And this rectangle then has a length one plus root five over two and a width one, this is a golden rectangle. And then we draw the top line over using a straight edge. Or parallel to this line what you can also do in classical construction, so we draw the line here up. We need to draw a perpendicular line to this bottom line, either perpendicular to this line, which you can do in classical construction. All we have to do now is complete the square. And then this total length, one-half plus root five over two is one plus root five over two, which is the golden ratio, right? So now we've drawn a bottom line that has the length of the golden ratio. And now what do we get? Well, this means here the square root of five over two, and this red line at the bottom here is also a square root of five over two because it's the radius, right? It's the radius. Okay, and then we can take our square side and use our straight edge to extend the side, all the way to the arc, okay all the way to the arc. So we can use the compass to draw this circle, arc of the circle. We can make this line segment the radius, okay? Make that the radius and then draw the arc. At the midpoint, at the bottom side of the square. The golden ratio is the square root of five plus one over two, okay? So what do we do next? Well, we can put our compass point here. So you see the reason of drawing this line segment is to introduce the square root of five. So one squared plus a half squared is one plus a quarter is five quarters, and we take the square root to get the hypotenuse, so we have a square root of five over two. Now we have Pythagorean's theorem, so we have a right triangle with a side of 1 and 1/2. And then you can use your straight edge to draw the line to the opposite corner here, okay. So we assume that you can find he midpoint. There's a classical construction using the compass that allows you to find the midpoint of any line segment. So first, you have to find the midpoint of this bottom side of the square. Then the next step is to draw this red line here. We just called the size of unit left so we put a 1 on the sides. So I won't go into that here because our goal is really the golden rectangle. So by classical construction using a straight edge of a compass, you can construct a square. Okay, so how do we construct a golden rectangle? We start by constructing a square. But nevertheless, it's kind of a fun thing to do particularly for students secondary school students, even university students. Not so popular these days because we have computers and we can just program the computer to draw anything we want. And the idea of classical construction was also very popular during the renaissance and after. So a lot of these common figures can be drawn using classical construction. Okay, fix the radius and you can draw an arch or a circle. A compass is that device that has a point that you can draw a circle off of this point. The idea of a classical construction is to construct this figure using just a straight edge which is like a ruler without the markings and a compass. So classical constructions of these planar geometrical figures goes back to the ancient Greeks. So the length, the longest side of the rectangle, divided by the width, the shortest side of the rectangle Is equal to capital Pi one plus the square root of five over two. What's a golden rectangle? A golden rectangle is a rectangle that has sides in the ratio of the golden ratio. In this lecture, I want to show you how to do a classical construction of a golden rectangle. Bat Sheva consults to the Ministry of Education and to organisations that develop books, games and teaching materials.Isaac Nativ studied mathematics and mathematics education at Monash University as well as at Melbourne University.Welcome back. Bat-Sheva has been teaching at various universities and teacher training colleges, and has supervised many PhD and MSc students. She has published many books, papers and teaching material for primary, secondary and tertiary levels. He conducted original research on this topic and wrote extensively about it for students at various levels.Dr Bat-Sheva Ilany is a distinguished mathematics educator. Opher was particularly interested in the Fibonacci sequences. Opher organised and convened several mathematics conferences and wrote numerous books and papers for both teachers and students of mathematics. Biography: Opher Liba The late Opher Liba was a distinguished teacher of mathematics at all levels.Publisher: Springer Nature Switzerland AG.Imprint: Springer Nature Switzerland AG.
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