Constructions for the Golden Ratio

Phi and Rectangles

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 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.

So what was the shape of the isosceles triangle which had its top part removed to make the traphizium?

traphisium triangle 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
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.


/ Things to do /

  1. Here is another trapezium PQRS that is constructed using the other gold point on a the equal sides of an isosceles 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 isosceles 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?

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

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

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 √(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

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!

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