Two-dimensional Geometry and the Golden section
Fascinating Flat Facts about Phi

On this page we meet some of the marvellous flat (that is, two dimensional) geometry facts related to the golden section number Phi. A following page turns our attention to the solid world of 3 dimensions.

Contents of this Page

The Things To Do icon means there is a Things to do investigation at the end of the section.

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Phi and 2-dimensional geometry

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

A---B 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.)
A----G--B 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.. = sqrt(5)-1
   AB   AG                           2
Here's how to construct point G using set-square and compasses only:
  1. Midpoint 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. L 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. construct C on perp line at B length AB/2 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 AB.  Mark D on AC where CD=CB 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. Mark G on AB where AD=AG 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?
A--G-B with all construction lines
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
AT2 = 12 + (1/2)2
AT2 = 1 + 1/4 = 5/4
Now, taking the square-root of each side gives:
AT = (sqrt5)/2
Point V was drawn so that TV is the same length as TB = AB/2 = 1/2.
So AV is just AT - TV = (sqrt5)/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. find midpoint M on AB 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. Mark T on perp line at B so AT=AB 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. FIND G on AB with MG=MT 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. A---B--G 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 = (sqrt(5)/2 + 1/2) times AB.
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

Hofstetters 4 circles construction 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).
Article: 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!

Hofstetter circles and line construction 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 (center A) at C;
  4. Label the lower point on both circles as D;
  5. With center C draw another circle through B;
  6. Where the larger circle meets the circle centered 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.

Article: 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

hofstetter2003 = Lemoine's construction 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.

Article: A 5-step Division of a Segment in the Golden Section K Hofstetter in Forum Geometricorum Vol 3 (2003) 205-206.
Book: 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 the Equilateral Triangle

equilateral+phi 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.
hex and phi 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.
Article: Equilateral Triangles and the Golden ratio J F Rigby, Mathematical Gazette vol 72 (1988), pages 27-30.

Phi and the Trapezium (Trapezoid)

traphesium Scott Beach has invented the trisoceles trapezoid (or trapezium as we say in the UK). It is a tall isoceles 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=0.618... times as long as the base or, equivalently, the base is Phi=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.
So what was the shape of the isoceles triangle which had its top part removed to make the traphizium?
traphisium triangle The diagram shows the whole isoceles 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 phi and the sides are x.
Therefore the ratio of 1 to (1+x) is the same as the ratio of phi to x.
or 1/(1+x) = phi/x
i.e. x=(1+x)phi
Collecting the x's on one side we have (1 - phi)x = phi
so that x=phi/(1-phi)
which is the same as 1/(Phi-1) = 1/phi = Phi
So x is the larger golden section number and the cut-off point on the side of the isoceles 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+Phi)) = 1/( 2Phi2) = phi2/2 = (1-phi)/2.
This makes it about 78.98984..° = 1.37863... radians.


/ Things to do /

  1. Here is another trapezium PQRS that is constructed using the other gold point on a the equal sides of an isoceles triangle RQT.
    Also, the gold point makes the three equal-length sides SP=PQ=QR in the resulting trapezium so it is trisoceles (to use Scott's phrase) and so is a special isoceles triangle.
    PS is Phi times as long as ST and QR is Phi times as long as RT.
    Note that PQT is not the same shape as ABE above!

    What is the length of the top edge, x, in this new trapezium?

  2. Can you find any other properties of the angles or lines in this trapezium?

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Phi and the Root-5 Rectangle

If we draw a rectangle which is 1 unit high and sqrt5 long, that is, about 2·236 units 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 to phi(=0·618) is the same as Phi(=1·618..)to 1, then the two rectangles are Golden Rectangles.


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.

Pentagons and Pentagrams

Pentagram 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.

pent and angles 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:

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:

The external angles in any polygon sum to 360°.
The angles on a straight line sum to 180°.
The angles in a triangle sum to 180°.
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

Isoceles 36-36-72 angle 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 isoceles 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.

r is Phi or -phi and since lengths are positive, we therefore have that r is Phi!

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). 72,72,36 degree triangle has sides P,P and 1

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 (pi/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!]

Pentagram with fib lengths

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.
:-)This is my favourite method since it involves a Knot(t)!
paper knot

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

Guinea Bissau flag Puerto Rico Flag
Here are two flags with just one 5-pointed star: Guinea-Bissau (left) and Puerto Rico (right) from the FOTW Flags Of The World website at site.

Its Colouring Book link has small pictures of the flags useful for answering the questions in this Quiz.

/ Things to do /

  1. How many five-pointed stars are there on the USA flag? Has this always been the case? What is the reason for that number?
  2. Many countries have a flag which contains the 5-pointed star above. Find at least four more.
  3. Which North African country has a pentagram on its flag?
  4. Some countries have a flag with a star which does not have 5 points: Which country has a six-pointed star in its flag?
  5. Find all those countries with a flag which has a star of more than 6 points.
  6. 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.
    Book: 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.


The decagon and the 36°-72°-72° triangle

The smallest angle in the "pentgram 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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The shape of a piece of paper

Modern paper sizes have sides that are in the ratio 1:sqrt(2). 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

A4 paper folding
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 sqrt(1/2) of the originals, or, an A4 sheet has sides sqrt(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:sqrt(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 is sqrt(2).

Fibonacci paper

fib size paper folding
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 the Pentagon Triangle

72,72,36 degree triangle has sides P,P and 1 Pentagram 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 onthe 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

36,36,72 degree triangle with sides 1,1 and P 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.


decagon 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

Penrose rhombs

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 glueing two of the flat pentagon triangles together along their long sides and the other from glueing two of the sharp pentagon triangles together along their short sides.
kite and dart

Note that the top "flat" diagram shows the sharp and flat triangles have the same height and that their bases are in teh ration Phi:1 (or 1: Phi-1 which is 1:phi):
The sharp triangle is Phi times the area of the flat triangle
if the shortest sharp triangle's side = the longest flat triangle's side
Since the Kite and Dart are made of two identical triangles, then
The dart is Phi times the area of the kite
The diagram on the right shows the relationship between the kite and dart and a pentagon and pentagram.
The tiling picture was made with Quasitiler 3.0 which is 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.

I plan more to follow here, but in the meantime, here are some interesting links to the Penrose tilings at other sites.

WWW: Here are some ready-to-photocopy Penrose tiles for you to photocopy and cut-out and experiment with making tiling patterns.
WWW: The Geometry Junkyard has a great page of Penrose links
WWW: Ivars Peterson's ScienceNewsOnline has an interesting page about quasicrystals showing how Penrose tilings are found in nature.
WWW: Pentaplex sell puzzle tiles based on a Penrose tiling.
WWW: Eric Weisstein's Penrose Tilings entry in his World of Mathematics online encyclopaedia.
Book: 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.
Article: 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.

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A Rectangle-Triangle dissection Problem

tri in rect 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:
x(w+z) = yw and x(w+z)= z(x+y).
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

Both sides of the rectangle are divided in the same proportion.

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:

Each side of the rectangle is divided in the same ratio
This ratio is Phi = 1·6180339... ie 1:1·618 or 0·618:1.
The Golden Section strikes again!

Article: 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.
Article: 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!

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Three Geometrical Gems

Here are three little gems of geometry showing Phi in unexpected places.
The first 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.

Balancing an "L" shape

full square 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.
small sq removed 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 center of gravity this time will be close to the centre but down a little on one of the square's diagonals.
large square removed 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!
phi L 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!
Article: 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:
full circle
The full circle balances at its centre point
small circle removed
Removing a small circle at the top moves the balance point down a little
big circle removed
Removing a large circle moves the balance point outside the crescent shape
the limit circle
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.
two circles 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). narrow ring If the ring is very narrow, the two circles are similar in size and the ellipse has a much bigger area than the ring. narrow ellipse 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
What ratio of circle sizes (radii) makes the ellipse equal in area to the ring between the two circles?
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 formula 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 = a/b, 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.

Article: Note 79.13 A Note on the Golden Ratio, A D Rawlings, Mathematical Gazette vol 79 (1995) page 104.
Book: 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.

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The Golden 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:

What is the equation of the Golden spiral?
Fib polar spiral 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 same shape as CDFH, but is phi times as large. Also it has been rotated by a quarter turn. Similarly with CDFH and HJEF. This applies 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+Pi/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,Pi/2), A is at (Phi2, Pi), and so on.
Similarly, G is at (phi,-pi/2), and I is at (phi2, -pi) and so on because phi is 1/Phi.

The points on the spiral are therefore summarised by:

r = Phin and theta= n Pi/2
If we eliminate the n in the two equations, we get a single equation that all the points on the spiral satisfy:
r = Phi2 theta / Pi
r = Mtheta where M = Phi2/Pi
Such spirals, where the distance from the origin is a constant to the power of the angle, are called equiangular spirals, that is, a line from the origin to any point on the curve always finds (the tangent to) the curve meeting it at the same angle.
Coxeter states that:
This true spiral is closely approximated by the artificial spiral formed by circular quadrants inscribed in the successive squares, as in [the figure above]. (But the true spiral cuts the sides of the squares at very small angles, instead of touching them.)
The above is adapted from H S M Coxeter's book Introduction to Geometry, 1961, page 165.]

design by Ned May Ned May has generated some beautiful pictures based on Fibonacci Spirals using Visual Basic.

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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:

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

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 pi/2 (radians) = 90°:
The graphs meet at 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+sqrt5)/2 or
sin x = (-1-sqrt5)/2
The second value is negative and our graph picture is for positive x, so we have our answer:
the third point of intersection is the angle whose sine is Phi-1=0·6180339...=phi
which is about 0·66623943.. radians or 38·1727076..°
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 sqrtphi = sqrt0·618033988... = 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)
1,t,sqrt(1+t^2) tri

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

Squaring both sides:
t2 = 1/(1 + t2)
Multiply both sides by 1 + t2:
t2 ( 1 + t2 ) = 1
t2 + t4 = 1
If we let T stand for t2 then we have a quadratic equation in T which we can easily solve:
t4 + t2 – 1 = 0
T2 + T – 1 = 0
T = ( –1 ± √(1+4) ) / 2
Since T is t2 it must be positive, so the value of T we want in this case is
T = ( √5 – 1 ) / 2 = phi
1-rootPhi-Phi triangle Since T is t2 then t is √ phi.
Since 1+Phi is Phi2 then the hypotenuse √(1+T2) is Phi as shown in the triangle here.
From the 1-√Phi-Phi triangle, we see that
tan(B)=1/sin(B) or tan(B)=cosec(B)

What is the angle whose tangent is the same as it cosine?
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 tanglent 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

Phi2 = 1 + (√Phi)2 = 1 + Phi
which is one of our classic definitions of (the positive number) Phi. 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:

tan(A)=cos(A) is the same as arctan( cos(A) ) = A
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.
Opp,Adj,hyp tri 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: You'll notice that these six names divide into two groups: 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:

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

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

/ Things to do /

  1. Above we solved cos(x)=tan(x) using the 1,t,&radic(1+t2) triangle. Use the same triangle and adapt the method to find the value of sin(x) for which sin(x)=cot(x).
    If you use the formulae above then remember that you will find t, the tangent of the angle for which sin(x)=cot(x). Since we want the cotangent, just take the reciprocal of t to solve sin(x)=cot(x)=1/t.
  2. On the previous page we saw two ways to find Phi on your calculator using the 1/x button, square-root button and just adding 1. Here's how we can do the same thing to find √phi and √Phi.
    From tan(x)=cos(x) we found t=phi is a solution to t=1/√(1+t2)
    So, to uncover this value using your calculator, follow these steps:
    1. Enter any number to start the process
      1. Square it
      2. Add 1
      3. Take the square-root
      4. Take the reciprocal (the 1/x button)
      5. Write down the number now displayed
    2. Repeat the previous step as often as you like.
    Each time, given t, we compute 1/√(1+t2) and keep repeating this transformation.
    Eventually, the number we write down does not change. It is √phi = 0.7861513777... . In fact, no matter how big or small is your starting value, you'll get √phi to 4 or 5 dps after only a few iterations. Try it!
    If you want √Phi, just use the 1/x button on your final answer.

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

Other angles related to Phi

Half of 1,1,Phi is 54-90-36; half of Phi,Phi,1 is 18-72-90 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:

cos(72°) = cos(2 pi) = sin(18°) = sin(pi) = phi = 1




2 Phi

cos(36°) = cos(pi) = sin(54°) = sin(3 pi) = Phi = 1




2 phi

We have sin(18°) but what about cos(18°)? This has a somewhat more awkward expression as:
cos(18°) =
Phi sqrt5

Now we know the sin and cos of both 30° and 18° we can find the sin and cos of their difference using:
cos(A–B) = cos(A)cos(B) + sin(A)sin(B)
and get:
cos(12°) =
phi +
3 Phi sqrt5

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:

cos(24°) =
Phi +
3 phi sqrt5

   cos(48°) =
phi +
3 Phi sqrt5

   cos(84°) =
Phi +
3 phi sqrt5


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

cos(60°) =
sin(30°) = 1

cos(30°) =
sin(60°) = sqrt3

and from a 45-45-90 degree triangle we can derive:
cos(45°) = sin(45°) = 1 = sqrt2


and not forgetting, of course:
sin(0°) = cos(90°) = 0
sin(90°) = cos(0°) = 1
Article: 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.

Can you find any more angles that have an exact expression (not necessarily involving Phi or phi)?

Let me know what you find and let's get a list of them here.
Here is my initial attempt - can you add any more angles?


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


use sine and cosine functions.

A Simple Trig formula for Fib(n)

Above we saw that cos(Pi/5) = cos(36°)=Phi/2 and cos(2 Pi/5) = cos(72°) = phi/2.
Since cos(180°-A)=–cos(A), we have cos(108°) or cos(180° - 72) = –phi/2. So
Phi = 2 cos( Pi/5 ) and –phi = 2 cos( 3Pi/5 )
We can use these in Binet's formula to rewrite it as
Fib(n) = 2n [ cosn(Pi/5) – cosn(3Pi/5) ]
Article: 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:
sin(pi) sin(3pi) = sqrt5     and    sin(3pi) sin(9pi) = – sqrt5






and so Binet's formula becomes this time:
Fib(n) = 2n+2(  cosn(pi) sin(pi) sin(3pi)  +  cosn(3pi) sin(3pi) sin(9pi)  )







or, if you prefer degrees rather than radians:
Fib(n) = 2n+2 ( cosn(36°) sin( 36°) sin(108°)  +  cosn(108°) sin(108°) sin(–36°) )


Article: 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 2pi 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

cos( x ) = 1 – x2 + x4 – x6 + ...




sin( x ) = x – x3 + x5 – x7 + ...



Here, n! means the factorial of n which means the product of all the whole numbers from 1 to n.
For example, 4! means 1x2x3x4 which is 24.

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

phi =
2 sin(pi)

2 (pi 
 – pi3
 + pi5
 – pi7
 + ...)




pi – pi3
 + pi5
 – pi7
 + ...





Phi =
2 cos( pi

2 ( 1 – pi2
 + pi4
 – pi6
 + pi8
 – ...)




2 – pi2
 + pi4
 – pi6
 + pi8
 – ...




In the upper formula, going to up to the pi9 term only will give phi to 9 decimal places whereas stopping at the pi8 term in the lower formula will give Phi 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 Phi and phi. To compute the next term from the previous one, multiply it by (pi/5)2 [or (pi/10)2 for phi] 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 Pi. Here is Pi 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 
With thanks to John R Goering for suggesting this connection between Phi and pi.

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Links and Bibliography

Book: The Changing Shape of Geometry : Celebrating a Century of Geometry and Geometry Teaching edited by Chris Pritchard, Cambridge, 2003, 557 pages is a collection of many articles from the Mathematical Association's Mathematical Gazette to commemorate 100 years of this intereting journal aimed at school teachers, lectureres and pupils aged 15-20. It has many articles on Geometry and a whole section on the Golden ratio.
Book: New Visual Perspectives on Fibonacci Numbers by K Atanassova, V Atanassova, A Shannon and J Turner, World Scientific (Oct 2002)
A book at the undergraduate level of mathematics that introduces the idea of two intertwined Fibonacci-type series (2-Fibonacci series), recurrence trees and Gray codes, and a new Fibonacci vector as well as John Turner's goldpoint geometry (well known from his papers and presentations at the International Fibonacci Conferences and printed volumes) and fractals and tilings. Have a look at the publisher's description and the chapter titles.
WWW: Steven Finch has an excellent page of Golden Section references at the MathSoft site.
WWW: Kyungsoon Jeon at the University of Georgia has an nice little article about Phi and the Fibonacci series and how to investigate it using a Spreadsheet.
WWW: Domingo Gómez shows a construction for the Isosceles-Fibonacci partition to construct points on any line AB which divide it into AB/2, AB/3, AB/5, AB/8, and so on, where the denominators are the Fibonacci numbers.

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