On any straight line S, pick two points X and Y.Hofstetter's 3 Constructions of Gold Points Using only circles Kurt Hofstetter has found a beautifully simple construction for a line and its golden section point using only compasses to draw 4 circles: ![]() ![]() Since BT is now also 1, how long is MT? This is the same length as MG so you can now find out how long AG is since AG=AM+MG. Let AB have length 1 again and so AM=MB=1/2. ![]() If you followed the reasoning for why the first construction (for phi) worked, you should find it quite easy to prove that AG is Phi times the length of AB, that is, that AG = (√5/2 + 1/2) times AB. This point is now divides the original line AB into two parts, where the longer part AG is phi (0♶1803.) times as long as the original line AB. Finally, put the compass point at point A, open it out to the new point just found on the diagonal and mark a point the same distance along the original line.Putting the compass point at the top point of the triangle and opening it out to point B (so it has a radius along the right-angle line) mark out a point on the diagonal which will also be half as long as the original line. Join the point just found to the other end of the original line (A) to make a triangle.Now you have a new line at B at right angles to the AB and BC is half as long as the original line AB. Put your compasses on B, open them to the mid-point of AB and draw an arc to find the point on your new line which is half as long as AB.So first draw a line at right angles to AB at end B. This is where you use the set-square (but you CAN do this just using your compasses too - how?). Now we are going to draw a line half the length of AB at point B, but at right-angles to the original line.The two points where thesemicircles cross can then be joined and this new line will cross AB at its mid point. Repeat this at the otherend of the line without altering the compass size. To do this without a ruler,put your compasses on one end, open them out to be somewhere near the other endof the line and draw a semicircle over the line AB. Here's how to construct point G using set-square and compasses only: times the size of the longer segment AGtoo. This will also meanthat the smaller segment GB is 0♶1803. We want to find a point G between A and B so that AG:AB = phi (0♶1803.)by which we mean that G is phi of the way along the line. (In fact we can do it with just the compasses, but how todo it without the set-square is left as an exercise for you.) ![]() We can do this using compasses for drawing circles and a set-square for drawing lines at right-angles to other lines, and we don't need a ruler at all for measuring lengths! Constructing the internal golden section points: phi If we have a line with end-points A and B, how can we find the point which dividesit at the golden section point? John Turner has nicknamed the two pointsthat divide a line at the golden ratio (0.618 of the way from either end) as gold points. Constructions for the Golden RatioLet's start by showing how to construct the golden section points on any line: first a line phi (0♶18.) times as long as the original and then a line Phi (1♶18.) times as long. Contents of this page The icon means there is a You do the maths.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |