Working with the numbers 6 and 12 introduces us to hexagons, hexagrams, 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.
Dividing the Circle Into 6
We start with a circle.
Keeping the same compass opening, place the dry point at the top or bottom of the circle and draw another circle.
Move the dry point to either of the intersection points just created, and draw another circle.
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 12
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.
Hexagon and Hexagram
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.
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.
Note how the hexagram is formed of two equilateral triangles, but also of a smaller hexagon and six small triangles.
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.
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.
One More Hexagram
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.
The Seven-Circle Grid
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.
Draw six more circles centered on the outermost intersection points. The compass opening never changes throughout the construction.
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.
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.
Finish with the last missing arcs.
Here is the finished grid with the seven circles proper highlighted.
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.
Grid of Hexagrams
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).
Now the next set, same idea.
Finally the horizontals, which complete the hexagrams. Note they do not pass through the centre of any of the seven main circles.
Here's the finished grid, made up of hexagons and triangles.
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.
Too angular? Here's how to make this grid/pattern curvy!
Curved Hexagrams Pattern
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.
Repeat with the point mirroring the first.
Continue all around the hexagram.
Repeat with each hexagram in the grid.
Grid of Equilateral Triangles
This is an additional step to break the grid down further, which creates even more flexibility in patterns. Start with the grid of hexagrams:
Add one set of diagonals...
... then the other.
Finish with the missing horizontals.
Here's the finished grid.
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).
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:
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.
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...
... then the other...
... and finally the horizontals.
Working in the highlighted circle, connect the six new points so as to form a second hexagram.
Now ink the outline of the hexagram.
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.)
Pattern With Squares, Triangles, and Hexagons
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.
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.
If you got the correct points, joining them produces a perfect square.
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.
Join the points:
The same squares can be found in the last four of the seven circles, all static.
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.
Joining the points, here are the squares:
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.
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.
You would then get the following, and the way it is coloured can change the shapes dramatically.
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.