THE VEDIC GEOMETRY - A film based on research about Ancient Indian Geometry || Project SHIVOHAM

[Music] temples are the heart and soul of indian culture and heritage and they stood the test of time since thousands of years the grandeur of the sculptures on this temples its architecture the way they are built thousands of years ago just unparalleled well it's very difficult to just draw a perfect circle with a free hand what could have been the geometrical expertise of people back then who built these grand structures that was the question we had and this video is a result of our attempt to understand what has been the vedic geometrical expertise of ancient

indians to set the context geometry is a very natural thing almost all the civilizations express their culture and heritage in some form or shape in which geometry takes the center stage this video is purely to understand from an ancient indian standpoint what was their expertise with respect to geometry back then for ease of expression we picked up the word vedic geometry as broadly to encompass geometry as described in all the ancient indian scriptures so the first place where we find the vedic geometry applications is in kalpa it deals with executing the vedic rituals of yagnam

for different purposes the second place where we find geometry is in judiciary deals with astronomical studies that's essential for a vedic culture moving next in the third place where we find geometry is in shilpa sastra branch of knowledge that deals with the temple construction and sculpting the fourth and the last area where we find geometry is in srivijya it is a specialized branch of divine worship which is broadly categorized as tantra but for ease of expression we'll use the word srividya deals with construction of complex two-dimensional and symmetrical geometrical diagrams that are the manifestations of

different gods and this documentary is about how geometry thrived in these four areas since times immemorial let's briefly flip over these four starting with shilpa shastra shilpa sastra is a very broad term which is generally used for categorizing all the scriptures the deal with the temple construction sculpting metallurgy architecture and a lot more there are many scriptures under shilpa sastra like manasara vastu sastram silparatna kosa and many more all the geometrical principles that are needed for triple construction are explained in these scriptures up next is jyotisha this is one of the most misunderstood branches of

knowledge that it's about future prediction etc etc but jyotisha the word literally means study of light it deals with computation of planetary positions and as part of it there are a lot of geometrical principles that are detailed out in the scriptures we'll see them as well in this video in india when it comes to divine manifestation gods are always depicted in forms of sculptures or murtis and that's what we generally see in the temples but there is another practice where the divine incarnations of gods are manifested in the form of a dimensional symmetrical diagrams so

sri vidya details about construction of these complex geometrical figures which deals with different geometrical shapes and the principles behind the construction and the fourth one is kalpa kalpa deals with setting the guidelines for execution of the vedic rituals like yagnam and you might have seen the fire altar which is typically in a square shape like it's shown here but there are a lot many other shapes which needs to be constructed depending upon the intent of the ritual so because of that there are a lot of geometrical concepts detailed out in kalpa as well now let's

see each of these four in detail starting with the first one kalpa how the vedic geometry is defined in kalpa so kalpa is the code of yagnyam it has all the rules and regulations and bylaws related to how you execute a vedic ritual and it has two parts to it the first one is this talks about the rules of the logistics that are needed foreign as to how you construct these fire altar the yagnasala and a lot more and the second one is shroud the sutras which contain the mantras and the vedic hymns which are

needed for execution of the ritual so sulba sutra is about preparation and shroud sutra is about execution and you'll get to know in a minute why we are talking about these sulba sutras it's two words suluva and sutras shulva means rope and sutras means principles so basically these are the mathematical instructions that are crucial in construction of the fire altars in the process of yagnam thousands of years ago they did not have scales like what we have now so a rope is used as an instrument to measure any surface and hence with that name goes

sulba sutras in simple words these are the geometrical principles the concept of yagnum is a very very ancient practice which has been thriving in india since no one knows when you take any kind of ancient indian scripture you will not find it without reference to some kind of a yagnyam such as the antiquity of this ritual so what you're seeing on the screen is a relic of one of such as part of one of these rituals there are three fire altars that should be maintained in every household back then it is called as threat agni

or in simple words three fire altars they are aha graham and dakshin agni as you're seeing in the markings as you see the shapes there are circles square and semi-circle the most important aspect here is there is a very strong constraint which says that all these three firealters should have exactly the same area now i don't know what is the religious significance behind that constraint but all these three if constructed at a place should be exactly off the same area this might not sound as a big deal to you now because we have all the

formulas invented for calculating the areas of these shapes but thousands of years ago geometry was totally in a different format so how did the ancient indian russians manage to measure areas of these kind of fire altars thousands of years ago now let's see talking of measurement the first thing is unit of measurement so what was the unit of measurement back then that is angular this is the basic unit of measurement as per sulphur 34 sesame seeds stacked up against each other in a line that is the length of one angular that was basically the unit

of measure of people back then so that comes around four to five inches as per our inches today also sesame seeds are one of the oldest known sources of oil to humanity so that also explains the antiquity of the vedic rituals and the time they belong to let's take a very simple problem to understand how the ancient indian people constructed these fire altars can you draw a square with its sides perfectly aligned with the four cardinal directions you would need definitely a pencil compass paper and a scale without these we cannot draw a square with

its sides perfectly aligned with the four cardinal directions the simple thing right but back then none of these were available no compass pencils or scales so how did they go about it let us see so first things first identifying the four cardinal directions these are not fixed because of the solstice the sun movement varies from season to season right so we need to get the directions which are applicable on that day of the execution of this ritual so here is how they did they took a long stick and erected on a plane ground and then

tie a string to the stick and the other end of the string tied to a nail then draw a circle around the stick like this and this has to be done before the sunrise and once sun rises the shadow of this stick is casted on the opposite direction so the longest shadow will be in the early morning so the shadow protrudes out of the circle that was drawn and as the sun progresses over the day proportionally the shadow reduces and in the evening again it protrudes towards the other end so these two spots which cross

over this circular path these two points give the east-west line now that we have the east-west line the next step is to construct a square i'm just translating directly the process given in sulphur sutras first take the east west line and draw it to the length of the square that you want say for instance 10 angulars so ew should be of 10 angulars and then with that as a diameter draw a circle again to draw a circle like i said measure 10 angulars with a rope and with e w as a diameter draw a circle

around it now taking the same ew as a diameter with e as a center draw a circle around it after that take a point on the circle which is perpendicular to ew and draw a circle around it with the same diameter of ew repeat this process again for with w as a center and now with the opposite point the second one with the same diameter so you now have four circles and these four intersection points of these four circles need to be plotted a b c d now joining a b c d gives you a

perfect square with its sides aligned with the four cardinal directions and the length that you desire which here is ten anglers so this is how geometry was back then very primitive but still precise as well so once you construct the square on the ground that forms as the marking for construction of the square shaped fire altar or the yagna kunda so coming back to three tag where we started we now have the markings needed for construction of a square shaped fire altar as you can see in this picture now they have to construct a circular

one which has exactly the same area as the square so how is that possible let's see we cannot talk the geometry of a circle without pie but back then the pie was not discovered yet the credit of pie goes to greeks or arguably egyptians one of these two civilizations indians don't know about pai during those times remember we are talking about couple of thousands of years ago don't just go with current ratings of 500 bc or thousand bc it is much much older than that so back then pi was not discovered yet so here is

a beautiful postulate given by sulbasutra on how to convert square which we already have to transform that into a circle of exactly the same area and here is the process given in silver sutras draw a circle a b c d and then mark x and y extending o so you have x and y as part of x and y divided into three parts so that mark z as one third of x y so y z is one third and zx is 2 3 of x y and finally draw a circle with o z as the

radius so this is the most important thing with o z as the radius the circle that you're drawing here is exactly of the same area as the square abcd so this was a postulate that was given which can be geometrically proved today as well if you want try it out it's really great to see how organically geometry has evolved over the times so we just saw the process of how we construct the circular fire altar as well which is exactly of the same area as the square like this there are many many geometrical principles explained

in srila sutras but for now let's move on these fire altars are called as chitti in samskritam and depending on the intent of the agnyam the shape of the chiti varies for instance to obtain prosperity a falcon-shaped chitis constructed to destroy the enemies in war a triangular or rhombus kind of a chitti is constructed and to pray for food and prosperity of the kingdom the hemisphere shape of chiti is constructed so it depends on the intent the process given for the ignom is different and so is the shape of the chitti this is very important

because this has defined the dynamics of vedic geometry so these chitis are not just made out of mud they are made out of brick arrangements and this is how the blueprints of couple of chitis like shayna chitty radha chakra chitty and kurmachiti are so this is from kalpa which ascribes to yazuru them so here the most important aspect is the bricks need to be baked according to the dimensions given here and that needs to fit exactly in the shape of the chiti if you observe the shapes of the bricks there are squares rectangles triangles rhombus

trapezium so here is where the vedic geometry gets totally into another level here is an example of shayna chitty which shows a miniature model about how the bricks needs to be cut observe the tiles how they are shaped there are right angle triangles isosceles triangles squares rectangles everything stacked up to form the shape of a garuda or a falcon which is an exact replica of the blueprint that you're seeing here down below and all the wooden equipments that you're seeing around it are the ones that are needed for executing a gum the interesting part here

is now what is the geometry behind cutting and baking of these bricks that gives us some incredible facts about vedic geometry let's see what they are to calculate the dimensions of this bricks based on the shapes there are a lot of formulas or principles given in sul vasudras here i just picked up one example which is very relevant to you and me based on what we learned in our school and this is how it reads basically talks about a rectangle but let's break it down word by word to understand the exact meaning of it which

is a rectangle akshaya means diagonal rajuhu here means rope for measurement let's put that aside pashwamani means perpendicular or straight and tiring which is horizontal which is equals the area of diagonal so basically what it says is the area made by the horizontal and the vertical lines of a rectangle is equal to the area made by the diagonal this is called as pythagoras theorem in today's world but is actually written by mahashiri bowdhyana who was conservatively dated as 800 bc but again like i said how old is yangyam that old is this kalpa so you

cannot frankly date how old are these scriptures so a safe and a fair statement would be as old as yesurvedam so old is this principle given by maharishi baadhayana which is commonly called as pythagoras theorem today take a look at these pictures this is shaina chitty which is in the form of a garuda or a falcon look at the size and the shape of the bricks they are in triangular forms stacked up into five layers now the geometry behind cutting and baking of this bricks is fully based on sulbasutras even today in the so called

pythagoras theorem which you just saw is just one of the many geometrical principles given by maharishi baudhina if you observe it from this direction you can see clearly how the entire fire altar is in the shape of a falcon like we have seen in the blueprint so on top of this the entire yagnam will be performed now because of these five layers the complexity behind the geometrical construction of these bricks even gets multiplied it takes to a level even above let's see what it is let's take the isometric view of the blueprint of shine here

if you see it has five layers all exactly in the same shape and there is an additional constraint that is imposed by the sulfur sutras that there should be only five layers in this one and each layer can have only 200 bricks and in five layers so that is one thousand bricks so you have to finish this entire construction of shainachiti in just 1000 bricks with exactly 200 bricks per layer and if you're thinking that's complex it's still not done yet the five layers of the shainachiti the layout for one three and five the odd

numbers layout is different from that of the even numbers so towards left what you're seeing here is the layout for the odd numbers and the right side is for the even numbers so 200 bricks two layouts five layers no scale no equipment nothing in a very primitive way the entire geometry has been executed and this is just an example to show how complex problems sulba sutras dealt with and this is the reason why sula sutras are considered as world's oldest applied geometry and after sulba sutras we have the euclidean geometry that came in 300 bc

99 of indians today are not aware of sulbasutras just for one reason we don't care about our own history and heritage talking of history here is a bit more about the pythagoras theorem the pythagoras theorem or the pythagoras triplets which satisfy the pythagoras theorem are first found in babylon which is modern day iraq inscripted on a stone tablet called as plimpton 322 that's the code name given to the tablet but it is dated to 1800 bc which is first found in babylon so it is almost 1300 years before pythagoras discovered the pythagoras theorem and this

is the tablet what you're seeing on the screen the pythagorean triplets are carved into three columns so these are like three numbers which satisfy the pythagoras rule and these combinations of numbers are very crucial and they have a lot of applications you can search for yourself another ancient source of the pythagoras triplets comes from egypt a mathematical document called rend papers which is a mathematical worksheet of the ancient egyptians apparently it also has the pythagorean triplets calculated and this is almost dated to approximately 1500 bc and as you know egyptians are very famous for their

triangular structures and finally from the greeks in 500 bc pythagoras is the one who generalized the property of a right angle that the base square plus side square is equal to hypotenuse square however the pythagoras theorem what we know today is given to us by euclid the greek mathematician 200 years later after pythagoras he's the one who is documented and that's the manuscript what you're seeing here it is possibly a copy of euclid's original or could be a recopy of a latin version but nonetheless it's a manuscript explaining about the pythagoras theorem now if we

take a macro look it is babylonians in the 1800 bce egyptians in the 1500 bce and greeks in approximately 300 to 500 bce and maharshi bhaina who is again conservatively placed at 800 bc gave the formula much before it is generalized by pythagoras it is not just about the antiquity but also about the continuity since many many thousands of years the sulba sutras have been defining the way a yagnam needs to be organized until today it is in practice such is the rich heritage when it comes to vedic geometry yet there is a high probability

that most of you would have never heard of it i just took this one as an example so that most of us can connect and understand but like this there are bowdhein apastamba manava kathyaina sulbasutras talking about the geometrical principles so that's about the geometrical principles in kalpa and now let's move on to jyotishia jyotisha misunderstandings misconceptions pseudoscience what not it is totally entangled in a lot of preconceived notions and allow me to set the record straight here there is astrology which deals mostly with the cultural element of predicting the future and there is astronomy

which is the study of the celestial bodies now getting straight to the point is astrology or the future prediction business is that scientific or not well the answer is i don't know and i'm not bothered either let's take a worst case assumption that it is totally a pseudoscience but the fact of the matter is because of astrology astronomy existed in india historically there have been a belief that the planetary positions impact people's lives now again i repeat we are not at all discussing about that aspect here and we are least interested about that what we

are rather interested is in order to compute the planetary positions jyotishia gave birth to incredible fields of mathematics and this is not a joke or i'm just trying to brew a conspiracy here if at all there is any branch of knowledge that is highly misused and highly misunderstood then it is jodicia misused because of one set of people who commercialized this entire jyotishastra and misunderstood because of another set of people who don't know a thing about it but just outright say that it's a pseudoscience now let's forget about these two highly polarized versions let's try

to find out what is in the middle with respect to geometry and at large with mathematics time competition it is the heart and soul of jyotisha calculating the planetary positions to the minute is of top-notch importance and to do so the ancient indian astronomers and mathematicians developed a fleet of mathematical concepts in these three areas of geometry trigonometry and calculus let me put all these four blocks in a simple perspective the time competition is of the top-notch importance and in order to do that the rate of change of planetary motion is also an integral part

of it and because of that calculus originated and is a precursor to calculus to calculate the ratio of observations of different planets trigonometry as well took birth and in order to construct the astronomical equipments geometry as well thrived as part of jyotisha these three topics are very much intertwined with each other and it is difficult to explain one and leave the rest out we're planning for a separate detailed documentary on jyotisha but for now i'll try my best to cover only the geometry aspect of it maybe i have to repeat myself as well not saying

indians founded all these branches of mathematics i'm saying there are significant contributions that came in from india for each of these fields which many of the people are not aware hope i'm clear now speaking of jyotisha in india ujjain holds the topmost importance for two main reasons number one it is positioned right on the tropic of cancer and for many many centuries the prime meridian of india used to cut through ujjain as you're seeing it here so counting of the day starts from ujjain as per ancient indian traditions however the british after the colonization in

1884 they kind of consolidated all the meridians across the world where people are having their own primary readings and mode it to greenwich otherwise historically ujjain is the place where time competition of a day starts second thing historically ujjain produced a lot of astronomers and mathematicians who contributed greatly to indian mathematics and here lord shiva is worshiped as mahakal which means the lord of time and death coming to geometry in jyotisha where all it played its path here is that follows it is used in construction of these astronomical observatories what you're seeing are different kinds

of structures for different purposes which we have in places like jaipur and ujjain the next it also played its part in calculating the radiuses of different planets and also its role in trigonometrical observations to calculate the relative positions of the planets and also in construction of certain astronomical equipments like astrolabes what you're seeing here which are very indigenous to india so we'll just see very few interesting topics related to geometry and move on as part of jyotisha there are many indian astronomers like aryabhatta varahamihira or brahmagupta who gave incredible contributions towards it here is just

one example from aryabhatiya given by aryabhatta in 500 ce he gave a formula for calculating the area of a circle without using pi today without pi we can't imagine calculating the area of a circle it's going to be very very hectic not impossible but very hectic but back then like i said there has been a tradition of measuring things using a rope in india so with that concept in mind this formula is easily applicable what aribata gave arya butter says that the area of a circle is half of the circumference multiplied by the radius now

circumference is something that you can measure with a row and the radius as well so using these the area of a circle can be calculated very easily without touching the value of pi because pi back then was still in a development mode you might not get an accurate result if you incorporate pi in your equations and when you're talking about astronomical calculations even the fourth or fifth place of pi has huge impact aryabhatta himself gave a better value for pi but at that time the value given by the chinese was far more accurate than the

one which was given by indians talking about geometry and jyotisha together the most incredible contribution that came from india was in the year 1200 from madhava people were struggling to find the value of pi an accurate one adding up one decimal after other so the development right from the discovery of the days when greeks first identified that yes there is a constant like pi which is a ratio of the circumference to the diameter that can be used in the calculations the value of pi has been quite irrational so madhava is the first one to give

the value of pi as an infinite series that means it's up to your patience you can calculate as much as you want this is no exaggeration but this invention is almost as good as landing man on mars if it's tough for you to believe read the history of the pie and its evolution and you'll get to know so madhava gave the value of pi in this small samskratam slokam so he says that so decoding this samskritam slocum this is what we get the value of pi is equal to 4 multiplied by an alternating series of

1 minus 1 by 3 plus 1 by 5 minus 1 by 7 and so on till infinite in fact he is the first one to express a finite quantity like pi in an infinite series this is one of the fundamental stepping stones of calculus integration by parts and this is why i said calculus also has its roots in jyotish but unfortunately we cannot discuss all that now maybe for some other day and this series of pi is called as madhava gregory series which was initially invented by madhava in the 14th century and reinvented by gottfried

leibniz in 17th century so in his honor the series is named after madhava as well as gregory saying that madhvaa gregory series again a vast majority of us are not aware of this for obvious reasons up next is geometry in shilpa sastra temples the epicenter of indian art culture and heritage so beautiful are the carvings and the sculptures but what's actually behind them let's quickly have a look unlike jyotishastra and kalpa the geometrical concepts of shilpa shastra is not really something that we could possibly use or try to understand it is so intricate and blended

with the way the sculptings are done so here you see as an example the walls of a temple there are different varieties of designs depending on the deity of the temple there are specified designs not just any random ones to be carved in accordance with the scriptures of agama manasara or shilpa sastra these are different books which describe how a temple construction needs to be done shilpa shastra also gives the geometrical guidelines about constructing and carving different kinds of pillars as you see here it gives the structural guidelines as well as the aesthetic guidelines in

building the pillars and other parts of the temple and here are some blueprints about the arches that would be constructed in the temple the interesting part is all these designs are explained in form of samskritam slocums it's quite interesting to see how such great details are encoded in samskritam here are the most catchy parts of the temple the vimanas or the gopuram shilpa shastra explains great detail about how these needs to be layered and sculpted both for its structure as well as for its aesthetics and in all geometrical principles are well blended in each of

them there we have the last category srividya which also has complex geometrical procedures blended into it sri vidya is a part of devotion in which sri lanka parameshuri is worshipped as the epitome of the nature and the cosmos as part of this practice what you're seeing on the screen called is yamtras these yantras are manifestation of the gods themselves so that's what is the practice is all about people meditate on these yantras as part of their spiritual process but as we are looking from a geometry standpoint we'll see it from that angle to understand this

geometrical diagrams called yantras basically they are revered as a layout in which each vertex line corner holds a deity and it's like a labyrinth where people start meditating through their paths navigating towards the center of the yantra it's a bit complicated and i myself don't have a clear-cut understanding about the spiritual aspects of it so let's limit ourselves from the geometry standpoint the most important topic here is as you see we have a lot of deities in each and every space of this diagram so that translates to incredible precision in the geometrical structures which are

the yantras so to create these geometrical structures of the yantras lot of techniques have originated in building these diagrams and this is a very very ancient culture in there are many temples where you have these yantras carved into either the floor or the ceiling or in other parts of the temples here is a manuscript detailing out some yantra and describing the way one has to follow it it's just a snippet from a very ancient manuscript we see here the diagrams how they are drawn these diagrams represent a specific deity which one chooses to worship in

his or her path of devotion so here is a specific example of ganesha yantra which is a manifestation of lord ganesha himself so that's what and how it is revealed and this is just a rough sketch but when it comes to actually consecrating such kind of a yantra a high geometric procession is ensured in this part of it there are a lot of construction techniques as part of the vedic geometry the designs these yamtras and this one is a bhairava yantra manifestation of lord shiva in the form of bhairava one of the most ferocious forms

of lord shiva and this yantra is mahakali yantra which is a manifestation of carlino these look like a simple diagrams but the proportions and precession and the symmetricity they matter a lot when it comes to consecrating them this one is sri mahalakshmiyantra again a manifestation of goddess shri mahalakshmi and of all the yantras sri yantra is revered as the most important yantra when it comes to srivia and it's a highly complex geometrical structure where the precision matters to the pixel it has nine triangles which shows as a confluence of shiva and shakti holding the topmost

importance in the spiritual process there is a lot of mathematical analysis that was done in many universities across india europe and us on sri chakra and its symmetricity and other geometrical properties at this point in time i don't fully understand it and the day i understand will definitely do a detailed documentary about it well now that's a very brief overview about vedic geometry and how it pans across all these four areas i'd like to come back to sulba sutras again i was really surprised to learn the fact that it is the world's oldest applied geometry

that is still running in the tradition till today broadly a very derogatory term called mythology is often attributed to anything that you talk about ancient indian history well that's not true there are many many mathematical or scientific concepts like sulva sutras i do agree that there are certain pseudo-scientific topics as well that's a very small portion of it but largely the history of india has always been tightly coupled with both science and devotion together and vedic geometry is an example of scientific heritage of india and as always thanks for watching you