Spiral Patterns
Jon Ingram shares some of his creative ideas on investigating spiral patterns. Much of what John offers here has been created with ‘freeware’, and he demonstrates the potential for generating images that will both interest and motivate learners in the mathematics classroom.
Take a circle
Take a circle, and draw twelve equally-spaced points, from the centre to the circumference:
Moving out from the centre, rotate the first point by 30 degrees, the second by 30 more, and so on. A spiral pattern appears.
If we join consecutive points, and vary the angle, we get a hint that the picture we get might not always be as simple as a spiral.
See it in action
These Geogebra images may not work in some versions of Firefox.
For an even better picture of what happens, we need to use more than just twelve points, and smaller angle increments than 30 degrees. Click the ‘play’ button on the bottom left to see what happens with 180 points.
...and with 180 points connected up with line segments
The two views emphasise very different aspects of the layout, with the dots often seeming to form rays out from the centre, and the lines starting off as spirals, but quickly turning into more interesting objects as the angles increase.
Explore for yourself
These Geogebra images may not work in some versions of Firefox.
This applet will let you explore what happens as you change the number of points, and the angle, in both the 'point view' and the ‘line view’.
Exploring the patterns
What is it possible to say about the angle used, just from the picture produced?
All of these patterns were produced using 360 points.
Is it possible to work out the exact angle used just by looking at the pattern?
Why do the rays appear, and how many of them will there be?
Is there a relationship between the angle used, the number of points used, and the number of ‘rays’ which seem to appear when we look at the points?
It seems obvious that the divisors of 360 should give us very simple patterns.
We often get these rays appearing, though, even when the angle is not a factor of 360:
Is there any way to predict how many rays there will be, and how quickly they appear?
Does this pattern model anything in the real world?
It has been claimed that spiral patterns like this can be used to model the formation of leaves and seed heads in flowers. Is has also been claimed that there is a connection between the angles observed in the seed-head of the sunflower, the Golden Ratio, and the Fibonacci numbers.
Here is a close-up image of a sunflower seed-head
A spiral pattern with angle chosen to match the sunflower.
Classroom resources
Blank grids
The quickest way to start exploring the types of pattern you can generate is to start to generate them by hand. Here are some blank worksheets which can be used, with 12, 15, 20 or 30 radial divisions from the centre to the circumference.
It is very easy, with these sheets, to explore patterns beyond those generated by the applet above. For example, what does it look like if you repeat a rule like: ‘go out two and round one, then in one and right three’?
Posters
Poster with all the images from this article, and with several more
Poster with four large examples of spiral patterns, each created using 360 points
To get the full benefit of these large examples, print them at A3 or larger, and put them somewhere they can be admired!
Interactive Environments
All the interactive environments in this article were created using Geogebra, a free dynamic geometry system written in Java. Here are two more, similar to those above, which are optimised for full-screen display on a projector or interactive whiteboard.
Alternative version with the ‘point view’ and the ‘line view’ at the same time
John Ingram teaches mathematics at Rugby School
Discuss this article
Scan this QR code using your mobile or handheld and connect to the MTi Forum anywhere.
