| The regular pentagon (the shape with just 5 straight sides of the same length and 5 corners of the same angle) is related to the golden mean. If you add a straight line to a regular pentagon joining 2 corners which are not already connected by one side the new line has a length phi times the length of the sides. |
| This can be shown as follows: |
| Imagine walking along the outline of this pentagon: at each corner you would have to turn through the angle marked between the red line and the next side; when you reached your starting point you would have turned 5 corners and be pointing the way you started. All the corners of a regular pentagon are the same so each angle must be 1/5 of a circle |
| The other angle outside of the pentagon at the corners is a 1/2 of a circle, which, added to the 1/5, makes 7/10 of a circle outside of each corner leaving 3/10 inside at each. The triangle which is made by two adjoining sides and one of the long green lines has one 3/10 corner of the pentagon and two pointier corners. The pointier corners must be equal to one another because the blue sides are equal. It is easy to show that the angles inside any triangle must add up to 1/2 a circle (Link) so the pointier corners must together be 1/5 so they are a 1/10 each. |
| Here are some pentagons with some lines and angles drawn in. You can see that the smaller triangle shown yellow above has the same angles (3/10, 1/10 & 1/10) as the larger triangle shown yellow below, and that the pink triangle has two corners the same (1/5) and so two sides the same. |
| Triangles which have all three angles the same are the same shape so their sides have the same proportion to one another. Now, take a deep breath... This means that the proportion of the side of the pentagon to the long line in the big yellow triangle, L (for line) for short, is the same as the proportion of the short side of the small yellow triangle, S (for short) for short, to the side of the pentagon (its' long side is one of the sides of the pentagon). L (the line joining two corners of the pentagon not joined by the sides) can be thought of as the side of the pink triangle (which is equal to the side of the pentagon) plus S. Putting these together gives; the proportion of S to the side of the pentagon, is equal to, the proportion of the side of the pentagon to S plus the side of the pentagon If we say the pentagon has a side of length 1, we can say 1/S=(S+1)/1 or 1/S=L but L=S+1 so L=1/(L-1) or L^2-L=1 or L+1=L^2 That is L squared is equal to L plus 1. The proportion which has this property is one plus the square root of five to two which is otherwise called phi or the golden mean. Back to sacred geometry |
