The pattern we are exploring today, also known as "the Heavenly City", is not one that appears in the art of any culture. Long-forgotten, and reconstructed by geometer John Mitchell in the 1970s, it is said to be the ground plan for several sacred sites built according to principles of sacred geometry. The reason for this would be that it encodes several number ratios making up the universe. Without going into the numbers themselves, which are quite literally mind-boggling, here is how Mitchell describes it:
"The Heavenly City is a geometer's name for the traditional diagram that represents the order of the universe and the numerical code that underlies it. [It] contains the numbers, measures, shapes, proportions, and musical harmonies that are constant in nature." — How the World Is Made: The Story of Creation According to Sacred Geometry, pXVI.
The reason it is called the Heavenly City, and also New Jerusalem, is that it was reconstructed from the detailed description of St John's vision in Revelations, where an angel showed him "the perfect pattern of creation". He described it as a city with, among other things, three gates on each of its four walls (the full description is in Revelation 21:9-14).
Anyone keen on finding out more about the mathematics behind the diagram will find them detailed at length in the book quoted above. Here, we are going to concern ourselves with its construction, which begins with the division of a circle into 28, and can be finished in a number of ways. The pattern has the unusual feature of twelve circles arranged in four groups of three, rather than equally distributed. This could make it, for instance, a geometric backdrop for art on the theme of the twelve months, or the zodiacal signs, and so on.
And now, grab your compass and a large sheet of paper, as this is going to involve many construction lines.
1. Prepare the Inscribed Circle
Draw a large circle on your paper, with one diameter through it, and find its bisector.
Proceed to divide the circle in 8 (see Working with 4 and 8).
Draw the surrounding square and clean up all lines that are no longer necessary. This is the basis we will work on.
2. Divide the Circle in 28
To divide the circle in 28, we will need to divide it in 7, four times.
Draw the equilateral triangle whose base is one side of the square.
Join the points where the triangle cuts the circle, to the middle of the side. These are two sides of a heptagon (seven-sided polygon).
Use your compass to transfer the length of that side and mark another two points of the heptagon.
Repeat once more to mark the last two points of the heptagon.
Join the points.
Number the points as follows. We are not used to working with a seven-fold division, so it's quite important to do this now, and exactly as shown.
To draw the second heptagon, start with the equilateral triangle opposite the first, which defines two sides in the same way.
Walk the measurement around the circle.
Join the heptagon and number the new points as follows.
Now repeat step 7 with one of the vertical sides of the square.
Find all the points of this third heptagon, join them, and number them.
Finally, repeat with the triangle on the last side of the square.
Complete the heptagon and number the points.
3. Draw the Heptagrams
For the first heptagram (seven-pointed star), join the numbers 1 to 7 only, in that order. Then join 7 back to 1.
Now join 8 to 14, ending with 14 back to 8.
Now join 15 to 21, ending with 21 back to 15.
Finally, join 21 to 28, ending with 28 back to 21.
The circle, divided by the power of 7 and 4.
4. Add the Circles
The twelve circles, or "fruits", are not tangent to each other, but to the nearest sides of the heptagrams, as shown in the first four drawn below. Now, on an aesthetic basis, you could make them tangent if you wanted. But there is a deeper reason for this specific circle size, and it will be revealed further down.
Draw the remaining circles. The base grid is complete.
5. Two Simple Ways to Finish
Here are examples of two different finished versions of the diagram, achieved through two different inking patterns, without additional construction.
Ink the heptagrams in full, then the circles as if they were behind the stars. Finish with the square in the far background, marking the tips of the triangles in the corners.
Only ink the outer outlines of each heptagram, taking it one at a time so they look clearly layered. Then ink the "fruits", and finally the original circle at the very back.
6. "Earth and Moon" Version
Even though the end result is (deceptively) simple, this requires a little more construction.
Taking out all the heptagram lines and the four triangles, we are working with the original circle-in-square, and the twelve fruits.
Draw the central circle tangent to the twelve fruits.
Construct the square around this circle.
Ink as follows.
What's remarkable about this diagram is that the central circle symbolizes the earth, and the twelve small ones the moon. This is quite literal, because their
ratio, which is of 3 to 11, is exactly the ratio of the size of the moon to that of the
earth. This is a same-scale diagram of the two physical planets.
7. Full Detail Version
Finally, one more version for the hardcore construction enthusiasts:
I've highlighted the triangles to show their intersection. Draw a circle that passes through the intersection shown below. The lines of the triangles contained in that circle form two squares.
To draw a third, static square, mark its corners on the circle using the diagonals of the original square.
Ink parts of the three squares to achieve the effect below.
Draw another smaller circle passing through the point below, which is an intersection of heptagrams.
Complete the inking. The ring with the three squares, interrupting the lines of the heptagrams, gives the effect of separate, layered shapes.
Here's how it would look without the moons, if you fancy.
Colour to taste!
Awesome Work, You're Done!
Today we've had just a taste of the dimension of geometry that is reserved for architecture and may end up inhabited, rather than looked at. Despite handsome results such as that exemplified above, it leans more towards hard maths than art, so we will leave it there.
In our next lesson, we will return to infinite tiling patterns, with a lesser-known Islamic design, light-hearted and flowery.