On the construction of the Śrī Yantra

2013

Some Geometrical Concepts in earlier Indian works will be discussed. This talk includes a brief discussion on few topics including: the Sulba Sutras, Pythagorus Theorem, Transfer a Square into a Circle, Square Root of 2, Some trigonometry in ancient Indian works, the Vastu Shilpa Shastra, use of Geometry in Sulbha sutras to calculate the roots of the quadratic equation.

The Sulba Sutras, part of the Vedic literature of In-dia, describe many geometrical properties and constructions such as the classical relationship a2 +b2 : c2 between the sides of a right-angle triangle and arith-metical formulas such as calculating the square root of two accurate to five decimal places. Although this article presents some of these constructions, its main purpose is to show how to consider each of the main Sulba SDtras as a finely crafted, integrated manual for the construction of citis or ceremonial platforms. Certain key words, however, suggest that the applications go far beyond this.

International Journal of Statistics and Applied Mathematics, 2025

This article explores the geometric knowledge from the Sulba Sutra and its relevance to school geometry using historical research methods collecting historical documents and analyzing them. Key assumptions, theory, theorems and practices were taken from Vedic texts Sulba Sutra and school mathematics texts. It examines key assumptions (Postulates) of Vedic ritual geometry, such as the construction of altars using precise proportions, alongside the Jatya Tribhuj (early right-angled triangle) and the Bhoja Koti Karna Nayana (Sulba theorem on right-angled triangles). Additional topics include shape transformations, symmetry, and the use of mean proportion. The research traces the mathematical content in various Sulba Sutras and their historical development. Similarly, a correlation of contents in school mathematics was established logically. The study concludes that Vedic ritual geometry's principles align with fundamental school geometry topics. Incorporating these concepts into school curricula can enhance students' engagement and understanding by linking abstract mathematical ideas to historical and cultural practices. This approach promotes hands-on, inclusive learning, blending ancient geometric knowledge with modern education for a culturally responsive teaching framework.

JOURNAL OF MECHANICS OF CONTINUA AND MATHEMATICAL SCIENCES

Coordinate geometry is a particular branch of mathematics where geometry is studied with the help of algebra. According to the concept of Bhattacharyya's Coordinate System, plane coordinate geometry consists of Four Positive dimensions or axes. In four-dimensional coordinate geometry, the position of a point can be determineda uniquely by two real positive numbers on a plane. Here, we shall discuss only the plane coordinate geometry consisting of four dimensions or axes. The author introduced four positive dimensions or axes to solve the problems with the help of 'Rectangular Bhattacharyya's Coordinate System' instead of the Rectangular Cartesian Coordinate System. This is the new concept which has been developed by the author. The author determined not only the distance between two points but also the direction of the line segment between two points on the plane.

I have argued for the existence in (what Western Europe sees as) the Near East of a long-lived community of practical geometers -first of all surveyors -which was not or only marginally linked to the scribe school traditions, and which (with branchings) carried a stock of methods and problems from the late third millennium BCE at least into the early second millennium CE. The arguments for this conclusion constitute an intricate web, and I shall only repeat those of them which are of immediate importance for my present concern: the links between the geometrical section of Mahāvīra's Ganita-sāra-saṅgraha and the practical mathematics of the Mediterranean region in the classical ages.

The objective of this paper is to provide a provable solution of the ancient Greek problem of trisecting an arbitrary angle employing only compass and straightedge (ruler). (Pierre Laurent Wantzel, 1837) obscurely presented a proof based on ideas from Galois field showing that, the solution of angle trisection corresponds to solution of the cubic equation; í µí±¥í µí±¥ 3 − 3í µí±¥í µí±¥ − 1 = 0, which is geometrically irreducible [1]. The focus of this work is to show the possibility to solve the trisection of an angle by correcting some flawed methods meant for general construction of angles, and exemplify why the stated trisection impossible proof is not geometrically valid. The revealed proof is based on a concept from the Archimedes proposition of straightedge construction [2, 3].

Workbook 9 : The Platonic Solids 98-1 02 'What is God? He is length, width, height and depth.' ST BERNARD OF CLAIRVAUX, On Consideration 'Geometry' means 'measure of the earth'. In ancient Egypt, from which Greece inherited this study, the Nile would flood its banks each year, covering the land and obliterating the orderly marking of plot and farm areas. This yearly flood symbolized to the Egyptian the cyclic return of the primal watery chaos, and when the waters receded the work began of redefining and re-establishing the boundaries. This work was called geometry and was seen as a re-establishment of the principle of order and law on earth. Each year the areas measured out would be somewhat different. The human order would shift and this was reflected in the ordering of the earth. The Temple astronomer might say that certain celestial configurations had changed so that the orientation or location of a temple had to be adjusted accordingly. So the laying of squares upon the earth had, for the Egyptian, a metaphysical as well as a physical and social dimension. This activity of 'measuring the earth' became the basis for a science of natural law as it is embodied in the archetypal forms of circle, square and triangle. Geometry is the study of spatial order through the measure and relationships of forms. Geometry and arithmetic, together with astronomy, the science of temporal order through the observation of cyclic movement, constituted the major intellectual disciplines of classical education. The fourth element of this great fourfold syllabus, the Quadrivium, was the study ofharmony and music. The laws ofsimple harmonics were considered to be universals which defined the relationship and interchange between the temporal movements and events of the heavens and the spatial order and development on earth. The implicit goal of this education was to enable the mind to become a channel through which the 'earth' (the level of manifested form) could receive the abstract, cosmic life of the heavens. The practice of geometry was an approach to the way in which the universe is ordered and sustained. Geometric diagrams can be contemplated as still moments revealing a continuous, timeless, universal action generally hidden from our sensory perception. Thus a seemingly common mathematical activity can become a discipline for intellectual and spiritual insight. Plato considered geometry and number as the most reduced and essential, and therefore the ideal, philosophical language. But it is only by virtue of functioning at a certain 'level' of reality that geometry and number can become a vehicle for philosophic contemplation. Greek philosophy defined this notion of 'levels', so useful in our thinking, distinguishing the 'typal' and the 'archetypal'. Following the indication given by the Egyptian wall reliefs, which are laid out in three registers, an upper, a middle and a lower, we can define a third level, the ectypal, situated between the archetypal and the typal. T o see how these operate, let us take an example of a tangible thing, such as the bridle of a horse. This bridle can have a number of forms, materials, sizes, colours, uses, all ofwhich are bridles. The bridle considered in this way, is typal; it is existing, diverse and variable. But on another level there is the idea or form of the bridle, the guiding model of all bridles. This is an unmanifest, pure, formal idea and its level is ectypal. But yet above this there is the archetypal level which is that of the principle or power-activity. that is a process which the ectypal formrand typal example of the bridle only represent. The archetypal is concerned with universal processes or dynamic patterns which can be considered independently of any structure or

Indo Nordic Author's Collective, 2021

The sages of India conceived a comprehensive picture of the reality of all forms of creation which are also manifestations of the Supreme Consciousness. They also understood that the almighty consciousness gave birth in the process of creation to numerous gods and goddesses. Each god has a distinct purpose to fulfill. The tantra sastra similarly, teaches that apart from the material world in which we live, there exist other worlds and universe. The Supreme Godhead controls these systems through the medium of a hierarchy of gods and goddesses (devatas). These entities exist in various planes on the rising tier of consciousness. These devatas help man in his uphill journey of liberation. The tantra developed this line with an eye on practical utility and this spiritual science is acclaimed as a great sadhana sastra or practical science. The concept of Sri Chakra Though the Supreme Consciousness is formless, nameless or timeless, in manifestation it has to limit itself to a form. The formless great radiance has to radiate rays of definite forms and channels them as the various gods thus creating out of ONE, MANY with specific and distinguishable forms ad features. These lines of light create the form-patterns of the goods which are known as yantra or charkas. A yantra is an instrument, a machine or a storehouse of power. It contains rather, in itself in a controlled form, the uncontrollable power of the deity. Tantrics believe that by worshipping the charka of a deity, the worshipper realizes the same deity or in other words he merges with it. Mandala an important aspect of a yantra which is frequently drawn or made with powders of various colours. The mandala is used in the case of any deity whereas the charka is intended for a specific deity. In south India it is a practice in most of the homes to draw a mandala in front of the house every morning to bring in auspiciousness and to ward off evil. The Tamil word for mandala is kolam (guise), as it contains in disguise the divine power. The mandala employed in tantra is no decorative imagery for a ritual. It provides a potent Pages 289 - 291 material focus for the operation of subtler forces within and without. The Chakra like the mantra leads one to direct perception of the Divine form and that is the reason why so much emphasis is given to the Chakra in tantric worship. In the Chakra are caught the lines of beauty, harmony and symmetry on which the eternal geometrician fashions the universe. And they are therefore, drawn with lines, triangles, circles and squares. All these are symbolic. The Symbolism The circle represents the principles with no beginning and no end. The triangle represents the triple principles of creation, the lower triple worlds as well as higher. It points out to the one that is threefold, the yoni, the origin of all matter. A triangle with apex turned up indicates a broad – based one – pointed aspiration rising from the depths to the heights. In tantric paralance it is called vahni kona, the cone of fire. This is the fire of aspiration which is ever burning in the heart of the worshipper and which carries (vah=to bear) on its crest his surrender to the Divine. The triangle with apex downwards in Sakti, the grace of the Divine Mother. The well-known satkona formed by superimposition of a triangle with apex downwards over a triangle with apex upwards signifies a rising aspiration and a responding grace, the ascent of the being and the descent of the deity or in other words the dynamic Sakti superimposing on the heart of the static and supine Siva. The lotus flower signifies the gradual unfolding of the latent powers in the being. When the Chakra is conceived as the material manifestation of the Deity, all the emanations of the Deity are also conceived as stationed in the Chakra. The main deity (pradhan) takes abode in the centre of the Chakra while its emanations gather round the pradhana as the parivara devats. Worship is done to the parivars and then to the pradhana. The Sri Chakra is a configuration of nine triangles, five triangles with apex downwards superimposed on four triangles with apex upwards. It consists of nine chakras. There are nine Chakreshwaris, nine classes of yoginis and nine mudras. Like the ninefold of Sri Chakra, the mantra is also nine fold as it contains only nine letters. The masters are nine in number. The human body has nine apertures. Therefore, an identity is sought to be established between Pages 289 - 291 the Chakra, Deity, Mantra, Guru and the sadhaka’s body. Sri Chakra is most auspicious and worshipped to get all the auspicious things in life, finally culminating in the attainment of liberation. It is also worshipped for the “six acts” viz., appeasement, attraction, stoppage, enemity, removal and death. The techniques for these specialized acts are described in various tantric texts. CONCLUSION The tantra deems it essential to inculcate the doctrine that no worship of the Deity is complete without the worship of the Deity’s body in a material image-an idol or a picture or some symbol in the physical world. Inner worship leads one on the path of yoga and knowledge.

The Mathematical Gazette, 2004