THE TALE OF THE SUNFLOWER CLOCK

Transcription

1 THE TALE OF THE SUNFLOWER CLOCK If you like stories about forgotten empires you certainly heard about the Mali Empire and the hidden city of Timbuktu. You also probably heard some of the stories brought to Europe by Leo Africanus, but you might not have heard of the Tale of the Sunflower Clock which was written by Leo Africanus in a letter to Pope Leo X. You see, between XIII and XV century Timbuktu was quite a lively place. Many merchants would come to Timbuktu as this was a very rich city and the commerce of salt and gold was flourishing. So you might not be surprised that in the middle of the city there is a big clock that helped with organizing the activities in the city. The boundary of the clock consisted of a big golden circle. Supported on a smaller concentric circle was a boat in the shape of a triangle. Inside the boat there was a mirror shaped like a parabola and painted to look like a sunflower. Timbuktu is a very dry place, few plants grow around there. It is said that long ago a merchant brought a Sunflower with him. The clock maker was fascinated, and spent hours looking at how the flower turned after the sun. In the end he decided to build this clock. From the middle of the Sunflower mirror starts the hand of the clock pointing the hour. At the end of the hand there is a tube filled with water. The clock is made in such a way that the mirror reflects all the sunlight into one point on the water tube. The tube becomes very hot and the steam pushes the handle of the clock together with the Sunflower mirror and the boat. The bright point on the water tube where all the light was reflected could be seen from miles away showing the hour of the day. Rain in Timbuktu was very rare but when it happened it could rain heavily for a long time and it was a time for joy. All the activities where interrupted and people would dance and sing. It is said that the clock maker really understood the soul of the Mali people because the only time when the clock would stop its hand was when the clouds and the rain were coming. They used to say that when joy and happiness start the time is at its end. Leo Africanus loved so much that clock that he stole the sketches used by the clock maker to build the clock and brought them to Morocco. Later he reproduced them for Pope Leo X. They are still kept in the Vatican library and because I am not good at drawing I will write their content in words and present them as geometry problems. The sketches number 5 and 9 are related to another legend. We don t know the name of the clock maker but people use to call him the light-tamer. It was said that he kept light in a triangular box made out of mirrors and he could release it in whatever direction he wanted. Some of the concepts and constructions in these drawings were rediscovered later in Europe. Whenever this is the case I will mention also the European names associated with them. 1. Part I : The story of the light ray in the mirror The story of the light ray in the mirror can be told because the light ray is very lazy, so whenever it has to get somewhere it always takes the shortest path, to do less effort. This is true even if before reaching its true destination it has to visit some mirrors. So, if the light ray leaves a candle and is reflected in a mirror before getting in your eye it has to find the shortest path from the candle to your eye that is also getting to the mirror in its way. 1

2 2 THE TALE OF THE SUNFLOWER CLOCK 1. a) Show that the angle made by the ray of light with the mirror when it comes to the mirror is the same with angled formed when it leaves the mirror. Hint: The ray of light goes straight toward the reflection of your eye in the mirror. Why? b) Let M 1 denote the point on the mirror closest to the candle and M 2 the point on the mirror closest to your eye. If M is the middle point between your eye and the source of light show that MM 1 MM 2. Hint: Let N be a point such that M 2 is the middle between N and your eye. Show that the distance from the candle to N is equal to the distance traveled by the ray of light reflected in the mirror to get from the candle to your eye. Also use a scaling transformation with center L and scaling factor 2. We now put the candle, which we will denote by L, between 3 mirrors. To simplify the problem we will assume that the 3 mirrors are 1 dimensional and they touch each other forming a triangle. 2. There is exactly one point P inside the triangle such that there exist 3 rays of light that leave B reflect in each of three mirrors and get to P at the same time. If we put a candle at the point P there are also three light rays departing from P and reaching the initial candle at the same time. We will say that L and P are izogonal points. Helping problem: We will consider a similar problem when only two mirrors are considered. 3. Let m 1 and m 2 denote the 2 mirrors and A denote the point where they meet. The point L denote the candle and P is a point which is reached simultaneously by 2 rays one reflected in m 1 and one in m 2. The angle formed by AL with m 1 is equal with the angle formed by AP and m 2. Moreover the 4 points on the mirrors which are closest to L and P are all situated on a circle with the center at the middle point of the segment LP. Hint: Look from P at the reflection of L in m 1 and m A special case of the problem 2 consists when the point P coincides with the candle. There are 4 such points which are center of circles tangents to the supporting lines of the triangle. They are situated of intersection of lines bisecting the interior and exterior angles of the triangle. We now assume that the triangle formed by the three mirrors is acute and the vertexes of the triangle are denoted by A,B and C. 5.a)Wherever the candle is inside the acute triangle, there are rays leaving the candle that come back to it after reflected by all three mirrors. (Hint: From the candle you look at the reflection of the reflection of the reflection of the candle.) b)show that you position the candle in such a way that there is a ray of light that gets back to the candle after reflecting in all three mirrors and then continues on the same path again and again. The triangle on the sides of which this ray of light travels is called the ortic triangle. Show that between all the triangles inscribed in a circle the ortic triangle has the smallest perimeter. Hint: The candle has to be positioned on one of the sides of the triangle having as vertexes the projections from A, B and C to the opposite sides of the triangle ABC. We will denote the projections from A, B and C by M, N and P. Check that MA, NB, CP, AB, BC,CA bisect the interior and exterior angles of the triangle MNP. From the previous problem we can conclude: 6. For any triangle ABC the lines passing through the A,B and C and perpendicular on the opposite sides are all passing through the same point H. H is called the orthocenter of the triangle. Show that H and the center O of the circle passing through A, B and C are izogonal points. There

3 THE TALE OF THE SUNFLOWER CLOCK 3 is a circle that contains the middle points of the sides of the triangle ABC and the vertexes of the ortic triangle. This circle is called Euler s circle and its center is the middle of the segment OH. The line OH is called the Euler s line and contains also the weight center of the triangle ABC. In the next sketch the source of light is the sun. As the sun is very far from us, we will consider the rays of sun parallel to each other. This problem could be seen as a limit case of the Problem a) We consider a triangle formed by 3 1-dimensional mirrors as before, which is situated in the same plane as the sun all day long. For each moment during the day there exists a unique point where photons that leave simultaneously the sun may meet after reflected in the mirrors. As the sun moves from east to west on the sky this point moves on the circle passing through the vertexes of the triangle constructed from the 3 mirrors. b) Let P be a point on the circle passing through the vertexes of the triangle ABC. The points M, N and Q situated on the lines AB, BC, respectively CA which are closest to P are all on the same line known as Simpson s line. 8. Let A,B,C and D be four points in plane and E = AB CD and F = AD BC. The circles circumscribed to the triangles ABF, BCE, CDF and ADE have all a point in common M, called Miquel s point. The points on the lines AB BC, CD, DA which are closest to M are situated on the same line. 9.The triangle ABC constructed by the 3 mirrors is now equilateral. A candle is placed inside the triangle. Any ray of light that departs from the candle on a direction parallel with the sides of the triangle will come back to the candle after it is reflected twice by each mirror and it will continue on the same path after that. 2. Part II :Looking at the circle In the previous section in proof of Problem 2, and Problem 3 we introduced a circle which seem a bit disconnected to the reflection of the light in the mirror. This was due to the fact that the natural curve associated to the reflection in the mirror, the conic, was yet to be discovered. The circle from Problem 2 is just an approximation of such a conic. What we mean be approximating a conic by a circle will become clear in Problem 3. The conic is plane curve for which are seen as a circle when looked at from certain points of the space. In other words a conic is a plane section through the cone over a circle. Looking from the vertex of the cone the circle and any other section of the cone is seen as a circle. A cone over a circle is the surface obtained by joining by lines all the points of a circle to a point P outside the plane of the circle. Any cone C can be written as union of 2 surface C 1 and C 2 where C 1 is obtained by joining by half-lines all the points of the circle with P. C 2 = (C \ C 1 ) P There are three types of conics: An ellipse is conic obtained by intersecting the cone C with a plane H such that H intersects all the half-lines contained in C 1. A parabola is a conic obtained intersecting C with a plane parallel to one of the lines contained in C. A hyperbola is a conic obtained by intersecting C with a plane that intersects both C 1 and C 2 and doesn t contain P.

4 4 THE TALE OF THE SUNFLOWER CLOCK For the problems that follows we will consider the cone C such that the perpendicular from P to the plane of the circle intersects this plane in the center of the circle. 1. We say that a sphere S is tangent to a cone if the intersection of the cone with the sphere is a circle. Show that for any plane H passing through P, H C is a pair of lines tangent to the circle S H. Let H be a plane such that H C is not a parabola. Show that there exists 2 spheres tangent to C and H. If H C is a parabola there is only one such sphere. Hint:Start be constructing a sphere tangent to the cone and then expend it or shrink it till becomes tangent to H. 2. Let S and S the 2 spheres tangent both to C and to a plane H. Let F and F be the the tangent points of S and S with H. F and F are called the focal points of the conic. Let C and C the 2 circles of tangency of S and S with C. Show that the length of all segments contained in C which have their ends on C and C is the same. We will denote this length by d. a) Let H C be an ellipse. Show that for any point M H C d(mf ) + d(mf ) = d. b) Let H C be a hyperbola. Show that for any point M H C d(mf ) d(mf ) = d. c) Let H C be a parabola.in this case only one of the 2 spheres exists, let s call it S. Let F = S H. Let l be the line contained in C parallel with H, and let l be a line in H perpendicular on l. Show that for any point M on the parabola the sum d(m, F ) + d(m, l) is independent of the position of M on the parabola. Notice that Problem 2 provide alternative definition to ellipse, parabola and hyperbola. A line is tangent to a conic C H if and only if is viewed from the vertex of the cone as the tangent line of a circle. In particular a line is tangent to an ellipse if and only if it intersects the ellipse in only one point. 3. Show that with the notations from Problem 2 from Part I there exists an ellipse E with focal points P and L which is tangent to all three mirrors. The tangency points, are the points where the light coming from L and going to P reaches the mirrors. Recall that the 6 points on the mirrors which are closest to L and P are all situated on a circle C with the center at the middle point of the segment LP. The segment having as end points E LP is a diameter of C. 4.The last sketch presents how the flower-clock works: A parabola mirror was situated on a triangular boat which we call ABC, such that the parabola was tangent to all the supporting lines of the triangle AB,BC,CA,. However there is a twist in the story: If this would have been the whole construction the quadrant of the clock should have been divided in 24 hours. For it to be divided in 12 hours as it was the hand of the clock had to rotate twice as fast as the sun. As we said in the beginning of the story the rays of light where warming a tube with water. What the clock maker did was to use the steam coming from the hot water to move the sides of the triangle forming the boat such that such that they still meet on the same circle and still be tangent to the same parabola. However the point where the light is focused can move now at a different speed than the sun. As the sketch shows, if one of the vertexes of the triangle ABC moves to a new point A on the circumscribed circle of ABC, then if B and C are on the same circle and A B, and A C are tangent to the parabola then B C is also tangent to the parabola. However no reason for this fact was suggested. The earliest explanation given may be found hundreds of years later in the sketches of a frenchmen, Poncelet, but this is another story and will be told another time.

5 THE TALE OF THE SUNFLOWER CLOCK 5 3. Epilog I told this story to many people and some pointed out that if you look through the Leo Africanus writings or if you search in Vatican library there is no mention of the Sunflower clock. They also said that they started to doubt that the story is true. But how could it not be true if the description of the sketches presented above is still with us?