Relevant artwork by M.C.Escher
Carved Spheres and spherical tessellations, pg 244-247 Shells and Starfish (Verblifa candy tin), pg 295 Lizard/Fish/Bat (carved ivory sphere and related studies), pg 307 Sphere Surface with Fish on Pg 322 Sphere with Angels and Devils on Pg 41 Sphere with Fish on Pg 97 Sphere with Fish on Pg 90-91 Sphere with Angels and Devils on Pg 92 Planetoids, pg 94-97 Solids, pg 98-99 Stars, pg 100 Gravitation, pg 101 Crystal, pg 102 Order & Chaos, pg 103 Waterfall (detail), pg 114 Concentric Rinds, pg 165 Reptiles (detail), pg 175 Sphere spirals, pg 178
So far we have looked at what is commonly called Euclidean geometry. There are occasions where this type of geometry doesn’t get one very far. Suppose we look at this sphere and want to measure the distance between the centers of two 5-pointed stars. You can’t just use a ruler, because you can’t put the ruler flat on the sphere to measure the length. If measuring length is already tricky, how would you find area?
Geometry was initially developed to measure things: length, area and volume. These things were used all the way back in ancient Egyptian and Babylonian times to measure the level of the Nile, to build temples, to construct the pyramids, measure how much land you had to compute taxes, etc. To make everything work we need lines, line segments, circles, angles and an understanding of parallel lines. These five things are the topic of the so called axioms of geometry. Also called the postulates of geometry. We have already seen that lines are a bit tricky. What do lines and segments do for us? The answer is that they measure the shortest distance between two points. Have you ever seen how builders construct straight lines in the field? The put two stakes in the ground and pull a piece of string taut between them. So if you wanted to measure the distance between two points on a sphere you could do the same thing: put two stakes on the sphere and pull a piece of string taut between them. If you do this you are constructing great circles on a sphere. Great circles look like the equator or longitudes on the globe. We sometimes call these great circles geodesics. Technically geodesics are distance minimizing on the sphere. They play the role of straight lines on the sphere. We will use these geodesic to create spherical triangles and other spherical polygons.
Here you see a sphere with three geodesics (solid great circles). One is the equator, and the other two cross the equator and each other at 90 degree angles. The dotted lines are two more geodesics meeting the equator at a 90 degree angle. (The continuation of the equator on the back of the sphere is also drawn as a dotted curve.) If you take the intersections point on the left, on the top, and on the front of the sphere and look at the geodesic segments that connect them, you will see a spherical triangle. Spherical triangles can behave in very strange ways. This is a 90-90-90 equilateral triangle. Such a triangle ONLY exists on the sphere! There is no way you could draw a 90-90-90 triangle on a piece of paper. Remember how the sum of the angles of a EUCLIDEAN triangle always has to be 180? Well, here you see an example of a spherical triangle with angle sum 270. The most drastic difference between Euclidean and spherical geometry however lies in the parallel lines. You have all seen parallel lines in Euclidean geometry (any straight piece of train track is a good example). Remember that two lines are parallel if they never meet. It is easy to convince yourself though that on the sphere any two great circles will intersect in two points. This means that one the sphere is not possible to draw parallel lines. They simply do NOT exist. This causes all kinds of problems for the theory of polygons. One obvious problem is that parallelograms cannot exist one the sphere. After all, if there are no parallel lines then you cannot construct a 4 sided polygon with the property that opposite sides are parallel. The previous comments show that we have to throw the theory we developed for Euclidean geometry out the window, and start anew. You will see though that working your way through the rules and theorems of geometry a second time is somewhat easier after you have done it once before.
Famous Early Geometers
Pythagoras (ca. 540 BC) Showed that in a right triangle the sum of the squares of the sides equals the square of the hypotenuse. Plato (ca 380 BC) Laid the basis for formal geometry. His name is associated with the Platonic solids. Above the entrance to his school of Philosophy (the Academy) was engraved : “Let no one ignorant of geometry enter my doors” Aristotle (ca 340 BC) The tutor of Alexander the Great, also trained many of the great geometers of the time. Euclid (ca 300 BC) The first to write down the postulates for what is now known as Euclidean geometry. He was associated with the famous School of Alexandria. Archimedes (ca 225 BC) Pliny called him “the God of Mathematics”. He was also associated with the School of Alexandria. His name is now associated with the Archimedean solids. He was killed during the Siege of Syracuse. He was so immersed in his math that he supposedly did not notice the city being taken over by the Romans. Eratosthenes (276-194 BC) Realized that the earth was round, and was able to compute a fair approximation of its circumference. Plutarch : “God eternally geometrizes” The geometry taught in elementary, middle, and high-school is all Euclidean geometry. As you can see above, many of the results we learn about have been known for over 2000 years. Mathematicians have investigated properties of spherical triangles for almost 2000 years, but no systematic theory for geometries other than Euclidean geometry were developed until the 18th and 19th centuries. Contributors include Lambert, Saccheri, Lobachevsky, Bolyai, Gauss, and Riemann. At the end of the 19th century it was recognized that there were different systems of geometry. We will investigate spherical geometry (the geometry of the sphere) and hyperbolic geometry (as illustrated by Escher’s prints Circle Limit I-IV).