Geometric Design: Working With 6 and 12


Final product image
What You'll Be Creating

Working with the numbers 6 and 12 introduces us to hexagonshexagrams,dodecagons (twelve-sided polygons) and a number of dodecagrams, but it also includes triangles as a matter of course. 
The first operation we're looking at is likely the oldest geometric construction known to mankind, because it only requires a compass or its ancient equivalent, a rope. I remember discovering it spontaneously as a child, while playing idly with my compass, an experience of astonished delight which I'm sure many have had.
We start with a circle.

Dividing the circle into 6 starting point

Keeping the same compass opening, place the dry point at the top or bottom of the circle and draw another circle.

Dividing the circle into 6 step 1

Move the dry point to either of the intersection points just created, and draw another circle.

Dividing the circle into 6 step 2

Walk around the whole circle in this way. The sixth circle passes through the top of the first (black) circle, and so completes the figure. The circle is now divided into 6.

Dividing the circle into 6 step 3

Start with the steps shown above to divide it into 6. Then connect the furthest intersection points as shown here. These lines cut the circle at six more points, so that it is now divided in 12.

Dividing the circle into 12

They are created on a circle divided into six, by joining the six points marked on the circle. If we start the division at the top or bottom of the first circle, as we did here, the shapes are dynamic. 

Dynamic hexagon and hexagram

To make them static, either start the division from the left or right of the circle, or carry on till it's divided in 12, and connect this second set of six dots instead.

Static hexagon and hexagram

Note how the hexagram is formed of two equilateral triangles, but also of a smaller hexagon and six small triangles.

Tiles from Egypt or Syria
Tiles from Egypt or Syria, 15th century.

Simply join the twelve points on the circle.


A different way of dividing a circle in twelve is one we have already learned in our lesson on Working With 4 and 8: Follow the steps for drawing a static square, and stop short of drawing the square itself. Join the points on the circle and you'll discover a perfect dodecagon.

Alternative method

How is this possible? This is a glimpse of the magic of numbers. 12 is a multiple of 4 as well as of 6 or 3, and therefore it is "related" to square constructions as much as to triangular ones. The dodecagon is where the hexagon and the square can meet. This is even more visible in the next shapes:
Four different twelve-pointed stars are generated, depending on whether we join every second, third, fourth or fifth point, and they are respectively formed of hexagons, of squares, of triangles, and finally of a single continuous line.

Four dodecagrams

The last dodecagram can serve as a grid to draw a different six-pointed star, either static or dynamic depending on which points are omitted.

Different hexagrams

Previously we learned to draw the five-circle grid, which generates a grid of squares. For 6 and 12, we will construct a seven-circle grid, to create root three (√3) patterns.
Follow the steps to divide the circle into six.

Seven-circle grid steps 1-3

Draw six more circles centered on the outermost intersection points. The compass opening never changes throughout the construction.

Seven-circle grid step 4

Repeat at the new intersections for six more circles. These circles, plus the original one that they surround, are the seven circles of the grid, but we need to complete the "flower" inside each of them so that the grid is fully functional.

Seven-circle grid step 5

Again place the dry point on each of the new, outermost intersection points (there are now 12), but only draw the arc that is inside the previously drawn circles.

Seven-circle grid step 6

Finish with the last missing arcs.

Seven-circle grid step 7

Here is the finished grid with the seven circles proper highlighted.

Seven-circle grid finished

Without adding anything else, the grid itself provides many simple patterns. Simply pick out some lines (or colour in areas) to create repeating shapes. Underlying circles can be added ad infinitum to extend the pattern.

Patterns from the seven-circle grid

Drawing a hexagram in each of the circles produces a derivative grid that is equally versatile. The hexagrams don't need to be drawn one by one—they appear when certain overall lines are added.
Start with one set of diagonals. To avoid confusion, remember they pass through the top or bottom of the seven circles (ignore intermediate circles).

Grid of hexagrams step 1

Now the next set, same idea.

Grid of hexagrams step 2

Finally the horizontals, which complete the hexagrams. Note they do not pass through the centre of any of the seven main circles.

Grid of hexagrams step 3

Here's the finished grid, made up of hexagons and triangles.

Grid of hexagrams finished

Selective colouring creates all sorts of patterns, including unusual ones like the third one below. The grid can be extended indefinitely to extend the pattern.

Patterns from grid of hexagrams
Detail from Topkapi palace
Detail from Topkapi Palace in Istanbul. Picture by Giovanni Dall'Orto.

Too angular? Here's how to make this grid/pattern curvy!
Place the dry point as indicated and draw just the two arcs show. Think of it as pushing out the sides of that hexagram into curves.

Curved hexagrams pattern step 1

Repeat with the point mirroring the first.

Curved hexagrams pattern step 2

Continue all around the hexagram.

Curved hexagrams pattern step 3

Repeat with each hexagram in the grid.

Curved hexagrams pattern step 4
Curved hexagrams pattern

This is an additional step to break the grid down further, which creates even more flexibility in patterns. Start with the grid of hexagrams:

Grid of equilateral triangles step 1

Add one set of diagonals...

Grid of equilateral triangles step 2

... then the other.

Grid of equilateral triangles step 3

Finish with the missing horizontals.

Grid of equilateral triangles step 4

Here's the finished grid.

Grid of equilateral triangles finished

Such a basic grid offers infinite possibilities. It's like pixel painting, but with triangular pixels. The popular puzzle game of Tangram is based on this. Two examples are shown below, with the grid showing and without it (which considerably attenuates the presence of the triangles).

Patterns from grid of equilateral triangles
Patterns from grid of equilateral triangles

Here's a different use of the seven-circle grid. Instead of dividing the surface into tiles to be individually filled, we're going to construct more complex shapes. 
Our starting point is a grid of hexagrams:

Dodecagram pattern starting point

Ink or at least darken the outer outline of each hexagram, as these are final lines. The details in the seven circles can be rubbed out for clarity, but we need their outlines.

Dodecagram pattern step 1

We need to divide each circle into 12. To do this in bulk, add the diagonals as if constructing a grid of triangles, first one way...

Dodecagram pattern step 2

... then the other...

Dodecagram pattern step 2b

... and finally the horizontals.

Dodecagram pattern step 2c

Working in the highlighted circle, connect the six new points so as to form a second hexagram.

Dodecagram pattern step 3

Now ink the outline of the hexagram.

Dodecagram pattern step 4

Repeat steps 3 and 4 in each circle. This final design can be filled in, or given a woven effect (how to do this will be covered in an upcoming lesson.)

Dodecagram pattern step 5

Let's finish with a slightly tricky, but different-looking pattern. As a basis, it requires the seven-circle grid plus the full triangular grid. This pattern looks best on an extended grid, where it can repeat more, but we'll stick to the seven circles for the demonstration.
Here is our starting grid, using different colours for the circles and triangles, for clarity.

Pattern with squares triangles and hexagons starting point

We'll start with the central circle. Look for and mark the intersections below. They are tricky because they are not the intersection of straight lines, or of circles, but four points on the outline of this central hexagon where it is cut by arcs.

Pattern with squares triangles and hexagons step 1

If you got the correct points, joining them produces a perfect square.

Pattern with squares triangles and hexagons step 1b

All we're going to do for this pattern is draw these squares. It is finding them that is tricky, with this complex grid, and the latter needs to have been drawn very accurately.
Let's now spot the squares in the top and bottom circles. They are also static, so the points are easy to spot.

Pattern with squares triangles and hexagons step 2

Join the points:

Pattern with squares triangles and hexagons step 2b

The same squares can be found in the last four of the seven circles, all static.

Pattern with squares triangles and hexagons step 3
Pattern with squares triangles and hexagons step 3b

Now we move on to the intermediate circles below, which produce tilted squares. Knowing that two of the four points coincide with corners of the dynamic squares, we can locate the other two points for each of these. The intersections are the same as before, but at an angle.

Pattern with squares triangles and hexagons step 4

Joining the points, here are the squares:

Pattern with squares triangles and hexagons step 4b

It is now easy to visualize the remaining squares when we turn to the remaining intermediate circles—in a fuller pattern, their four corners would be already defined, as they are in the two closer to the centre below. 

Pattern with squares triangles and hexagons step 5
Pattern with squares triangles and hexagons step 5b

Here is the pattern without the grid (I have added four outer squares to "close" it, so it's more clear). Notice that where three squares touch, they enclose an equilateral triangle, and each circle of six squares creates a hexagon. Of course, you can also achieve the same result by finding the hexagons and/or triangles instead.
It is just a portion of pattern, really: the proper way to draw this one is to continue the grid of circles to the edge of the surface you're filling, draw an equally extensive grid of triangles, and then find all the squares involved.

Partial pattern

You would then get the following, and the way it is coloured can change the shapes dramatically.

Pattern with squares triangles and hexagons

That will do for 6 and 12! We've not only learned to work with triangles, hexagons and dodecagons (and related stars), we've also seen how to bring them together with squares, and how a seven-circle grid can yield not only various patterns, but other grids.
Next month we will work with a final pair of numbers, 5 and 10, and also some odd numbers such as 7 and 9.


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