Ever Had the Feeling You're Going Round in Circles?
by
Lyndon Baker

If only.... ...we had some compasses and... there was time in 2H's busy timetable....

'Cos on Friday afternoon we could learn to use compasses properly along the way, see some interesting curves and develop the idea of an envelope. (envelope: n. a curve or surface that is tangential to each of a family of curves or surface, according to Collins Dictionary of Mathematics by Borowski E.J. & Borwein J.M. (1989), Harper Collins, Britain) Well, in the first place we could...

Draw a base circle.
Using the 'star' mark off (in red) 18 points evenly spaced around the circumference.

circles

Choose one of the points, colour it blue.

From each of the red points as centres in turn - draw the circles that pass through the blue point.

cardoid

What results is the CARDIOID. (Which could be coloured in later...)

circles

Secondly we could...

Repeat all of the above: base circle, 18 points, but with the blue point away from the base circle.

circles circles

What results this time is the LIMACON

circle

Or we could...

Draw a base circle. Using the 'star' mark off (in red) 36 points evenly spaced around the circumference. Draw in the diameter. Draw in the circles with centres at the red points such that EACH circle touches the diameter.

NEPHROID

What results is the NEPHROID.

two Nephroids

Or ...

Draw a base circle of radius 100mm.
Mark off at 50mm along the diameter.
At each point, draw in chords perpendicular to the diameter.

Chords

With each chord in turn - using it as a diameter, draw circles.

Elipse Elipse

What results is the ELLIPSE.

Elipse

If only we had the time for ...

...associated ideas
...mathematical terms in context
...gaining genuine insights
...

If only...

...we still had those compasses and
...there was time in 2H's busy timetable...

...we could have looked at some transformations and a few other things en route!

We could have used the ray diagram (fig 1) to produce 12 equally spaced dots around the circumference of any circle (fig 2).

star circle

We could have chosen any point, counted on 5 more points and joined them together. Then counted on 5 more points and joined them...

star in circle

The transformation being considered was

n \to n + 5

The resulting diagram could them be coloured in...
Discussion could have been encouraged about how many times we counted round the circle and why.

Other transformations could also have been considered:

n \to n + 3

$n \to n + 4$

$n \to n + 9$

$n \to n + 1$ etc.

or $n \to n - 4$

$n \to n - 7$ etc.

Discussion could have developed about which transformation produced the most interesting/ most boring patterns. Which produced the same patterns?

We could have used more points around the circle...
We could have combined transformations...
We could have....
We could.......
We...........

While if there had really been time we could have considered joining each point to every other point!

If only....

...while en passant...
...en passant ...Shades of opinion.

A colleague and I had been talking about the merits of children colouring in their work when the following question was unearthed.

With a simple Venn diagram (this gives an indication of time!) there are four ways of shading it:

venn diagrams

Convince yourself there are 24 ways of shading a more complex Venn diagram:

venn diagrams