| | #### Question : Where do you buy polygons ?
Answer : The corner shop.
### Richard Ashbery is a productive artist. Each time a new collection on his exhibits goes online, RISCOScode gives him a mention. His name has poped up on the NEWS page so frequently that it seemed appropriate to look at his work in more detail. His art typically features a strong mathematical component. In this article, written exclusively for RISCOScode, Richard takes us behind his art and guides us through the mathematics he is currently exploring. At the end of the article, I've added several links to other articles he has written over recent years.
Figure 1
A slide show of Richard Ashberry's kaleidoscope art
Why I explored drawing polygons by hand
I am a graphic artist who enjoys drawing many types of geometric shapes, the polygon being one of the most flexible. Using a variety of reference books and web based sources for ideas, I then use my own interpretation to create designs using a commercial RISC OS vector drawing package, ArtWorks.
Modern drawing software enables the user to create polygons instantly with a couple of mouse clicks and therefore it is easy to dismiss the complexity required to produce them. With this in mind I decided to experiment by *hand-drawing* the most common polygon shapes. The consequence of this has been to improve my basic understanding of geometric principles. It was also interesting and fun. This tutorial shows how to draw several regular polygons, starting from the triangle, up to the octagon. It contains a small amount of simple mathematics with some questions in the text to get you thinking.
Regular Polygons
Regular polygons are a group of geometric shapes having sides of equal length and angles that are all equal. Techniques have been developed over the centuries and therefore there is *no* one method of drawing these interesting shapes. A straight edge, pencil, paper and a pair of compasses are all you need to create regular polygons.
Equilateral Triangle
One of the easiest polygons to draw is the equilateral triangle. All edges are equal in length and all angles are 60°. It is created by drawing two circles of fixed diameter as shown in figure 2.
Figure 2
How to hand draw an equilateral triangle
Hexagon
Having drawn a three sided polygon, you may be expecting four sides next. However, six is more logical! The reason is that the hexagon comprises six equilateral triangles, we've already looked at how to draw one, and adding five more is straightforward. It is a simple matter of adding another circle as shown in figure 3.
Figure 3
How to hand draw a regular hexagon
Square
A regular polygon with four sides, a square, is a harder shape to make than the equilateral triangle or the hexagon because it involves quite a few intersection points. The instructions below show its construction.
Figure 4
How to hand draw a perfect square
Angles made by a square and triangle
By superimposing a triangle on a square it is immediately obvious why a triangle can only have angles that add up to 180° where a square has angles that add up to 360°. Look at Figure 5. In this example an isosceles triangle has two sides of equal length, one angle at 90° and the other two at 45° making a total of 180°.
Figure 5
An isosceles triangle inside a square
Mathematically, triangles are extremely useful when it comes to calculating the angles of polygons and this is explained in figure 7.
Pentagon
The regular pentagon is more challenging than any of the polygons tackled so far - there are a significant number of steps to work through, and they must be carried out accurately to get a good result.
Figure 6
How to hand draw a regular pentagon
Finding the internal angles of regular polygons
How do we find the angle (known as the internal angle) of, say, a pentagon? Regular polygons can be made up of triangles and we have already established that the internal angles in a triangle always add up to 180° (figure 5). Knowing this means that we can calculate the internal angles of any regular polygon using a simple formula. One of the simplest, the square can be made from two isosceles triangles. Figure 7 shows this and also a pentagon, divided into triangles.
Figure 7
Calculating the internal angle of a regular polygon
The diagram in figure 8 shows one method for constructing a heptagon, also called a septagon. Figure 9 shows all the steps.
Figure 8
Overview of hand drawing a heptagon
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Figure 9
How to hand draw a regular heptagon
A heptagon is also known as a septagon
### Octagon
Have a go at trying to draw the octagon. Detailed instructions are not provided but simply following figure 10 should be sufficient to see how it is drawn. Observe how the octagon consists of a square divided into eight segments.
Figure 10
How to hand draw a regular octagon
### Regular polygons having 9 sides and above
There are numerous web resources and books providing information for anyone wanting to try their hand at drawing polygons with more sides. However, once you've had a go at doing the examples presented here, you will probably want to investigate a vector drawing package where the polygons can be drawn very rapidly. Such specialised programs allow you to experiment with more involed shapes. Figure 11 shows some shapes from this tutorial being used in art-like ways.
The illustrations here were all drawn with the superb RISC OS, *ArtWorks 2* vector drawing package. Other commercial packages like the excellent Xara, Photo and Graphic Designer and the ubiquitous Adobe Illustrator can be used to create effective polygon patterns.
Figure 11
Starting to think of how to use polygons in art
### Answers
Figure 2 : 60° (Equilateral triangle)
Figure 3 : 120° (Hexagon)
Figure 7 : Four triangles / 120°
Figure 9 : 128.6° (Heptagon)
Figure 10 : 135° (Octagon)
Other online articles by Richard Ashbery
Kkaleidoscope art using Photodesk
Tiled Shapes
Holding a candle to ArtWorks
Creating the Olympic Rings ArtWorks
Canvas effects in ArtWorks
Creating a 3D Neon Star
Copyright materal
This document is copyright of Richard Ashbery. Feel free to copy this material but **please** keep my name with any hand-out.
Richard's artworks can be seen at http://artworks.riscos.org.uk
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