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Geometric Design: The "Heavenly City" Diagram

This post is part of a series called Geometric Design for Beginners.
Geometric Design: Create a Pattern from the Cordoba Synagogue
Geometric Design: How to Draw a Flowery Tiling Pattern
Final product image
What You'll Be Creating

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

Step 1

Draw a large circle on your paper, with one diameter through it, and find its bisector.

Heavenly City step 1

Step 2

Proceed to divide the circle in 8 (see Working with 4 and 8).

Heavenly City step 2

Step 3

Draw the surrounding square and clean up all lines that are no longer necessary. This is the basis we will work on.

Heavenly City step 3

2. Divide the Circle in 28

To divide the circle in 28, we will need to divide it in 7, four times.

Step 1

Draw the equilateral triangle whose base is one side of the square.

Heavenly City step 4

Step 2

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).

Heavenly City step 5

Step 3

Use your compass to transfer the length of that side and mark another two points of the heptagon.

Heavenly City step 6

Step 4

Repeat once more to mark the last two points of the heptagon.

Heavenly City step 7

Step 5

Join the points.

Heavenly City step 8

Step 6

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.

Heavenly City step 9

Step 7

To draw the second heptagon, start with the equilateral triangle opposite the first, which defines two sides in the same way.

Heavenly City step 10

Step 8

Walk the measurement around the circle.

Heavenly City step 11

Step 9

Join the heptagon and number the new points as follows.

Heavenly City step 12

Step 10

Now repeat step 7 with one of the vertical sides of the square.

Heavenly City step 13

Step 11

Find all the points of this third heptagon, join them, and number them.

Heavenly City step 14

Step 12

Finally, repeat with the triangle on the last side of the square.

Heavenly City step 15

Step 13

Complete the heptagon and number the points.

Heavenly City step 16

3. Draw the Heptagrams

Step 1

For the first heptagram (seven-pointed star), join the numbers 1 to 7 only, in that order. Then join 7 back to 1.

Heavenly City step 17

Step 2

Now join 8 to 14, ending with 14 back to 8.

Heavenly City step 18

Step 3

Now join 15 to 21, ending with 21 back to 15.

Heavenly City step 19

Step 4

Finally, join 21 to 28, ending with 28 back to 21.

Heavenly City step 20

Step 5

The circle, divided by the power of 7 and 4.

Heavenly City step 21

4. Add the Circles

Step 1

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.

Heavenly City step 22

Step 2

Draw the remaining circles. The base grid is complete.

Heavenly City step 23

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.

Version 1

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.

Finished version 1
Finished version 1 coloured

Version 2

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.

Finished version 2
Finished version 2 coloured

6. "Earth and Moon" Version

Even though the end result is (deceptively) simple, this requires a little more construction.

Step 1

Taking out all the heptagram lines and the four triangles, we are working with the original circle-in-square, and the twelve fruits.

Earth and Moon step 1

Step 2

Draw the central circle tangent to the twelve fruits.

Earth and Moon step 2

Step 3

Construct the square around this circle.

Earth and Moon step 3

Step 4

Ink as follows.

Earth and Moon finished

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.

Earth and Moon coloured

7. Full Detail Version

Finally, one more version for the hardcore construction enthusiasts:

Step 1

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.

Full detail step 1

Step 2

To draw a third, static square, mark its corners on the circle using the diagonals of the original square.

Full detail step 2

Step 3

Ink parts of the three squares to achieve the effect below.

Full detail step 3

Step 4

Draw another smaller circle passing through the point below, which is an intersection of heptagrams.

Full detail step 4

Step 5

Complete the inking. The ring with the three squares, interrupting the lines of the heptagrams, gives the effect of separate, layered shapes.

Full detail finished

Here's how it would look without the moons, if you fancy.

Full detail without moons

Step 6

Colour to taste!

Full detail coloured

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.

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