Constructions for the Golden Ratio

Phi and other Polygons


decagon 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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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 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.

Sharp Triangle dissection
Dissecting the Sharp and Flat Triangles:

Here the sharp triangle is dissected into two smaller sharp triangles and one flat triangle, the flat triangle into one smaller flat and one sharp triangle. At each stage all the triangles are dissected according to this pattern.
Repeating gives rise to one version of a Penrose Tiling.
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 the 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.
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.

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.