Though a full discussion of phi could go on for many pages yet, I shall wrap up this thread with a look at the use of the golden ratio in developing curves. Before I do that, I wanted to bring up a very useful traditional design tool known as the vesica piscis. All you will need is a compass and straightedge.
Draw a line, and then place the compass on the line and draw a circle:
Keeping the compass set at the same distance, place the point on the intersection of the first circle and the line. Draw a second circle, overlapping the first:
Highlighted in red is the overlap between the two circles of identical size - this is termed the vesica pisces (Latin for, "bladder of the fish"):
The vesica pisces has long been a symbol in Christian iconography, present in much artwork and, in regards to the fish motif, is a symbol sometimes seen on the back of some believer's cars, termed in that case 'Ichthys'.
Immediately it is apparent that the vesica piscis, as a form, is the origin of the Gothic pointed arch:This arch, compared to a semi-circular arch, reduces the horizontal load displacement, thus allowing for lighter construction. Lighter construction is of course less expensive and more efficient with materials, but there was another aspect to lighter as well: the pointed arch allowed a greater amount of the suns rays to penetrate to interior spaces, and thus became the favored arch form, used also in vaulting, for European church building of the medieval period, 1200~1400:
Returning to the geometry, the vesica piscis is a form which generates many of the irrational number values used in design, mathematics, art, and building. We construct a rectangle having a base equal to the distance between our two circle centers, equal to our radius in other words. We'll set that distance as one. Making the long sides of our rectangle equal to the diameter of the circle, 2.0, then produces the diagonal with the first incommensurable number, √5:
Cut this rectangle in half, and construct a diagonal, and we have the second incommensurable number, √2:
Now divide the square in half, and construct a diagonal from the apex of the equilateral triangle present in the upper half of the vesica. This forms a right-angled triangle, with a hypotenuse equal to our circle radius of 1.0, a 'run' of 0.5, and a 'rise' of some unknown amount. While this can be readily calculated, it is a little easier to show the derivation of the value though a graphical approach. Project the 'rise' and hypotenuse of this triangle as shown:
Since we have doubled the 'rise' of this triangle, and kept the same angular relationships, the principle of similar triangles say that the hypotenuse and 'run' of our larger triangle must also have double the values of the originating triangle. A little computation now, using the Pythagorean method, will reveal that the length of the 'rise' of this triangle is equal to another incommensurable number, √3, or approximately 1.73205....:
We can convert these dimensions over with our compass and straightedge to produce the √3 rectangle, which may be used as a design tool in much the same manner as I showed for the Golden Rectangle in part IV of this thread:
Speaking of the Golden Number, it can be readily produced from the vesica piscis - here's one method already familiar to readers:
And the Golden Rectangle thus produced:
Here's another method to produce phi by division instead of addition, also familiar from previous postings in this thread:We'll return to phi in a short while, but first I thought I'd make a brief exploration of how the vesica piscis can generate design. First, we project lines from the equilateral triangle found within the vesica, both top and bottom:
Connect the dots formed, and voila!, you have the hexagon:
Many of the polygons can be readily constructed within the vesica piscis.
Repeating the previously illustrated process on both sides, we produce the following:
Using each of those connecting points as a center to place the compass upon, we draw six new circles, forming 7 altogther:This produces a pattern in the center known as the "Seed of Life", which serves as a component of The Flower of Life. Leonardo da Vinci made study of such forms in his work.
Returning to our Golden Rectangle (it helps to have a large sheet of paper, and draw this first rectangle quite small) now:
Take the long side of the rectangle, measuring ø, and swing a continuation of our arc downward:You can see that the new square formed has sides that are a multiple of phi longer than the original generating square. Repeat this process again:
And again. Each new square formed has a Golden relationship to the previous one: And a few more times to produce the Golden Spiral: This looks a bit like the form for the human ear, does it not? There are many natural forms in the shape of a spiral. Many of these are in the form of logarithmic spirals, of which the Golden Spiral is but one type.
The logarithmic spiral has also found plenty of application in architecture, like these stairs:
This concludes my look at phi, the Golden Number, 1.6180339887.... My visit to this topic has been a necessarily shallow one - there are many resources out there for further explorations of this topic, should a reader be interested. The development of a logarithmic spiral can be carried out from other rectangles as well, for example, the √3 rectangle described above. A worthwhile read in that regard is an article written by the geometer Rachel Fletcher, which explains the process in depth.
Designers and builders looking for inspiration can find it in abundance in the worlds of both nature and mathematics.