The simplest way to draw such a spiral is to start from its outer boundaries, contrary to the previous one. The golden spiral is a type of logarithmic spiral with a growth factor linked to the Golden Number. Such spirals, found in the growth of many organisms, are self-similar: the size of the spiral increases but its shape is not altered (for this it was also named spira mirabilis, the "miraculous spiral"). In contrast to the regular spirals above, the distance between successive turnings in logarithmic spirals grows in a geometric sequence. ![]() When these spirals are placed side-by-side, we can appreciate how much smoother and more perfectly circular they are when the base has a higher number of points. The critical part is drawing the bases and the extension of their sides very accurately. With a hexagon as base, the construction is really the same. Move to the second point, adjust the compass opening and draw the next quarter-circle. As the angle of the turnings becomes smaller (first it was 180º for each, then 120º, now 90º), the spiral becomes smoother. Our base is now a square, and we are still working clockwise. Move to the next point, adjust the opening and draw the next arc.Īfter a few turnings, the spiral looks like this: If the sides are extended as shown here, the spiral turns clockwise (and the compass moves from point to point in a clockwise direction). The compass will be moving from point 1 to 2 to 3 then back to 1, and so on. The method is the same but we start with an equilateral triangle, the sides of which are extended. Move the compass back to the first point, open it to meet the end of the curve, and draw another semicircle.Ĭontinue in this vein, moving the compass from one of the construction points to the other and adjusting the opening each time to take up the curves where you left off.Ĭarry on as much as desired. The two semicircles make a continuous curve. Place the compass on one of the points, open it to meet the other, and draw a semicircle on the other side of the line. This is the first turning of the spiral, and the two points where it cuts the line are the construction points. On a horizontal line, draw a semicircle that's as small as possible. The more points, the tighter and more perfect the spiral, but as that also makes construction more tedious, a hexagon is the highest one usually goes. It is drawn by moving the compass point from one point to the other in a base figure that can be a segment (two points), a triangle, a square, etc. This spiral is defined by an equal distance between turnings, so that it has a concentric appearance. Some can be defined using a mathematical equation, which translates, for specific spirals, into easy geometric constructions-approximate, but quite good enough for the eye. The distance between turnings, and the angle of each turning, determines their appearance. In this lesson we are using circles for their own sake, namely in two types of constructions: spirals and inscribed circles. ![]()
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