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Working with 5 and 10, we'll be constructing pentagons, decagons, and stars that go with them, but also the versatile decagram grid used in Moroccan zillij (ceramic mosaics) to create a great variety of shapes. We'll also work with numbers for which no truly accurate division of the circle exists.
Dividing the Circle Into 5
Step 1
Draw a circle centered on a horizontal line, and the perpendicular to the centre.
Step 2
Keeping the same compass opening, move the dry point to one side and mark the two points as shown.
Step 3
The line that connects these two points cuts the horizontal diameter at point C.
Step 4
Place the dry point on C, with the compass opening CA. This arc cuts the diameter at point D.
Step 5
Open the compass to distance AD, and draw the arc centered on A. It will not go through C even if it comes very close.
Step 6
Set the compass opening to the distance from D to the centre, but place the dry point on B to draw an arc. The five points on the circle divide it into five.
Dividing the Circle Into 10
Follow the steps 1-6 above, and then carry on:
Step 7
Keep the compass opening as in step 6, and draw an arc centered on A.
Step 8
Now return the compass to the opening AD, and draw an arc centered on B.
Shapes
Pentagon and Pentagram
Join the points in a circle divided into five.
Decagon
Join the points in a circle divided into ten.
Decagrams (ten-pointed stars)
Different decagrams are obtained depending whether we join every second, third or fourth point. They are respectively made up of two pentagons, a continuous line, and two pentagrams.
The Decagram Grid
This is a grid formed by overlaying the latter two decagrams shown above: the one formed with a single line, and the one formed of two pentagrams.
Many different shapes, both regular and irregular, can be drawn using the grid lines. Here are just a few that recur in traditional art:
More elaborate ten-pointed stars can also be built upon it. Here is one of them.
Interlaced Ten-Pointed Star
Step 1
Start with a decagram grid. Pick out the shape shown here.
Step 2
Repeat with the next shape: they overlap. Carry on all around the grid.
Step 3
Now pick out the angle highlighted here, that connects the inner point of two overlapping shapes. Do this all around: the pattern now looks as if it were drawn with a single continuous line (which indeed it can be).
This pattern can then be coloured in various ways, or given a woven effect (this will be detailed in our sixth lesson).
Approximate Constructions
By now, we have learned to divide a circle and draw polygons with every number up to 12. Only three numbers are left to study: 7, 9 and 11. It is actually not possible to draw a true heptagon, enneagon, or hendecagon using geometry, but a few methods have been developed to create good approximations, quite accurate enough for the naked eye.
Heptagon (7 Sides)
Step 1
Follow the construction steps for a static square (see Working With 4 and 8).
Let's name the relevant points to make what follows easier.
Step 2
With the dry point on A and the compass opening at AB, mark point F on the vertical.
Step 3
The line DF cuts the circle at G; the line EF cuts the circle at H. AG and AH are the measures of the sides of the heptagon, so all we have to do now is walk these measures around the circle.
Step 4
With the point on G and the opening set to GA, mark point I on the circle. Then move the point to H (the compass opening HA is equal to GA) and mark point J.
Step 5
Now use I as a centre to find K, and J as a centre to find L. KL may not be quite the same measure, but that's normal in an approximate construction. When creating patterns with the hepta- family, geometers were not above cheating a little to make it fit!
Step 6
Join the points on the circle. Careful! Of the points made on the circle by the horizontal and vertical lines, only A is involved in the final shape.
The heptagon has two corresponding heptagrams, both made of a continuous line:
Enneagon (9 Sides)
Step 1
Draw a circle centered on a horizontal line, and the perpendicular to the centre.
Step 2
Return the compass opening to the radius of the circle, and with the point on F, draw an arc that cuts the circle at G and H.
Step 3
Set the compass to the distance AE. With the point on C, draw an arc that cuts the vertical at I.
Step 4
With the point on E, draw the arc EI to cut the circle at two points. Repeat with the point on G and then on H.
Step 5
E, G, H, plus the six points marked in step 4, are the nine points of the enneagon. Again, of the points made on the circle by the horizontal and vertical lines, only E is involved in the final shape.
Three enneagrams correspond to the enneagon. The central one is made of three equilateral triangles, the others of one continuous line.
Hendecagon (11 sides)
Step 1
Draw a circle on a vertical line.
Step 2
With the same opening and the dry point on A, draw an arc that cuts the circle at B and C.
Step 3
Connect B and C to find point D on the vertical.
Step 4
With the opening set to DO, place the dry point on O and G respectively to draw two arcs. Connect their intersections to find G (in other words we have bisected DO).
Step 5
With the point on D, draw the arc DG that cuts the upper arc at two points.
Step 6
Join the two points to find H.
Step 7
Now set the opening to AH and place the point on A to draw an arc that cuts the circle at I and J. The distance AH=AI=AJ is the division of the circle we need, so all we have to do now is walk this distance around the circle.
Step 8
Dry point on I and J, respectively, to mark two more points on the circle...
Step 9
... then two more, and so on till done.
Step 10
Join the 11 points, which do not include the point where the vertical line cuts the bottom of the circle.
A hendecagon produces four possible hendecagrams, all made of a single line:
With this, we have completed our basic shape constructions, covering all numbers from 3 to 12, related grids and patterns that can be modified ad infinitum. But in a sense, we have not worked with 1, which unfolds in space as a circle (a one-sided shape). This will be the subject of our next lesson, the last to cover basics before we move on to more complex ornamental designs.
