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Design

Geometric Design: Working With 5 and 10

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This post is part of a series called Geometric Design for Beginners.
Geometric Design: Working With 6 and 12
Geometric Design: Working With Circles
Final product image
What You'll Be Creating

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.

Dividing the circle into 5 step 1

Step 2

Keeping the same compass opening, move the dry point to one side and mark the two points as shown.

Dividing the circle into 5 step 2

Step 3

The line that connects these two points cuts the horizontal diameter at point C.

Dividing the circle into 5 step 3

Step 4

Place the dry point on C, with the compass opening CA. This arc cuts the diameter at point D.

Dividing the circle into 5 step 4

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.

Dividing the circle into 5 step 5

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 5 step 6

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.

Dividing the circle into 10 a

Step 8

Now return the compass to the opening AD, and draw an arc centered on B.

Dividing the circle into 10 b

Shapes

Pentagon and Pentagram

Join the points in a circle divided into five.

Pentagon and pentagram

Decagon

Join the points in a circle divided into ten.

Decagon

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.

Different decagrams
Ten-point star in the Tuman Aqa complex Samarkand
Ten-point star in the Tuman Aqa complex, Samarkand. Photo by Patrick Ringgenberg

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.

Decagram grid

Many different shapes, both regular and irregular, can be drawn using the grid lines. Here are just a few that recur in traditional art:

Decagram grid shapes

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.

Interlaced star step 1

Step 2

Repeat with the next shape: they overlap. Carry on all around the grid.

Interlaced star step 2

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).

Interlaced star step 3

This pattern can then be coloured in various ways, or given a woven effect (this will be detailed in our sixth lesson).

Interlaced star coloured

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).

Heptagon step 1

Let's name the relevant points to make what follows easier.

Heptagon step 1b

Step 2

With the dry point on A and the compass opening at AB, mark point F on the vertical.

Heptagon step 2

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.

Heptagon step 3

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.

Heptagon step 4

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!

Heptagon step 5

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.

Heptagon step 6

The heptagon has two corresponding heptagrams, both made of a continuous line:

Heptagrams

Enneagon (9 Sides)

Step 1

Draw a circle centered on a horizontal line, and the perpendicular to the centre.

Enneagon step 1

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.

Enneagon step 2

Step 3

Set the compass to the distance AE. With the point on C, draw an arc that cuts the vertical at I.

Enneagon step 3

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.

Enneagon step 4

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.

Enneagon step 5

Three enneagrams correspond to the enneagon. The central one is made of three equilateral triangles, the others of one continuous line.

Enneagrams

Hendecagon (11 sides)

Step 1

Draw a circle on a vertical line.

Hendecagon step 1

Step 2

With the same opening and the dry point on A, draw an arc that cuts the circle at B and C.

Hendecagon step 2

Step 3

Connect B and C to find point D on the vertical.

Hendecagon step 3

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).

Hendecagon step 4

Step 5

With the point on D, draw the arc DG that cuts the upper arc at two points.

Hendecagon step 5

Step 6

Join the two points to find H.

Hendecagon step 6

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.

Hendecagon step 7

Step 8

Dry point on I and J, respectively, to mark two more points on the circle...

Hendecagon step 8

Step 9

... then two more, and so on till done.

Hendecagon step 9
Hendecagon step 9b
Hendecagon step 9c

Step 10

Join the 11 points, which do not include the point where the vertical line cuts the bottom of the circle.

Hendecagon step 10

A hendecagon produces four possible hendecagrams, all made of a single line:

Hendecagrams

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.

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