life 15 viraj joshi (2011) The first step in forming the Seed of Life (or Flower of Life) is to begin with a circle (as in a 2D model) or a sphere (as in a 3D model). According to some religious beliefs, the first step in building the Seed of Life was the creation of the octahedron by a divine "creator" (or "God"). The next step was for the creator to spin the shape on its axes. In this way, a sphere is formed (see diagram). The creator's consciousness is said to exist within the sphere and the only thing that physically exists is the membrane of the sphere itself. This "first step" is not to be confused with the "first day", the latter being in reference to the seven days of creation.

16. Flower of Life 16 viraj joshi (2011) Derivation of Metatron’s cube and Platonic Solids from the flower of life.

17. Tree of Life 17 viraj joshi (2011) The concept of a tree of life as a many-branched tree illustrating the idea that all life on earth is related has been used in science, religion, philosophy,mythology, and other areas. A tree of life is variously; a motif in various world theologies, mythologies, and philosophies; A metaphor for the livelihood of the spirit. a mystical concept alluding to the interconnectedness of all life on our planet; and a metaphor for common descent in the evolutionary sense. According to the Encyclopædia Britannica, the tree of knowledge, connecting to heaven and the underworld, and the tree of life, connecting all forms of creation, are both forms of the world tree or cosmic tree. According to some scholars, the tree of life and the tree of the knowledge of good and evil, portrayed in various religions and philosophies, are the same tree.

18. Tree of Life 18 viraj joshi (2011)

19. Pythagorean Symbols 19 virajjoshi (2011) VESICA PISCIS- The vesicapiscis is a symbol made from two circles of the same radius, intersecting in such a way that the center of each circle lies on the circumference of the other. The name literally means the bladder of the fish in Latin. The shape is also called mandorla ("almond" in Italian).It is also known in the early Mesopotamian, African, and Asian civilizations. TRIAD - Enclosing the greater area with the smallest perimeter, the triangle, derived from the vesicapiscis, the Triad was considered by the Pythagoreans as the most beautiful number, as it is the only number to equal the sum of all the terms below it, and the only number whose sum with those below equals their product Triad VesicaPiscis

20. Pythagorean Symbols 20 virajjoshi (2011) TETRAD- The tetrad or number four is the first number formed by the addition and multiplication of equals. To the Pythagoreans this symbol and number represented justice as it is the first number that is divisible every way into equal parts TETRACTYS - The Tetractys is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row. As a mystical symbol, it was very important to the followers of the secret worship of the Pythagoreans. Tetractys Tetrad

21. Pythagorean Symbols 21 virajjoshi (2011) PENTAGRAM- A pentagram (sometimes known as pentalpha or pentangle) is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek word πεντάγραμμον (pentagrammon), a noun form of πεντάγραμμος (pentagrammos) or πεντέγραμμος (pentegrammos), a word meaning roughly "five-lined" or "five lines".Pentagrams were used symbolically in ancient Greece and Babylonia. The Pentagram has magical associations, and many people who practice neopagan faiths wear jewelry incorporating the symbol. Christians once more commonly used the pentagram to represent the five wounds of Jesus, and it also has associations within Freemasonry.The pentagram has long been associated with the planet Venus, and the worship of the goddess Venus, or her equivalent. It is also associated with the Roman Lucifer, who was Venus as the Morning Star, the bringer of light and knowledge. It is most likely to have originated from the observations of prehistoric astronomers.[citation needed] When viewed from Earth, successive inferior conjunctions of Venus plot a nearly perfect pentagram shape around the zodiac every eight years.The word "pentacle" is sometimes used synonymously with "pentagram", although their technical usages are different, and their etymologies may be unrelated Pentagram

22. Pythagorean Symbols 22 virajjoshi (2011) PENTAD- The pentad, was used by the Pythagoreans as a secret sign to recognize each other. It represents the number five, life, power and invulnerability. Nicomachus explored the mathematical characteristics of the pentad as related to the Pythagorean saying "Justice is five". MONAD -Monad (from Greek μονάςmonas, "unit"; monos, "alone"), according to the Pythagoreans, was a term for God or the first being, or the totality of all beings. Monad being the source or the One meaning without division.For the Pythagoreans, the generation of number series was related to objects of geometry as well as cosmogony. According to Diogenes Laertius, from the monad evolved the dyad; from it numbers; from numbers, points; then lines, two-dimensional entities, three-dimensional entities, bodies, culminating in the four elements earth, water, fire and air, from which the rest of our world is built up. The term monad was later adopted from Greek philosophy by Giordano Bruno, Leibniz, and others. Pentad Monad

23. Pythagorean Symbols 23 virajjoshi (2011) DYAD- Dyad, according to the Pythagoreans, is the principle of "twoness" or "otherness", Nicomachus explored the mathematical characteristics of the pentad as related to the Pythagorean saying "Justice is five". DECAD -The number ten or decad was not considered a number by the Pythagoreans, it was seen as an "assembly point" and a symbol of earth and heaven Dyad Decad

24. Crop Circles 24 viraj joshi (2011) Crop circles have been appearing in grain fields all over the earth during the last few decades. They appear suddenly, usually at night. At first they were simple circles of bent-over grain stalks. Soon a new crop of more elaborate designs evolved—geometric forms reminiscent of profound mathematical theorems. Some cerealogists (people who study crop circles) say that these diagrams must be created by intelligent alien beings from elsewhere

25. Astrological Aspect 25 viraj joshi (2011) In astrology, an aspect is an angle the planets make to each other in the horoscope, and also to the ascendant, midheaven, descendant and nadir. The aspects are measured by the angular distance along the ecliptic in degrees and minutes of celestial longitude between two points, as viewed from the earth. They indicate focal points in the horoscope where the energies involved are given extra emphasis. The astrological aspects are said to influence affairs on Earth according to millennia of astrological tradition.

26. Mandala 26 virajjoshi (2011) Maṇḍala (मण्डल) is a Sanskrit word that means "circle". In the Hindu and Buddhist religious traditions their sacred art often takes a mandala form. The basic form of most Hindu and Buddhist mandalas is a square with four gates containing a circle with a center point.

27. Tesseract 27 virajjoshi (2011) In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8 cubicalcells. The tesseract is one of the six convex regular 4-polytopes. A generalization of the cube to dimensions greater than three is called a "hypercube", "n-cube" or "measure polytope". The tesseract is the four-dimensional hypercube, or 4-cube.

28. San Gaku 28 virajjoshi (2011) Sangaku or San Gaku (mathematical tablet) are Japanese geometrical puzzles in Euclidean geometry on wooden tablets which had dedicated to Shinto shrine during the Edo period (1603–1867) by members of all social classes. Modern Translation into English Traditional San Gaku

29. Ley Lines 29 virajjoshi (2011) Ley lines are alleged alignments of a number of places of geographical interest, such as ancient monuments and megaliths, natural ridge-tops and water-fords. Their existence was suggested in 1921 by the amateur archaeologist Alfred Watkins, in his books Early British Trackways and The Old Straight Track. Watkins theorized that these alignments were created for ease of overland trekking by line of sight navigation during neolithic times and had persisted in the landscape over millennia. An example of Ley Lines on a map (U.K.)

30. Influences on Music 30 virajjoshi (2011) The Fibonacci series appears in the foundation of aspects of art, beauty and life. Even music has a foundation in the series, as:There are 13 notes in the span of any note through its octave.A scale is composed of 8 notes, of which the5th and 3rd notes create the basic foundation of all chords, and are based on whole tone which is2 steps from the root tone, that is the1st note of the scale. Note too how the piano keyboard scale of C to C above of 13 keys has 8 white keys and 5 black keys, split into groups of 3 and 2. While some might "note" that there are only 12 "notes" in the scale, if you don't have a root and octave, a start and an end, you have no means of calculating the gradations in between, so this 13th note as the octave is essential to computing the frequencies of the other notes. The word "octave" comes from the Latin word for 8, referring to the eight whole tones of the complete musical scale, which in the key of C are C-D-E-F-G-A-B-C. In a scale, the dominant note is the 5th note of the major scale, which is also the 8th note of all 13 notes that comprise the octave. This provides an added instance of Fibonacci numbers in key musical relationships. Interestingly, 8/13 is .61538, which approximates phi. What's more, the typical three chord song in the key of A is made up of A, its Fibonacci & phi partner E, and D, to which A bears the same relationship as E does to A. This is analogous to the "A is to B as B is to C" basis for the golden section, or in this case "D is to A as A is to E."

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