The Geometry of
The West Rose Window
The Cathedral Of St. John The Divine
New York City

(c) Michael S. Schneider 2004
M.Ed. Mathematics

Intended Completion At Present

In 1992 I had the honor and pleasure of working with master stonecarver Simon Verity who was then carving the statues at the Portal of Paradise, the front entrance to the beloved Cathedral Church Of St. John The Divine in New York City. At that time, the Cathedral had been under construction for over a century, and should be completed later in this century. The above drawing depicts how it is intended to appear when finished. Like any well-designed Cathedral, it appears not to be standing on the ground, but rightfully hanging down from heaven.

My role was to study the uncarved blocks of limestone on the south side of the portal and suggest a geometric theme which would allow these blank blocks to be carved into statues which would be harmoniously integrated with each other and with the entire building. Rather than impose anything, I sought to discover what the architect had originally intended, speaking to me through the timeless language of mathematics. I measured the group of upright blocks and realized that they formed a square-root-of-3 rectangle and that it was proportionally integrated with the facade and the space in front of the building. I presented Simon with the unique geometric possibilities of this ratio in the context of the Cathedral in three dimensions. In this photo you can see some of the lines of that construction which he painted to guide his carving. You can still see parts of the two long diagonals, and other lines, some of which were extended by string beyond the scuplture to important points on, and in front of, the Cathedral.

One little-known but interesting fact deriving from this geometric scheme is that as you walk from the street up the stairs toward the entrance, notice how the eyes of the statues follow you in sequence as you pass them by, one by one, each welcoming you through a section of your approach to the Portal.


The Cathedral Of St. John The Divine is a marvel of design in whole and part. At 601 feet, it is the world's largest Gothic Cathedral. It was designed in modern times by architects familiar with the 13th century High Gothic style of northern France. Its role in the community, city, nation and world are priceless. In 2003, the Cathedral was designated a landmark by the New York City Landmarks Preservation Commission.

(The word Cathedral means "seat of the bishop" and is related to the mathematical word "polyhedral" ["many seats" or faces, a three-dimensional structure bounded by polygons] through hedron = "seat").

One of this Cathedral's key features is its west Rose Window above the main entrance facing the light of the afternoon and setting sun. From inside, the rose window is glorious in its design and colorful radiance.

Like all classically designed Rose Windows, it displays a very definite geometric design. If you stare at it as a kind of mandala, it may come to your attention that it is based on a four-fold symmetry which reaches towards eight and sixteen directions.


Begin with eight equally-spaced points around a circle.
Connect every third point to draw this octagram star.
The Rose Window will be inscribed within the octagon at its center.

Many people are surprised to see that a Rose Window is usually a small part of a much larger construction. But this is the case. The Rose Windows at Chartres is also designed this way, although not based on the number eight but on its close cousin, twelve.







The crossings of the octagram star become the centers of eight overlapping circles whose radii reach to the inscribed circle of the Rose Window.

These eight circles show us the curves of the window's outermost design.






This design is reminiscent of an idealized version of the retrograde pattern of the planet Mars as seen from Earth, where Mars appears to make eight loops over sixteen years.





Let's look now at only the Rose Window at the construction's center.

A square has been inscribed around the window.

The square's diagonals and horizontal and vertical bisectors divide the circle into eight equally spaced points.

These line segments and an octagram star inscribed within begin to show us more of the window's design.




Eight line segments radiating from the center to the crossings of the octagram star show us how the central portion is divided into sixteen divisions.





Connecting the crossings of this octagram star gives us another octagram (yellow), which further subdivides significant areas.

Notice how the central octagon circumscribes the circle of stained glass.






The crossings of this octagram star become the centers of four large circles (red) inscribed within the circle of the Rose Window.

A circle is inscribed at the center tangent to the four larger circles. This reveals the circle at the center of the Rose Window.





Curiously, this same lovely geometric construction has been shown by John Martineau in his great work "A Little Book Of Coincidence" as the relative orbits of Earth and Mars (the large inner and outer circles) with 99.9% accuracy.





Observe the sacred architecture of every religion and you will see many examples of the interplay of circle, square and octagon, the symbols of heaven, earth, and mediation between them.

An octagon mediates between the heavenly circle and earthly square because it partakes of both: it is a square which has turned as a circle. All the geometric details of this window may be found from the lines and points coming from the basic Octagram construction.

Through its geometric symbolism, the Rose Window announces the Cathedral as a place of mediation between heaven and earth.



A view of the Rose Window from the Cathedral's outside confirms this geometric structure, and more.

It has been my experience investigating the geometry of sacred architecture that a Rose Window displays in miniature the geometry of the larger Cathedral. In modern terms, the Rose Window is a fractal of the Cathedral. Below (left) we expand this last geometric scheme of the Window (minus the small corner circles) as a square extending from the Cathedral's base to the top of it's two smaller central towers. Notice that the central yellow circle, which is tangent to the four larger blue circles, and which in the Window encloses Christ, now becomes the size of the Window itself! In the illustration below, right, the lines which this geometry implies are drawn and extended. Notice that the diagonals of the square extend by arcs up to become the sides of a root-two rectangle in which further details of the Cathedral's plan may be found by drawing diagonals and arcs, and subdividing by squares and root-two rectangles.

You might like to do this geometric construction on a blank sheet of paper and then use it to design your own Rose Window, or a Mandala, a building, or anything you like.

To see an animated version of this Rose Window's geometry, click here (2.5 Mb).


Text and Images(c) 2004-2007 Michael S. Schneider

To see any of these geometric analyses, click on its name:

The North Rose Window of Chartres Cathedral

The East Rose Window of Grace Cathedral (SF)

A Herter Brothers Cabinet


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