Cosmology – Construction Assignment 3 –

Due Wednesday, October 25.

As a preliminary to this exercise look again at Chapter 5 in Schneider.

1. In this exercise we will cut a segment in such a way that we will produce the golden ratio between two segments.

a) First, with the paper placed horizontally draw a horizontal straight line about 5 inches long a little below the center of the paper. Call the endpoints A and B with A on the left.

b) Using the compass construct circles with the same radius and with centers A and B. Let the radius be about 3.5 inches. The two circles will intersect at two points, one above the line and one below the line. Call these points D and E with D above the original line.

c) Connect D and E with a straight line. The line intersects the original line in a point which is the midpoint of AB. Call that point C.

d) Draw circles with centers at A and at D with radii AC and DC respectively. These circles will intersect above A at a point. Call that point F.

e) ACDF should be a square. If it isn’t try again.

f) Connect F and B with a straight line.

g) Construct a circle with center B and radius BC. It intersects FB in a point Call that point G.

The ratio FG:AB is the Golden Ratio

2. We will now use the segments of Exercise 1 to construct a regular pentagon.

a) With a piece of paper placed vertically draw a straight line of length FG from the first exercise a bit below the middle of the page. Call these points U and V. Now from Centers U and V draw circles of radius AB (again from Exercise 1). These circles will meet at a point above the line UV. Call this point X. Now draw circles from centers U and X of radius FG. These will meet at a point. Call it Y. Repeat this at points X and V and call the point of intersection W. Connect U and Y, X and Y, V and W, and X and W. The five sided figure UVWXY is a regular pentagon (If you have been absolutely accurate in your drawing.)

3. There is an alternate form of this construction given on pages 105-106 of Schneider. Perform that construction.

4. If we denote the Golden Ratio by , then we know that or . In this construction we will test this equality.

a) With the paper horizontal draw a line about 6 inches long in the lower half of the page. From the left end of this line draw another line at an angle of about 30° .

b) Mark from the left end of the horizontal line a segment whose length is AB from Exercise 1. Mark off segments on both line starting from the left endpoint of length FG from Exercise 1. Let A be the vertex of the angle and let B be the other end of the segment of length AB. Let C and D be the end points of the other two segments (those of length FG) with C between A and B.

c) We will now construct a line through point C that is parallel to the line BD. Draw circles of the same radius of about 1.5 inches with centers B and C. The circle centered at B will cut the lines AB and BD at two points (Call them X and Y). It will cut AB at a point. (Call it Z) Use the compass to measure the distance between the two points X and Y. Now draw a circle with that measurement as radius with center Z. Draw a circle with center Z and radius XY. This circle will intersect the circle with center at C at a point. Call it T. We are almost done. Draw line CT. It intersects AD at a point we will call S. Then AS has length ².

d) Using the compass check that AS and AC together are the same length as AB. Since we are taking AB to be 1 unit and AC to be this should illustrate that .