John
Sharp20 The Glebe,Watford Herts WD2 6LR, England PART 2: SPIRALS FROM THE GOLDEN
RECTANGLE
Figure 8 You can, of course, do the reverse, and subtract. Each time, if you keep the same orientation of the figure, you need to rotate the drawing. This brings home the rotation and magnification (or dilation if you are subtracting). This is the so-called 'whirling squares' so named by the art historian Jay Hambidge [1926]. Figure 8 was drawn with a true Golden Section rectangle. You can draw a similar figure if you use a rectangle with sides in the ratio of two successive terms of the Fibonacci sequence. The rectangles and squares then have integral sides that are Fibonacci numbers. The most well-known Golden Section spiral is drawn from Figure 6, using arcs of circles. A quarter circle is drawn in each square so that the line joining the centres go though the touching point to give a smooth curve (Figure 9). Figure 9 The pole for the spiral is found by drawing the diagonals of the Golden Section rectangles. Figure 10 This allows us to unravel the mathematics of this spiral, but before doing that I would like to show that it is possible to use this technique for creating other spirals and designs using this set of whirling squares.
Figure 11 and then rotated by 45° on its own gives a smooth curve because of the way the arcs flow: Figure 12
Figure 13 Using the technique shown in Figures 12 and 13, the arcs no longer appear to be formed from quarter circles. Figure 14
Figure 15 The full drawing, with the three-quarter circles drawn for each square, then becomes: Figure 16 I called this the "wobbly" spiral, since it appears to wobble back and forth. Without the construction squares it looks like this. Figure 17
Figure 18
The two types of isosceles triangles having the Golden Section ratio of sides then become LLS and SSL. (An LLS triangle has angles of 72, 72, and 36 degrees; and an SSL triangle of 36, 36, and 108 degrees.) Dividing a side L in each of them in the Golden Section creates more isosceles triangles. Figure 19 With the SSL triangle at the top of Figure 19, each division produces a LLS and another SSL which when subdivided in the same way creates the spiral of triangles. Similarly the LLS triangle below it when subdivided creates an SSL and another LLS and so on. It is easier to draw the spiral using arcs of circles with the LLS triangle (Figure 20). They are not quarter circles as with the Golden Section rectangle but arcs subtending 108°. The centre and ends of the arcs are clearly defined, with the centre as the division point on one L side and the ends on the opposite L side. Figure 20 The equivalent arc-spiral on the SSL triangle looks like this: Figure 21 At first glance, the arcs seem to be drawn with centre on the point of division on the L side and ends at the endpoints of the S side. But the rule shown in figure 6 does not apply if this is the case because the line joining the centres does not pass though the joining point of the two arcs. If you try to draw a spiral this way, then the spiral looks odd because it is not smooth. Figure 22 In order to draw the correct spiral, the arcs must be drawn with centre at the 'centroid' (centre of mass) of the corresponding triangle, which is the point of intersection of the angle bisectors of the three angles (actually, just two angle bisectors will suffice); the ends of the arc are at the endpoints of the L side. Figure 23 The arcs in this case subtend an angle of 144°. Because it is a more complex diagram, it is well worth drawing and studying how the centres and ends of the arcs are related and how the bisectors come together.
Figure 24 Although you can see a set of spiralling triangles, there is not the symmetry of the standard Golden Section triangles, and thus creating a spiral using circular arcs is not possible.
Figure 25 A pair of spirals can be drawn using arcs of circles. The arcs used to create the spiral are drawn as follows. Consider the line on which an arc is drawn as the side of an equilateral triangle. Then the arc is part of the circumcircle of that triangle. Figure 26 Because triangles tile the plane very easily, it can be adapted to use other triangles, for example with Golden Section isosceles triangles it looks like this: Figure 27
Figure 28 There are a number of spirals that can be drawn in this way. The set of ten spirals is very much like natural forms composed of Golden Section spirals as you would see in the seeds of a sunflower, for example. Figure 29
top of
page |
NNJ HomepageNNJ Winter 2002 Index About the AuthorComment on this articleOrder
books!Research
ArticlesThe
Geometer's AngleDidacticsBook
ReviewsConference and Exhibit ReportsThe Virtual LibrarySubmission GuidelinesTop
of Page |