Let's start by showing how to construct the golden section points on any line: first a line phi (0·618..) times as long as the original and then a line Phi (1·618..) times as long. John Turner has nicknamed the two points that divide a line at the golden ratio (0.618 of the way from either end) as gold points .

 

Constructing the internal golden section points: phi

If we have a line with end-points A and B, how can we find the point which divides it at the golden section point?
We can do this using compasses for drawing circles and a set-square for drawing lines at right-angles to other lines, and we don't need a ruler at all for measuring lengths!

(In fact we can do it with just the compasses, but how to do it without the set-square is left as an exercise for you.)
We want to find a point G between A and B so that AG:AB = phi (0·61803...) by which we mean that G is phi of the way along the line. This will also mean that the smaller segment GB is 0·61803.. times the size of the longer segment AG too.

AG	=	GB	= phi = 0·618033.. =	√5 – 1
AB		AG		2

Here's how to construct point G using set-square and compasses only:

1. First we find the mid point of AB. To do this without a ruler, put your compasses on one end, open them out to be somewhere near the other end of the line and draw a semicircle over the line AB. Repeat this at the other end of the line without altering the compass size. The two points where the semicircles cross can then be joined and this new line will cross AB at its mid point.
2. Now we are going to draw a line half the length of AB at point B, but at right-angles to the original line. This is where you use the set-square (but you CAN do this just using your compasses too - how?). So first draw a line at right angles to AB at end B.
3. Put your compasses on B, open them to the mid-point of AB and draw an arc to find the point on your new line which is half as long as AB. Now you have a new line at B at right angles to the AB and BC is half as long as the original line AB.
4. Join the point just found to the other end of the original line (A) to make a triangle. Putting the compass point at the top point of the triangle and opening it out to point B (so it has a radius along the right-angle line) mark out a point on the diagonal which will also be half as long as the original line.
5. Finally, put the compass point at point A, open it out to the new point just found on the diagonal and mark a point the same distance along the original line. This point is now divides the original line AB into two parts, where the longer part AG is phi (0·61803..) times as long as the original line AB.
   

Why does this work?

It works because, if we call the top point of the triangle T, then BT is half as long as AB. So suppose we say AB has length 1. Then BT will have length 1/2. We can find the length of the other side of the triangle, the diagonal AT, by using Pythagoras' Theorem:

AT² = AB² + BT²

i.e.

AT² = 1² + (1/2)²

AT² = 1 + 1/4 = 5/4

Now, taking the square-root of each side gives:

AT = (√5)/2

Point V was drawn so that TV is the same length as TB = AB/2 = 1/2.
So AV is just AT - TV = (√5)/2 – 1/2 = phi.
The final construction is to mark a point G which is same distance (AV) along the original line (AB) which we do using the compasses.

So AG is phi times as long as AB!

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Constructing the external golden section points: Phi

This time we find a point outside of our line segment AB so that the new point defines a line which is Phi (1·618..) times as long as the original one.

Here's how to find the new line Phi times as long as the original:

1. First repeat the steps 1 and 2 above so that we have found the mid-point of AB and also have a line at right angles at point B.
2. Now place the compass point on B and open them out to touch A so that you can mark a point T on the vertical line which is as long as the original line.
3. Placing the compass point on the mid-point M of AB, open them out to the new point T on the vertical line and draw an arc on the original line extended past point B to a new point G.
4. The line AG is now Phi times as long as the original line AB.

Why does this work?
If you followed the reasoning for why the first construction (for phi) worked, you should find it quite easy to prove that AG is Phi times the length of AB, that is, that AG = (√5/2 + 1/2) times AB.

Hint:
Let AB have length 1 again and so AM=MB=1/2. Since BT is now also 1, how long is MT? This is the same length as MG so you can now find out how long AG is since AG=AM+MG.

Hofstetter's 3 Constructions of Gold Points

Using only circles

Kurt Hofstetter has found a beautifully simple construction for a line and its golden section point using only compasses to draw 4 circles:

1. On any straight line S, pick two points X and Y.
   With each as centre draw a circle (green) through the other point labelling their points of intersection G (top) and B (bottom) and the points where they meet line S as P and Q;
2. With centre X, draw a circle through Q (blue);
3. With centre Y, draw a circle through P (blue); labelling the top point of intersection of the blue circles A;

G is a gold point of AB (G is at the golden section of AB).
A Simple Construction of the Golden Section, Kurt Hofstetter in Forum Geometricorum Vol 2 (2002) pages 65-66 which has the proof too.

Another even simpler method!

Kurt Hofstetter has discovered a very simple constructions of the gold point on a line AB just using circles and one line to find the gold point(s) on a given line segment AB:

1. With centre A, draw a circle through B;
2. With centre B, draw a circle through A;
3. Extend BA to meet the first circle (centre A) at C;
4. Label the lower point on both circles as D;
5. With centre C draw another circle through B;
6. Where the larger circle meets the circle centred on B, label the upper point E;

Line DE crosses line AB at G.
G is a gold point of AB
The other gold point on AB can be constructed similarly.

Uli Eggermann of Germany points out that we also have A as the gold point of CG!
It is quite simple to prove too using the properties of Phi and phi that phi=1/Phi and 1+phi=Phi:

• If AB is 1 then so is CA: they are both radii of the same circle
• Above we constructed G, the gold point of AB (=1), so that AG is phi
• CG = CA + AG = 1 + phi = Phi
• Since CA is 1 and CG is Phi then CA/CG = 1/Phi = phi
• so A is the gold point of CG

Kurt Hofstetter's proof was published as Another 5-step Division of a Segment in the Golden Section in Forum Geometricorum Vol 4 (2004) pages 21-22.

Lemoine's Construction of the Golden Section

In this construction to find the gold point of a given line segment AB rediscovered by K Hofstetter but going back to 1902, we again need to construct only one line:

1. With A as centre, draw the first circle through B;
2. With B as centre, draw a second circle through A using labels C (top) and D (bottom) for the two points of intersection with the first circle;
3. Through the top point, C, draw a third circle through A and B and use label E for its other intersection with the first circle;
4. Draw the line CD and where it meets the third circle, label this point F;
5. With centre E draw a circle though F; where it meets the line AB label this point G;

G is a gold point of AB.
Also, if BA is extended to meet the larger circle with centre E at G', then
A is a gold point of G'B.

A 5-step Division of a Segment in the Golden Section K Hofstetter in Forum Geometricorum Vol 3 (2003) 205-206.
The earliest reference to this construction appears to be in Géométrographie ou Art des Constructions Géométriques by E. Lemoine, published by C. Naud, Paris, 1902, page 51.

Phi and Pentagons

There is an intimate connection between the golden section and the regular 5-sided shape called a pentagon and its variation - the pentagram - that we explore first.

Pentagons and Pentagrams

The pentagram is a symmetrical 5-pointed star that fits inside a pentagon. Starting from a pentagon, by joining each vertex to the next-but-one you can draw a pentagram without taking your pen off the paper.

The pentagram has 5 triangles on the edges of another pentagon at its centre. Let's focus on one of the triangles and the central pentagon as shown here.

All the orange angles at the vertices of the pentagon are equal. They are called the external angles of the polygon. What size are they? This practical demonstration will give us the answer:

• Take a pen and lay it along the top edge pointing right
• Turn it about the top right vertex through the orange angle so that it points down to the lower right
• Move the pen down that side of the pentagon to the next vertex and turn it through the next orange angle
• Repeat moving it along the sides and turning through the rest of the orange angles until it lies back on the top edge
• The pen is now back in its starting position, pointing to the right so it has turned through one compete turn.
• It has also turned through each of the 5 orange angles
• So the sum of the 5 orange angles is one turn or 360°
• Each orange angle is therefore 360/5=72°

The green angle is the same size as the orange angle so that the two "base" angles of the blue triangle are both 72°.
Since the angles in a triangle sum to 180° the yellow angle is 36° so that 72° + 72° + 36° = 180°.

The basic geometrical facts we have used here are:

So the pentagram triangle has angles of 36 °, 72° and 72°. Now let's find out how long its sides are.

The 36°-72°-72° triangle

In this diagram, the triangle ABC is isosceles, since the two sides, AB and AC, are equal as are the two angles at B and C.
[Also, angles ABC and ACB are twice angle CAB.]

If we bisect the base angle at B by a line from B to point D on AC then we have the angles as shown and also angle BDC is also 72°. BCD now has two angles equal and is therefore an isosceles triangle; and also we have BC=BD.

Since ABD also has two equal angles of 36°, it too is isosceles and so BD=AD. So in the diagram the three sides BC, BD and DA are all the same length.

We also note that the little triangle BCD and the whole triangle ABC are similar since they are both 36°-72°-72° triangles.

Let's call the smallest segment here, CD, length 1 and find the lengths of the others in relation to it. We will therefore let the ratio of the smaller to longer sides in triangle BCD be r so that if CD is 1 then BC is r.

In the larger triangle ABC, the base is now r and as it is the same shape as BCD, then its sides are in the same ratio so Ab is r times BC, e.d. AB is r².
Also, we have shown BC=BD=AD so AD is r (and CD is 1).
From the diagram we can see that AC=AD+DB.
But AC=1+r and in isosceles triangle ABC, AB (which is r²) is the same length as AC (which is 1+r), so
r² = 1 + r
and this is the equation which defined the golden ratio.

So the triangle with angles 36°, 72°, 72° has sides that are proportional to Phi, Phi and 1 (which is the same as 1,1,phi).

Pentagrams and the 36°-72°-72° triangle

If we look at the way a pentagram is constructed, we can see there are lots of lines divided in the golden ratio: Since the points can be joined to make a pentagon, the golden ratio appears in the pentagon also and the relationship between its sides and the diagonals (joining two non-adjacent points).

The reason is that Phi has the value 2 cos (π/5) where the angle is described in radians, or, in degrees, Phi=2 cos (36°).
[See below for more angles whose sines and cosines involve Phi!]

Since the ratio of a pair of consecutive Fibonacci numbers is roughly equal to the golden section, we can get an approximate pentagon and pentagram using the Fibonacci numbers as lengths of lines:
There is another flatter triangle inside the pentagon here. Has this any golden sections in it? Yes! We see where further down this page, but first, a quick and easy way to make a pentagram without measuring angles or using compasses:

Making a Paper Knot to show the Golden Section in a Pentagon

Here's an easy method to show the golden section by making a Knotty Pentagram; it doesn't need a ruler and it doesn't involve any maths either!
Take a length of paper from a roll - for instance the type that supermarkets use to print out your bill - or cut off a strip of paper a couple of centimetres wide from the long side of a piece of paper. If you tie a knot in the strip and put a strong light behind it, you will see a pentagram with all lines divided in golden ratios.

Here are 5 pictures to help (well it is a pentagram so I had to make 5 pictures!) - although it really is easy once you practice tying the knot!

1. As you would tie a knot in a piece of string ...
2. ... gently make an over-and-under knot, rolling the paper round as in the diagram.
3. (This is the slightly tricky bit!)
   Gently pull the paper so that it tightens and you can crease the folds as shown to make it lie perfectly flat.
4. Now if you hold it up to a bright light, you'll notice you almost have the pentagram shape - one more fold reveals it ...
5. Fold the end you pushed through the knot back (creasing it along the edge of the pentagon) so that the two ends of the paper almost meet. The knot will then hang like a medal at the end of a ribbon.
   Looking through the knot held up to the light will show a perfect pentagram, as in the diagram above.

 

Flags of the World and Pentagram stars

 

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:

1. 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.
2. Choose every other point to make an equilateral triangle ACE.
3. 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).
4. 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.

Phi and Right-angled Triangles

Pythagorean triangles are right-angled triangles that have sides which are whole numbers in size. Since the golden section, Phi, is not a pure fraction (it is irrational), we will not be able to find a Pythagorean triangle with two sides in golden section ratio.

However, there is as a right-angled triangle that does have sides in the golden ratio. It arises if we ask the question:

If there is such a triangle, let its shortest side be of length a and let's use r as the common ratio in the geometric progression so the sides of the triangle will be a, ar, ar².
Since it is right-angled, we can use Pythagoras' Theorem to get:

We can divide through by a²

:

and if we use R to stand for r² we get a quadratic equation:

which we can solve to find that

R =

1 ± √5 2

= Phi or –phi

Since R is r² we cannot have R as a negative number, so

The sides of the triangle as therefore

and any right-angled triangle with sides in Geometric Progression has two pairs of sides in the same ratio √Phi and one pair of sides in the Golden Ratio!

We will meet this triangle and its angles later on this page in the section on Trigonometry and Phi.

A trigonometric intersection D Quadling, Math. Gaz. (2005), Note 89.70

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Phi and Rectangles

Phi and the Trapezium (Trapezoid)

Scott Beach has invented the trisoceles trapezoid (or trapezium as we say in the UK). It is a tall isosceles triangle with its top part cut off to form a quadrilateral with the following properties:

1. the top and bottom edges are parallel
2. the angles at A and B are equal as are those at C and D
3. all the sides except the top are of equal length (hence trisoceles in Scott's name for this shape)
4. the top and bottom edges are in the proportion of the golden section: the top edge is phi = φ = 0.618... times as long as the base or, equivalently, the base is Phi = Φ = 1.618... times as long as the top edge.
   So we could call this a traphizium or, in the USA, a traphisoid!

It has the interesting property that the diagonals bisect the top two angles - the yellow angles in the diagram are all equal.

The diagram shows the whole isosceles triangle AEB with the missing part of the sides DE and CE given the length x.
Since the triangles AEB and DEC are similar (the angles are the same in each triangle) then their sides are in the same ratio (proportion) to each other.

In the large triangle AEB the base is 1 and the sides are 1+x.
In the smaller triangle DEC the base is φ and the sides are x.
Therefore the ratio of 1 to (1+x) is the same as the ratio of φ to x.
or 1/(1+x) = φ/x
i.e. x = (1+x)φ
Collecting the x's on one side we have (1 - φ)x = φ
so that x = φ/(1 – φ)
If we divide top and bottom by φ and using Φ = 1/φ we see this is the same as 1/(Φ – 1) = 1/φ = Φ
So x is the larger golden section number and the cut-off point on the side of the isosceles triangle is a golden section point!

If we split the triangles in half from E to the base, we can see that the sine of the green angle is 1/(2 (1+Φ)) = 1/( 2Φ²) = φ²/2 = (1 – φ)/2.
This makes it about 78.98984..° = 1.37863... radians.

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Phi and the Root-5 Rectangle

If we draw a rectangle which is 1 unit high and √5 = 2·236 long, we can draw a square in it, which, if we place it centrally, will leave two rectangles left over. Each of these will be phi=0·618.. units wide and, of course, 1 unit high.

Since we already know that the ratio of 1 : phi is the same as Phi : 1, then the two rectangles are Golden Rectangles (one side is Phi or phi times the other).

This is nicely illustrated on Ironheart Armoury's Root Rectangles page where he shows how to construct all the rectangles with width any square root, starting from a square.

This rectangle is supposed to have been used by some artists as it is another pleasing rectangular shape, like the golden rectangle itself.

The shape of a piece of paper

Modern paper sizes have sides that are in the ratio √2 : 1. This means that they can be folded in half and the two halves are still exactly the same shape. Here is an explanation of why this is so:

"A" series Paper

Take a sheet of A4 paper.
Fold it in half from top to bottom.
Turn it round and you have a smaller sheet of paper of exactly the same shape as the original, but half the area, called A5.
Since its area is exactly half the original, its sides are √(1/2) of the originals, or, an A4 sheet has sides √2 times bigger than a sheet of A5.
Do this on a large A3 sheet and you get a sheet of size A4.
The sides must be in the ratio of 1:√2 since if the original sheet has the shorter side of length 1 and the longer side of length s, then when folded in half the short-to-longer-side ratio is now s/2:1.
By the two sheets being of the same shape, we mean that the ratio of the short-to-long side is the same. So we have:
1/s = s/2 /1 which means that s² = 2 and so s = √2

Fibonacci paper

If we take a sheet of paper and fold a corner over to make a square at the top and then cut off that square, then we have a new smaller piece of paper.
If we want the smaller piece to have the same shape as the original one, then, if the longer side is length f and the short side length 1 in the original shape, the smaller one will have shorter side of length f-1 and longer side of length 1.
So the ratio of the sides must be the same in each if they have the same shape: we have 1/f = (f-1)/1 or, f²-f=1 which is exactly the equation from which we derived Phi.
Thus if the sheets are to have the same shape, their sides must be in the ratio of 1 to Phi, or, the sides are approximately two successive Fibonacci numbers in length!

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Phi and other Polygons

Decagons

The smallest angle in the "pentagram triangle" was 36° so we can pack exactly ten of these together round a single point and make a decagon as shown here.

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Here is a decagon - a 10-sided regular polygon with all its angles equal and all its sides the same length - which has been divided into 10 triangles.
Because of its symmetry, all the triangles have two sides that are the same length and so the two other angles in each triangle are also equal.
In each triangle, what is the size of the angle at the centre of the decagon?
We now know enough to identify the triangle since we know one angle and that the two sides surrounding it are equal. Which triangle on this page is it?

From what we have already found out about this triangle earlier, we can now say that

The radius of a circle through the points of a decagon is Phi times as long as the side of the decagon.

This follows directly from Euclid's Elements Book 13, Proposition 9.

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Penrose tilings

Recently, Prof Roger Penrose has come up with some tilings that exhibit five-fold symmetry yet which do not repeat themselves for which the technical term is aperiodic or quasiperiodic. When they appear in nature in crystals, they are called quasicrystals. They were thought to be impossible until fairly recently. There is a lot in common between Penrose's tilings and the Fibonacci numbers.
The picture above is made up of two shapes of rhombus or rhombs - that is, "pushed over squares" where each shape has all sides of the same length. The two rhombs are made from gluing two of the flat pentagon triangles together along their long sides and the other from gluing two of the sharp pentagon triangles together along their short sides.

Note that the top "flat" diagram shows the sharp and flat triangles have the same height and that their bases are in the ration Phi:1 (or 1: Phi-1 which is 1:phi):

Since the Kite and Dart are made of two identical triangles, then

The diagram on the right shows the relationship between the kite and dart and a pentagon and pentagram.
You can make similar tiling pictures with Quasitiler 3.0, a web-based tool and its link mentions more references to Penrose tilings.
A floor has been tiled with Penrose Rhombs at Wadham College at Oxford University.

Here are some interesting links to the Penrose tilings at other sites.

Here are some ready-to-photocopy Penrose tiles for you to photocopy and cut-out and experiment with making tiling patterns.
The Geometry Junkyard has a great page of Penrose links
Ivars Peterson's ScienceNewsOnline has an interesting page about quasicrystals showing how Penrose tilings are found in nature.
Pentaplex sell puzzle tiles based on a Penrose tiling.
Eric Weisstein's Penrose Tilings entry in his World of Mathematics online encyclopaedia.
Penrose Tiles to Trapdoor Ciphers, 1997, chapters 1 and 2 are on Penrose Tilings and, as with all of Martin Gardner's mathematical writings they are a joy to read and accessible to everyone.
A Near Golden Cuboid by Graham Hoare in Mathematics Today Vol 41, April 2005, page 53 gives the relationship between the pentagon/pentagram and Penrose's kite and dart.

Another Irregular tiling using Phi: the Ammann Chair

A single tile which produces an "irregular" tiling was found by Robert Ammann in 1977. Its dimensions involve powers of the square root of Phi (Φ):

which means it can be split into two smaller tiles of exactly the same shape. They all depend on the property of Φ that

By splitting the larger tile, and any others of identical size, in the same way, we can produce more and more tiles all of exactly the same shape, but the whole tiling will never have any periodic pattern.
Each time we split a tile, we make one that is

times the size and one that is

times the size. There are only two sizes of tile in each dissection.

The buttons under the tile show successive stages as the dissections get more numerous.

This tiling is irregular or aperiodic which means that no part of it will appear as an indefinitely recurring pattern as in the regular tilings.

Enrique Zeleny's Ammann Chair Mathematica demonstration shows these dissections either animated on the page or using the free Mathematica Player, and in colour too.

Tilings and Patterns B Gr�nbaum, G C Shephard (a Dover paperback of the original 1987 edition is due out in 2009) ISBN 0486469816 is encyclopaedic in its depth and range of tilings and patterns. It mentions this Ammann tiling on page 553.
Penrose Tiles to Trapdoor Ciphers: And the Return of Dr Matrix, M Gardner, (The Mathematical Association of America; Revised edition 1997), the second chapter gives more on Ammann's work but omits this simplest of aperiodic tilings given above. As with all mathematical books by Martin Gardner, they are excellent and I cannot recommend them highly enough!
Aperiodic Tiles, R.Ammann, B. Gr�nbaum, G.C. Shephard, Discrete and Computational Geometry (1992), pages 1-25.

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More Geometrical Gems

Here are some ore little gems of geometry showing Phi in unexpected places.
The first is about cutting three triangles from a rectangle. The next two are about balancing a flat shape on a pin head or pencil point. The final one is about areas of flat circular shapes.

First the two balancing puzzles.
The centre of gravity of a flat shape is the point at which the shape balances horizontally on a point such as the tip of a pencil. If you are careful, you can get the shape to spin on that point, also called the pivot(al) point.

A Rectangle-Triangle dissection Problem

The problem is, given a rectangle, to cut off three triangles from the corners of the rectangle so that all three triangles have the same area. Or, expressed another way, to find a triangle inside a given rectangle (any rectangle) which when it is removed from the rectangle leaves three triangles of the same area.
As shown here, the area of the leftmost triangle is x(w+z)/2.
The area of the top-right triangle is yw/2.
The area of the bottom triangle is (x+y)z/2.
Making these equal means:

The first equation tells us that x = yw/(w+z).
The second equation, when we multiply out the brackets and cancel the zx terms on each side, tells us that xw = zy. This means that y/x = w/z.

Or in other words, we have our first deduction that

Returning to xw = zy, we put x = yw/(w+z) into it giving yw²/(w+z)=zy.
We can cancel y from each side and rearrange it to give w² = z² + zw.
If we divide by z² we have a quadratic equation in w/z: (w/z)² = 1 + w/z
Let X=w/z then X² = 1 + X
The positive solution of this is X = Phi, that is, w/z = Phi or w = z Phi.
Since we have already seen that y/x = w/z then:

The Golden Section strikes again!

This puzzle appeared in J A H Hunter's Triangle Inscribed In a Rectangle in The Fibonacci Quarterly, Vol 1, 1963, Issue 3, page 66.
A follow-up article by H E Huntley entitled Fibonacci Geometry in volume 2 (1964) of the Fibonacci Quarterly on page 104 shows that, if the rectangle is itself a golden rectangle (the ratio of the longer side to the shorter one is Phi) then the triangle is both isosceles and right-angled!

Balancing an "L" shape

	In this first problem, suppose we take a square piece of card. Where will the pivot point be? That should be easy to guess - at the centre of the square.
	Now suppose we remove a small square from one corner to make an "L" shape. Where will the pencil point be now if the "L" is to balance and turn freely? The centre of gravity this time will be close to the centre but down a little on one of the square's diagonals.
	If we took off a very large square to make the "L" shape quite thin, then the centre of gravity would lie outside the L shape and we could not balance it at all!
	So we are left with the question What is the largest size of square that we can remove so that we can still just balance the L shape? Nick Lord found it was when the ratio of the big square's side and the removed square's side are in the golden ratio phi!

Note 79.59: Balancing and Golden Rectangles N Lord, Mathematical Gazette vol 79 (1995) pages 573-4.

Balancing a Crescent

Here is a similar problem but this time using circles: to remove the largest circle and still have a pivotal point:

The full circle balances at its centre point

Removing a small circle at the top moves the balance point down a little

Removing a large circle moves the balance point outside the crescent shape

The limit point is when the radii of the two circles are in the golden ratio and the centre of gravity is just on the edge of the inner circle

Ellipse and Ring

The third little geometrical gem is about a ring and an ellipse. An ellipse is the shape that a plate or anything circular appears when viewed at an angle.

Here are two circles, one inside the other. The yellow and red areas define an outer ring. Also the orange and red parts form an ellipse (an oval).

If the ring is very narrow, the two circles are similar in size and the ellipse has a much bigger area than the ring.

If the inner circle is very small, the ellipse will be very narrow and the outer ring will be much bigger in area than the ellipse.

So the question is

The answer is again when the inner radius is 0.618 of the larger one, the golden ratio.

This is quite easy to prove using these two formulae:

The are of a circle of radius r is π r²
The area of an ellipse with "radii" a and b (as shown above) is πab

(Note how that, when a = b in the ellipse, it becomes a circle and the two formulae are the same.)
So the outer circle has radius a, the inner circle radius b and the area of the ring between them is therefore:

Area of ring = π (b² – a²)

This is equal to the area of the ellipse when

π (b² – a²) = π a b
b² - a² - a b = 0

If we let the ratio of the two circles radii = b/a, be K, say, then dividing the equation by a² we have

K² - K - 1=0

which means K is either Phi or –phi. The positive value for K means that b = Phi a or a = phi b.

The equation of an ellipse is

(x/b)² + (y/a)² = 1

When a = b, we have the equation of a circle of radius a(=b)

:

(x/a)² + (y/a)² = 1 or
x² + y² = a² as it is more usually written.

You might have spotted that this equation is merely Pythagoras' Theorem that all the points (x,y) on the circle are the same distance from the origin, that distance being a.

Note 79.13 A Note on the Golden Ratio, A D Rawlings, Mathematical Gazette vol 79 (1995) page 104.
The Changing Shape of Geometry C Pritchard (2003) Cambridge University Press paperback and hardback, is a collection of popular, interesting and enjoyable articles selected from the Mathematical Gazette . It will be of particular interest to teachers and students in school or indeed anyone interested in Geometry. The three gems above are given in more detail in the section on The Golden Ratio.

1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More..

The Fibonacci Spiral and the Golden Spiral

The Fibonacci Squares Spiral

On the Fibonacci Numbers and Golden Section in Nature page, we looked at a spiral formed from squares whose sides had Fibonacci numbers as their lengths.

This section answers the question:

The Golden section squares are shown in red here, the axes in blue and all the points of the squares lie on the green lines, which pass through the origin (0,0).
Also, the blue (axes) lines and the green lines are each separated from the next by 45° exactly.
The large rectangle ABDF is the exactly the same shape as CDFH, but is (exactly) phi times as large. Also it has been rotated by a quarter turn. The same applies to CDFH and HJEF and to all the golden rectangles in the diagram. So to transform OE (on the x axis) to OC (on the y axis), and indeed any point on the spiral to another point on the spiral, we expand lengths by phi times for every rotation of 90°: that is, we change (r,theta) to (r Phi,theta + π/2) (where, as usual, we express angles in radian measure, not degrees).

So if we say E is at (1,0), then C is at (Phi,π/2), A is at (Phi², π), and so on.
Similarly, G is at (phi,–π/2), and I is at (phi², –π) and so on because phi = 1/Phi.

The points on the spiral are therefore summarised by:

If we eliminate the n in the two equations, we get a single equation that all the points on the spiral satisfy:

or

For ordinary (cartesian) coordinates, the x values are y values are generated from the polar coordinates as follows:

x = r cos(theta)
y = r sin(theta)

which we can then use in a Spreadsheet to generate a chart as shown here.
Such spirals, where the distance from the origin is a constant to the power of the angle, are called equiangular spirals. They also have the property that a line from the origin to any point on the curve always finds (the tangent to) the curve meeting it at the same angle.
Another name is a logarithmic spiral because the angle of any point from the x axis through the origin is proportional to the logarithm of the point's distance from the origin.

To see that the Fibonacci Spiral here is only an approximation to the (true) Golden Spiral above note that: at its start there are two squares making the first rectangle but the true golden spiral above has no "start" those two squares make a rectangle 1x2 but all rectangles in the true spiral are true Golden Rectangles 1xPhi. All the other rectangles on the right are ratios of two neighbouring Fibonacci numbers and are therefore only approximations to Golden Rectangles

The Golden or Phi Spiral

In the spiral above, based on Fibonacci squares spiralling out from an initial two 1x1 squares, we noted that one quarter turn produces an expansion by Phi in the distance of a point on the curve from the "origin". So in one full turn we have an expansion of Phi⁴.
In sea-shells, we notice an expansion of Phi in one turn, so that not only has the shell grown to Phi times as far from its origin (now buried deep inside the shell). Also, because of the properties of the golden section, we can see that the distances measured on the outside of the shell also have increased by Phi and it is often easier to measure this distance on the outside of a shell, as we see in the picture here on the left.
In this case, the equation of the curve is

Notice that an increase in the angle theta of 2 π radians (360° or one full turn) makes r increase by a factor of Phi because the power of Phi has increased by 1.
I will call this spiral, that increases by Phi per turn, the Golden Spiral or the Phi Spiral because of this property and also because it is the one we find in nature (shells, etc.).

Click on the Spreadsheet image to open an Excel Spreadsheet to generate the Golden Spiral in a new window.

1·61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 ..More..

Trigonometry and Phi

What is trigonometry?
We can answer this by looking at the origin of the word trigonometry.
Words ending with -metry are to do with measuring (from the Greek word metron meaning "measurement"). (What do you think that thermometry measures? What about geometry? Can you think of any more words ending with -metry?)
Also, the -gon part comes from the Greek gonia) meaning angle. It is derived from the Greek word for "knee" which is gony.

The prefix tri- is to do with three as in tricycle (a three-wheeled cycle), trio (three people), trident (a three-pronged fork).
Similarly, quad means 4, pent 5 and hex six as in the following:

• a (five-sided and) five-angled shape is a penta-gon meaning literally five-angles and
• a six angled one is called a hexa-gon then we could call
• a four-angled shape a quadragon
  (but we don't - using the word quadrilateral instead which means "four-sided") and
• a three-angled shape would be a tria-gon
  (but we call it a triangle instead)
  "Trigon" was indeed the old English word for a triangle.

So trigonon means "three-angled" or, as we would now say in English, "tri-angular" and hence we have tri-gonia-metria meaning "the measurement of triangles".
With thanks to proteus of softnet for this information.

Phi and Trig graphs

Here are the graphs of three familiar trigonometric functions: sin x, cos x and tan x in the region of x from 0 to π/2 (radians) = 90°:
The graphs meet at

• the origin, where tan x = sin x
• in the middle, where sin x = cos x i.e. where tan x = 1 or x = 45° = π/4 radians
• at another angle where tan x = cos x

What angle is at the third meeting point?

tan x	= cos x and, since tan x = sin x / cos x , we have:
sin x	= (cos x)2
	=1-(sin x)2 because (sin x)2+(cos x)2=1.
or	(sin x)2 + sin x = 1

and solving this as a quadratic in sin x, we find
sin x = (–1 + √5)/2 or
sin x = (–1 – √5)/2
The second value is negative and our graph picture is for positive x, so we have our answer:

On our graph, we can say that the intersection of the green and blue graphs (cos(x) and tan(x)) is where the red graph (sin(x)) has the value phi [i.e. at the x value of the point where the line y = phi meets the sin(x) curve].

Is there any significance in the value of tan(x) where tan(x)=cos(x)?

Yes. It is √phi = √0·618033988... = 0·786151377757.. .

Here's how we can prove this.

Take a general right-angled triangle and label one side t and another side 1 so that one angle (call it A) has a tangent of t. By Pythagoras's Theorem, the hypotenuse is √(1+t2). So we have: For all right-angled triangles: tan A = t cos A = 1 √(1 + t2) sin A = t √(1 + t2)

We now want to find the particular angle A which has tan(A) = cos(A).
From the formulae above we have:

Squaring both sides:

Multiply both sides by 1 + t²

:

If we let T stand for t² then we have a quadratic equation in T which we can easily solve:

SinceT is t² it must be positive, so the value of T we want in this case is

Since T is t² then t is √phi.
Since 1 + Phi = Phi² then the hypotenuse √(1+T²) = Phi as shown in the triangle here.
From the 1, √Phi, Phi triangle, we see that

 

In the triangle here, it is angle A:
A = 38.1727076° = 30° 10' 21.74745" = 0.1060352989.. of a whole turn = 0.666239432.. radians.
The other angle, B has its tangent equal to its cosecant (the reciprocal of the sine):
B = 90° – A = 51.8272923..° = 51° 49' 38.2525417.." = 0.143964701.. of a whole turn = 0.90455689.. radians.

Notice how, when we apply Pythagoras' Theorem to the triangle shown here with sides 1, Phi and root Phi, we have

which is one of our classic definitions of (the positive number) Phi.....
and we have already met this triangle earlier on this page!

The next section looks at the other trigonometrical relationships in a triangle and shows that, where they are equal, each involves the numbers Phi and phi.

Notation for inverse functions

A common mathematical notation for the-angle-whose-sin-is 0.5 is arcsin(0.5) although you will sometimes see this written as asin(0.5).
We can prefix arc (or a) to any trigonometrical function (cos, cot, tan, etc.) to make it into its "inverse" function which, given the trig's value, returns the angle itself.
Each of these inverse functions is applied to a number and returns an angle.
e.g. if sin(90°)=0 then 0 = arcsin(90°).

What is arccos(0.5)?
The angle whose cosine is 0.5 is 60°.
But cos(120°)=0.5 as does cos(240°) and cos(300°) and we can add 360° to any of these angles to find some more values!
The answer is arccos(0.5) = 60° or 120° or 240° or 300° or ...
With all the inverse trig. functions you must carefully select the answer or answers that are appropriate to the problem you are solving.

"The angle whose tangent is the same as its cosine" can be written mathematically in several ways:

Can you see that it can also be written as arccos( tan(A) )= A?

More trig ratios and Phi (sec, csc, cot)

In a right-angled triangle if we focus on one angle (A), we can call the two sides round the right-angle the Opposite and the Adjacent sides and the longest side is the Hypotenuse, or Adj, Opp and Hyp for short.

You might wonder why we give a name to the ratio Adj/Opp (the tangent) of angle A but not to Opp/Adj. The same applies to the other two pairs of sides: we call Opp/Hyp the sine of A but what about Hyp/Opp? Similarly Adj/Hyp is cosine of A but what about Hyp/Adj?

In fact they do have names:

• the inverse ratio to the tangent is the cotangent or cot i.e. Adj/Opp; cot(x)=1/tan(x)
• the inverse ratio to the cosine is the secant or sec i.e. Hyp/Adj; sec(x)=1/cos(x)
• the inverse ratio to the sine is the cosecant or cosec or sometimes csc i.e. Hyp/Opp; csc(x) = 1/sin(x)

You'll notice that these six names divide into two groups:

• secant, sine, tangent
• cosecant, cosine, cotangent

and show another way of choosing one representative for each of the 3 pairs of ratios (x/y and y/x where x and y are one of the three sides).

Here is a graph of the six functions where the angle is measured in radians:

This extended set of graphs reveals two more intersections involving Phi, phi and their square roots: e.g.

• if A is the angle where cos(A) = tan(A) then sin(A) = phi and cosec(A) = Phi;
  The value of A is A = 38.172..° = 0.666239... radians;
• if B is the angle where sin(B) = cot(B) then cos(B) = phi and sec(B) = Phi;
  The value of B is 51.827..° = 0.904556... radians.

Notice that A and B sum to 90° - as we would expect of any two angles where the sine of one is the cosine of the other.

Some Results in Trigonometry L Raphael, Fibonacci Quarterly vol 8 (1970), pages 371-392.

---

Some Results in Trigonometry, Brother L Raphael, The Fibonacci Quarterly vol 8 (1970) pages 371 and 392.

Other angles related to Phi

Look again at the sharp and flat triangles of the pentagon that we saw above. If we divide each in half, we have right angled triangles with sides 1 and Phi/2 around the 36° angle in the flat triangle and sides 1/2 and Phi around the 72° angle in the sharp triangle. So:

We have sin(18°) but what about cos(18°)? This has a somewhat more awkward expression as:

Now we know the sin and cos of both 30° and 18° we can find the sin and cos of the angle differences using:

and get:

AAAAGH! as Snoopy might have said.
Is there a neater (that is, a simpler) expression? Perhaps you can find one. Let me know if you do and it will be added here with your name!

There are several angles whose cosine is similar to this one:

What about other angles? From an equilateral triangle cut in half we can easily show that:

and from a 45-45-90 degree triangle we can derive:

and not forgetting, of course:

The form of cos(12°) above is derived from the expression on page 42 of
Roots of (H-L)/15 Recurrence Equations in Generalized Pascal Triangles by C Smith and V E Hoggatt Jr. in The Fibonacci Quarterly vol 18 (1980) pages 36-42.

Binet's Formula for Fib(n) in trig. terms

These two alternative forms of Binet's formula:

Fib(n) =	Phin – (–Phi)–n	=	Phin – (–phi)n
	√5		√5

use sine and cosine functions.

A Simple Trig formula for Fib(n)

Above we saw that cos(π/5) = cos(36°) = Φ/2 and cos(2 π/5) = cos(72°) = φ/2.
Since cos(180° – A) = –cos(A), we have cos(180° – 72) = – φ/2. So

We can use these in Binet's formula to rewrite it as

The Bee and the Regular Pentagon W Hope-Jones, Mathematical Gazette vol 55, 1971, pg 220 ff which is a reprint of the original 1921 version (vol 10).

Binet's Formula solely in Trig terms

This time we even replace the √5's by trig forms:

and so Binet's formula becomes this time:

or, if you prefer degrees rather than radians:

See Fibonacci in Trigonometric Form Problem B-374 proposed and solved by F Stern in The Fibonacci Quarterly vol 17 (1979) page 93 where another form is also given.

Phi and Powers of Pi

There is a simple (infinite) series for calculating the cosine and the sine of an angle where the angle is expressed in radians. See Radian Measure (the link opens in a new window - close it to return here) for a fuller explanation.
Basically, instead of 360 degrees in a full turn there are 2π radians. The radian measure makes many trigonometric equations simpler and so it is the preferred unit of measuring angles in mathematics.

If angle x is measured in radians then

Here, n! means the factorial of n which means the product of all the whole numbers from 1 to n.
For example, 4! = 1 × 2 × 3 × 4 = 24

.

So, using the particular angles above in sin(π/10) and cos(π/5) we have formulae for φ and Φ in terms of powers of π :-

In the upper formula, going to up to the π⁹ term only will give φ to 9 decimal places whereas stopping at the π⁸ term in the lower formula will give Φ to 7 decimal places.

These two formula easily lend themselves as an iterative method for a computer program (i.e. using a loop) to compute Φ and φ. To compute the next term from the previous one, multiply it by (π/5)² or by (π/10)² for φ and divide by two integers to update the factorial on the bottom, remembering to add the next term if the previous one was subtracted and vice versa. Finally multiply your number by 2.

You will need and an accurate value of π. Here is π to 102 decimal places:

     3. 14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
        58209 74944 59230 78164 06286 20899 86280 34825 34211 70679
        82..

With thanks to John R Goering for suggesting this connection between Phi and π.