Sri Yantra – A Study in Triangles

My wife and I are planning a big trip to India early next year. When I think of India and geometry, two things immediately come to mind: the Taj Mahal, and the Sri Yantra. Everyone knows what the Taj Mahal is, but have you heard of the Sri Yantra?

What is the Sri Yantra?

You have probably seen a Sri Yantra even if you haven’t heard the name for it. It looks like this:

And is often surrounded by additional ornamentation representing lotus petals, etc. like this:

This symbol (both the basic form and the full ornamented form above) is laden with many layers of meaning in Hindu philosophy, which you can read about in the Wikipedia article on Sri Yantra. However, I’m going to focus on the geometry, as you might expect. :-)

An Acknowledgement

Before I dive in I want to note that much of what I am presenting (including a number of the images) I found a wonderful site called Sri Yantra Geometry Research (www.sriyantraresearch.com).  If you enjoy this post, you must go to this site, where you can spend hours (as I did!) learning all the finer details about this amazing figure.

Sri Yantra Basic Definitions

So, what make a Sri Yantra a Sri Yantra?  In short, it is a figure composed of nine triangles and a central point (called the bindhi) such that:

  • Four triangles point up and five point down
  • The corners of the two biggest triangles all touch the same outer circle
  • For all the other triangles, the uppermost or lowermost point touches the base of an opposite-pointing triangle
  • At eighteen different points, three triangles cross at a mutual point!
  • The central bindhi is located at the geometric center of the innermost triangle (formed by the crossing of two of the downward-pointing triangles)
You would think that with all those requirements there would only be one way to draw the Sri Yantra. But it turns out that in fact there are many, many configurations that meet the requirements above. Here are just a few:

Sri Yantra Variations

So, which one is the “right one”??

The “Optimal” Sri Yantra

It turns out that an optimal figure has three important properties:

Concurrency = Everything Touches/Intersects Perfectly

The first property of an optimal Sri Yantra is concurrency, which is simply the property described above where all the triangles touch and intersect with each other perfectly at their meeting points.  It turns out that there are a LOT of figures out there that are “fudged” – in other words, where three triangles seem to meet in a point but they actually miss each other slightly:

A concurrency error - lines don't intersect

Concentricity = Centers Match

The second property is concentricity, which means “shared center.”  In this case, what we mean is that the bindhi should land in the same place as the center of the large surrounding circle. Here is an example of a concentricity error (the green lines show the center of the surrounding circle which is outside of this diagram):

A concentricity error - centers don't match

Equilaterality = Center Triangle Has Equal Sides

Finally, in an optimal Sri Yantra, that innermost triangle will be equilateral, meaning all three sides will have the same length and all three angles will be 60 degrees. Very often this is not the case:

An equilaterality error - center angles are not 60 degrees

Can It Be Done?

From everything I’ve read, it may be impossible to perfectly satisfy all of these requirements at the same time.  However, it is possible to come very, very close. If you want to experience the challange of balancing all of these various requirements, try playing with the Sir Yantra Geometry Research web site’s Interactive Sri Yantra.  Prepare to be frustrated, though! It’s easy to come close but I have yet to make an optimal Sri Yantra.  If you do, though, let us all know in the comments below!
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About Phil Webster

Phil is the creator of GeometricArts.com. You can reach him on the Contact page.
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