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Sacred geometry is the geometry used in the planning and construction of religious structures such as churches, temples, mosques, religious monuments, altars, tabernacles; as well as for sacred spaces such as temenoi, sacred groves, village greens and holy wells, and the creation of religious art. In sacred geometry, symbolic and sacred meanings are ascribed to certain geometric shapes and certain geometric proportions. According to Paul Calter:
In the ancient world certain numbers had symbolic meaning, aside from their ordinary use for counting or calculating; plane figures, the polygons, triangles, squares, hexagons, and so forth, were related to the numbers (three and the triangle, for example), were thought of in a similar way, and in fact, carried even more emotional value than the numbers themselves, because they were visual.
As worldview and cosmology
The belief that God created the universe according to a geometric plan has ancient origins. Plutarch attributed the belief to Plato, writing "Plato said God geometrizes continually" (Convivialium disputationum, liber 8,2). In modern times the mathematician Carl Friedrich Gauss adapted this quote, saying "God arithmetizes." 
At least as late as Johannes Kepler (1571–1630), a belief in the geometric underpinnings of the cosmos persisted among scientists.
According to Stephen Skinner, the study of sacred geometry has its roots in the study of nature, and the mathematical principles at work therein. Many forms observed in nature can be related to geometry, for example, the chambered nautilus grows at a constant rate and so its shell forms a logarithmic spiral to accommodate that growth without changing shape. Also, honeybees construct hexagonal cells to hold their honey. These and other correspondences are seen by believers in sacred geometry to be further proof of the cosmic significance of geometric forms.
Art and architecture
Geometric ratios, and geometric figures were often employed in the design of Egyptian, ancient Indian, Greek and Roman architecture. Medieval European cathedrals also incorporated symbolic geometry. Indian and Himalayan spiritual communities often constructed temples and fortifications on design plans of mandala and yantra.
Many of the sacred geometry principles of the human body and of ancient architecture have been compiled into the Vitruvian Man drawing by Leonardo da Vinci, itself based on the much older writings of the roman architect Vitruvius.
Stephen Skinner discusses the tendency of some writers to place a geometric diagram over virtually any image of a natural object or human created structure, find some lines intersecting the image and declare it based on sacred geometry. If the geometric diagram does not intersect major physical points in the image, the result is what Skinner calls "unanchored geometry." 
Pythagoras is often credited for discovering that an oscillating string stopped halfway along its length produces an octave relative to the string's fundamental, while a ratio of 2:3 produces a perfect fifth and 3:4 produces a perfect fourth. However the Chinese culture already featured the same mathematical positions on the Guqin and the tone holes in flutes, so Pythagoras was not the first. Pythagoreans believed that these harmonic ratios gave music powers of healing which could "harmonize" an out-of-balance body.
- dartmouth.edu: Paul Calter, Polygons, Tilings, & Sacred Geometry
- Cathérine Goldstein, Norbert Schappacher, Joachim Schwermer, The shaping of arithmetic, p235. 
- Skinner, Stephen (2009). Sacred Geometry: Deciphering the Code. Sterling. ISBN 978-1-4027-6582-7.
- Stephen Skinner, Sacred geometry: deciphering the code, p91
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