Constructions for the Golden Ratio - Flags of the World and Pentagram stars
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Flags of the World and Pentagram stars
Its Colouring Book link has small pictures of the flags useful for answering the questions in this Quiz.
Things to do
- How many five-pointed stars are there on the USA flag? Has this always been the case? What is the reason for that number?
- Many countries have a flag which contains the 5-pointed star above. Find at least four more.
- Which North African country has a pentagram on its flag?
- Some countries have a flag with a star which does not have 5 points: Which country has a six-pointed star in its flag?
- Find all those countries with a flag which has a star of more than 6 points.
- Project Make a collection of postage stamps containing flags or specialise in those with a five-pointed star or pentagram. You might also include mathematicians that have appeared on stamps too. Here's a Swiss stamp to start off your (virtual) collection.
Prof Robin Wilson has a Stamp Corner section in the Mathematical Intelligencer.
He has produced a book Stamping Through Mathematics Springer Verlag, 136 pages, (2001) which has got some good reviews, eg this and this. There is also a comprehensive web site called Sci-Philately with several sections on maths and also Jeff Miller's Mathematicians on Stamps page is a large catalogue of stamps with pictures. Jim Kuzmanovich also has a page on mathematical stamp collecting.
Phi and Triangles
Phi and the Equilateral Triangle
Chris and Penny at Regina University's Math Central (Canada) show how we can use any circle to construct on it a hexagon and an equilateral triangle. Joining three pairs of points then reveals a line and its golden section point as follows:
- On any circle (centre O), construct the 6 equally spaced points A, B, C, D, E and F on its circumference without altering your compasses, so they are the same distance apart as the radius of the circle. ABCDEF forms a regular hexagon.
- Choose every other point to make an equilateral triangle ACE.
- On two of the sides of that triangle (AE and AC), mark their mid-points P and Q by joining the centre O to two of the unused points of the hexagon (F and B).
- The line PQ is then extended to meet the circle at point R.
Q is the golden section point of the line PR.
Q is a gold point of PR
The proof of this is left to you because it is a nice exercise either using coordinate geometry and the equation of the circle and the line PQ to find their point of intersection or else using plane geometry to find the lengths PR and QR.
The diagram on the left has many golden sections and yet contains only equilateral triangles. Can you make your own design based on this principle?
Chris and Penny's page shows how to continue using your compasses to make a pentagon with QR as one side.
Equilateral Triangles and the Golden ratio J F Rigby, Mathematical Gazette vol 72 (1988), pages 27-30.
Phi and the Pentagon Triangle
Earlier we saw that the 36°-72°-72° triangle shown here as ABC occurs in both the pentagram and the decagon.
Its sides are in the golden ratio (here P is actually Phi) and therefore we have lots of true golden ratios in the pentagram star on the left.
But in the diagram of the pentagram-in-a-pentagon on the left, we not only have the tall 36-72-72 triangles, there is a flatter on too. What about its sides and angles?
Phi and another Isosceles triangle
If we copy the BCD triangle from the red diagram above (the 36°-72°-72° triangle), and put another triangle on the side as we see in this green diagram, we are again using P=Phi as above and get a similar shape - another isosceles triangle - but a "flat" triangle.
The red triangle of the pentagon has angles 72°, 72° and 36°, this green one has 36°, 36°, and 72°.
Again the ratio of the shorter to longer sides is Phi, but the two equal sides here are the shorter ones (they were the longer ones in the "sharp" triangle).
These two triangles are the basic building shapes of Penrose tilings (see the section mentioned previously for more references). They are a 2-dimensional analogue of the golden section and make a very interesting study in their own right. They have many relationships with both the Fibonacci numbers and Phi.
