Spherical Geometry

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Relevant artwork by M.C.Escher (Spherical geometry and Platonic Solids)

  1. [Concentric Rinds] Color Print
  2. [Concentric Rinds] Black and White Print (Easier to see detail)
  3. [Double Planetoid]
  4. [Gravity]
  5. [Order and Chaos]
  6. [Sphere with Fish]
  7. [Sphere with Spiral]
  8. [Stars]
  9. [More Stars]
  10. Sphere with Angels and Devils (Magic of M.C. Escher)
  11. Sphere with Fish (Magic of M.C. Escher)
  12. Sphere with Angels and Devils (Magic of M.C. Escher)


Introduction

Sphere1.png

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.


Sphere2.png


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

  1. Pythagoras (ca. 540 BC) Showed that in a right triangle the sum of the squares of the sides equals the square of the hypotenuse.
  2. 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”
  3. Aristotle (ca 340 BC) The tutor of Alexander the Great, also trained many of the great geometers of the time.
  4. 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.
  5. 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.
  6. Eratosthenes (276-194 BC) Realized that the earth was round, and was able to compute a fair approximation of its circumference.
  7. 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).


Points and Lines

Spherical geometry is nearly as old as Euclidean geometry. In fact, the word geometry means “measurement of the Earth”, and the Earth is (more or less) a sphere. The ancient Greek geometers knew the Earth was spherical, and in c235BC Eratosthenes of Cyrene calculated the Earth’s circumference to within about 15%. Navigation motivated the study of spherical geometry, because even 2000 years ago the fact that the earth was curved had a noticeable effect on mapmaking. Even more importantly, the sky can be (and often was) thought of as a spherical shell enclosing the earth, with sun, moon, and stars dancing about on its surface. Navigation and timekeeping required a thorough understanding of how the heavenly bodies moved, and that required spherical geometry. In geometry there are undefined terms. There are also first principles “the truth of which” as Aristotle affirmed, “it is not possible to prove”. These first principles are called postulates. In Euclidean geometry we assume that we know what is meant by “point” and “line” – these are undefined terms. To do geometry on a sphere, we need to make sense of these terms. In spherical geometry, the “points” are points on the surface of the sphere. We are not concerned with the “inside” of the sphere. A soap bubble makes a good mental image. When thinking about the Earth, it’s helpful to realize that if you shrunk the Earth and dried off the oceans with a towel, the planet would be as smooth as a pool ball, and ones elevation off the surface would be too small to notice. Lines in spherical geometry are more subtle. Since the surface is curved, there are no straight lines on it, in the usual sense of the word straight. Because of this, we use the word geodesic instead of line when talking about spherical geometry: Definition A geodesic in non-Euclidean geometry plays the role that a straight line plays in Euclidean geometry. We expect geodesics in spherical geometry to behave like straight lines in Euclidean geometry. In particular, there are two essential features of a straight line in Euclidean geometry that we expect geodesics to have: • The shortest distance between two points is a straight line. • A point that moves without turning will follow a straight line.

Definition A great circle is a circle on a sphere which divides the sphere into two equal hemispheres. The picture below shows four great circles (all of which wrap around the “back” side of the sphere.


A point that moves along the surface of a sphere without otherwise turning will follow a great circle. The shortest distance between two points on a sphere also lies along a great circle. Because of this: Geodesics in spherical geometry are great circles. We will use these in place of straight lines when doing spherical geometry. Consider the statement “two points determine a line”. This is a postulate of Euclidean geometry, which means we accept its truth without proof. In spherical geometry, it is not quite true. Consider the Earth’s North and South poles. These points are joined by many great circles, which are known as meridians or lines of longitude. In fact, leaving the North pole in any direction and heading straight will take you to the South pole along a geodesic. The North and South pole are not the only points with this property.

Definition Two points which are opposite each other on the sphere are called antipodal points.

In spherical geometry, we can say “two points determine a geodesic, unless they are antipodal points, in which case there are infinitely many geodesics joining them”. This is less elegant than Euclidean geometry but fairly typical for spherical geometry, where there are often exceptions for antipodal points.

Definition A geodesic segment is an arc of a geodesic and its two endpoints.

When saying “two points determine a line”, one usually thinks of the line segment joining the two points. On a sphere, two points lying on a geodesic create two geodesic segments since the geodesics are circles. Unless the points are antipodal, there will be a short segment and a long segment which “goes around the back of the sphere”.

Angle Sum and Area

Definition A polygon in spherical geometry is a sequence of points and geodesic segments joining those points. The geodesic segments are called the sides of the polygon.

A triangle in spherical geometry is a polygon with three sides, a quadrilateral is a polygon with four sides, and so on, as in Euclidean geometry. One fundamental result of Euclidean geometry is that the sum of the angles in any triangle is 180°. To see this, we used properties of parallel lines. However, in spherical geometry there are no parallel lines, because any pair of geodesics intersect at two (antipodal) points. Instead, in spherical geometry we have: The sum of the angles in any spherical triangle is more than 180°. To justify this statement, take a spherical triangle and then draw a flat triangle with the same vertices, as in the figure. The flat triangle has angle sum 180°, and since the spherical triangle bulges out from the flat one, its angles must be larger.

Definition The defect of a spherical triangle is (angle sum of the triangle) - 180°. The more area a triangle covers, the more it bulges, the more its angles differ from a Euclidean triangle, and the larger its defect. There is a direct mathematical relationship between a triangle’s area and its defect. We measure the area as a fraction of the total area of the sphere:

fraction of sphere’s area covered = defect/720°

To find the actual area covered by a triangle, you need to know the radius R of the sphere and then use the fact that the total surface area of a sphere of radius R is 4πR^2.

File:Sphere3 Spherical 90°-90°-45° triangle

Example The triangle shown in the figure has two 90° angles and one 45° angle. Its angle sum is 90°+90°+45° = 225°, and its defect is 225° – 180° = 45°. It covers 45/720 = 1/16 of the sphere. Can you see how 16 of these triangles would cover the whole sphere?


We saw in a previous Chapter that a Euclidean polygon with n sides has angle sum (n – 2)×180°, by cutting the polygon into n–2 triangles. A spherical polygon with n sides can be cut in the same way into n–2 spherical triangles, each of which has angle sum more than 180°, and so the angle sum of a spherical n-gon is more than (n – 2)×180°. Put another way, the angle sum of a spherical polygon always exceeds the angle sum of a Euclidean polygon with the same number of sides. The amount (in degrees) of excess is called the defect of the polygon. The fraction of the sphere covered by a polygon is equal to its defect divided by 720°, just as for triangles.

Spherical Tessellations And Polyhedra

A tessellation of the sphere is a covering of the sphere by tiles, with no overlapping tiles and no gaps. We focus exclusively on tessellations by tiles which are polygons. As a first step, we look for regular tessellations. Recall that a regular polygon is a polygon with all sides the same length and all angles equal. We keep the same definition in non-Euclidean geometry. In Euclidean geometry, the angle sum for a polygon with n sides is (n – 2)×180°, and this forces the corner angles of a regular n-gon to be (n – 2)×180°n . This means there is only one shape of Euclidean regular n-gon. In spherical geometry, however, there are many regular n-gons. There is a regular n-gon with any angle sum larger than (n – 2)×180° (up to a maximum size). So, there is a regular n-gon with any choice of corner angle larger than (n – 2)×180°n (again, up to some maximum size). The maximum sizes aren’t as important, and are left for the exercises. This table summarizes the corner angles of some regular polygons on the sphere:

Name Number of Sides Corner Angle
biangle
2 >0
triangle
3 >60°
quadrilateral
4 >90°
pentagon
5 >108°
hexagon
6 >120°
heptagon
7 >128.57...°
octagon
8 >135°


To make a regular tessellation of the sphere, we need to pick one regular polygon and use it to cover the sphere. As with regular tessellations of the plane, the difficulty is to fit corner angles around a vertex – it requires the corner angle to divide evenly into 360°. This means that the possibilities for corner angles are 360/2 = 180°, 360/3 = 120°, 360/4 = 90°, 360/5 = 72°, 360/6 = 60°, and so on.

Two strange “degenerate” types of regular tessellations show up in spherical geometry. The first is by polygons with corner angles equal to 180°. A 180° corner doesn’t look like a corner at all, and a regular n-gon with 180° corner angles simply looks like a hemisphere with n evenly spaced dots on its edge for the “vertices”. Two of these fit together to cover the sphere. One can argue about whether this should be a polygon at all, but we’ll see that it fits very nicely in a larger picture of regular tessellations and is worth including. The other degenerate tessellation is a “beach ball” made with k biangles which have 360°/k corner angles. The beach ball with k = 7 is shown below, and there is one for any choice of k > 1.

Sphere4.png Sphere5.png

Tessellation by 7 biangles. Two 12-gons with 180° “corner” angles

If you want to make non-degenerate spherical tessellations (using regular polygons with more than two sides, and which have corner angles that actually bend) there are only a few options. First, the corner angles must be 120° or less to divide evenly into 360°. This limits the choices of polygon to triangle, quadrilateral, and pentagon, and in fact leaves only five possibilities:

triangles with 72° angles, five meeting at a vertex triangles with 90° angles, four meeting at a vertex triangles with 120° angles, three meeting at a vertex quadrilaterals with 120° angles, three meeting at a vertex pentagons with 120° angles, three meeting at a vertex

When making tessellations of the Euclidean plane, it was not so surprising that once we had six 60°-60°-60° triangles around one vertex we were able to fill out the rest of the plane. With spherical geometry, we can fit five 72°-72°-72° triangles around a vertex, but as we fill up the sphere with triangles we have to hope that they come together on the back and actually close up. Amazingly, in all five cases listed above, the polygons do cover the sphere and we get a regular tessellation. Here’s what they look like:

Spheretess.png

The five non-degenerate regular tessellations have been known for thousands of years, although in their alter-egos as polyhedra. A polyhedron is a three dimensional solid with a surface made of polygons, known as the faces of the polyhedron. Using the same vertices as the spherical tessellations but replacing the spherical polygons with flat polygons (that lie inside of the sphere), you get five polyhedra known as the Platonic solids:


The Platonic Solids

Platonic5.png

From left to right we see: Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron.


Plato did not discover these solids, but in the dialogue Timaeus he discusses the construction of the universe and (at some length) associates the cube, tetrahedron, octahedron and icosahedron with the elemental ideas of earth, fire, air and water. The dodecahedron, he claims, “God used in the delineation of the universe”. The attachment of mystical or spiritual properties to the platonic solids is, in some sense, a tribute to their mathematical perfection, and continues to this day. The 17th century astronomer Johannes Kepler wrote extensively about them, and attempted (more or less unsuccessfully) to explain the orbits of the planets as coming from the radii of a nested set of platonic solids. His reasoning was that God must have created the universe according to the platonic solids because of their mathematical perfection. In Kepler’s time, this was a somewhat heretical stance since it suggested that God was bound by rules of mathematics discovered by science.


Euler Characteristic

In 1750, the Swiss mathematician Leonhard Euler discovered a remarkable formula involving the number of faces f, edges e, and vertices v of a polyhedron:

v – e + f = 2

As a first step to understanding this equation, we will calculate v, e, and f for the Platonic solids and check that v – e + f = 2. We can count the faces of each Platonic solid by considering the corresponding spherical tessellation. For example, in the tessellation corresponding to the dodecahedron there are three pentagons at each vertex, so that each pentagon has 120° corner angles. The five angles give an angle sum of 5*120° = 600°. Since a Euclidean pentagon has angle sum 540°, these spherical pentagons have defect equal to 600°-540° = 60°. Each pentagon therefore covers 60°/720° = 1/12 of the sphere, and so there are 12 faces on the dodecahedron. In the spherical tessellation corresponding to the octahedron, four triangles meet at a vertex. Therefore these are 90°-90°-90° triangles, which have defect 270°-180° = 90°. Each one covers 90°/720° = 1/8 of the sphere, so there are 8 faces on the octahedron.

To count the edges of the dodecahedron, notice that each of the 12 faces has 5 edges. Since each edge is shared by two faces, there are 12*52 = 30 edges on the dodecahedron. Another way to understand this calculation is to imagine each edge cut into a left and right half. Then each face contributes 5 half-edges, and 12 * 5 *1/2 = 30.

As another example, let’s count the edges of the octahedron (which you can probably do by inspection). There are eight triangles, each with three edges, so there are 8*32 = 12 edges in the octahedron.

We’ll count the vertices in a similar manner. For the dodecahedron, each of the 12 faces has 5 vertices. Since each vertex is shared by three faces, there are 12*53 = 20 edges on the dodecahedron. Another way to understand this calculation is to imagine each vertex cut into three 120° wedges. Then one corner of one face is 1/3 of a vertex, and 12*5*1/3 = 20. For the octahedron, there are 8 faces with 3 vertices each, and each vertex is shared by 4 faces. There are 8*34 = 6 vertices.

Similar calculations establish the number of faces, edges, and vertices on the tetrahedron, cube, and icosahedron. It is also possibly to simply count by inspection. We arrive at the following table:

Polyhedron # of vertices (v) # of edges (e) # of faces (f) v – e + f
Tetrahedron
4 6 4 2
Cube
8 12 6 2
Octahedron
6 12 8 2
Dodecahedron
20 30 12 2
Icosahedron
12 30 20 2


We calculate one more example, the “tetrakis hexahedron”, which is the basis for Escher’s Sphere with Angels and Devils (Magic #151). Each face is a triangle, but this is not a regular tessellation of the sphere since these are not equilateral triangles. In the corresponding spherical tessellation, two of the triangle’s corners are 60° angles, since six of them together at those points. The other corner is 90° since four triangles come together at that point. The defect is 60°+60°+90° - 180° = 30°. The area fraction is 30°/720° = 1/24, so 24 of these triangles cover the sphere. Since the tetrakis hexahedron has 24 faces and each face has 3 edges, it has 24*3/2 = 36 edges.

Tetrakis.png

The Tetrakis Hexahedron and the corresponding tessellation of the sphere.


The easiest way to count the vertices of the tetrakis hexahedron is to use v – e + f = 2. Since f = 24 and e = 36, v must be 14. As a check, we count another way. Each triangle contributes 1/4 of one vertex, and 1/6 each of two others. Since there are 24 triangles, the total number of vertices v = 24*(1/4 + 1/6 + 1/6) = 14. Having seen some evidence that v – e + f is 2, we try to make a convincing argument that it is always 2 for spherical tessellations. We write the Greek letter chi as a shorthand, χ = v – e + f. The plan is to start with a sphere with one dot on it. Since there is one face (the sphere) and one vertex (the dot), χ = v – e + f = 1 – 0 + 1 = 2 in this case. Now we build whatever tessellation we desire by using one of the following two moves: Move I: Add a new dot and an edge connecting it to an existing dot. Move II: Add an edge connecting two (different) existing dots. The key point is that neither Move I or Move II changes χ. Move I adds one vertex and one edge, which cancel in v – e + f. Move II adds one new edge, and cuts one face into two, creating a net increase of one face. Again, e increasing by 1 and f increasing by 1 cancel in v – e + f. To make this argument into a rigorous mathematical proof, we would need to argue that any spherical tessellation can be built from the one-dot-sphere via a series of Moves I and II. While not a difficult argument, it is too technical for this discussion. However, it is not hard to draw specific tessellations by hand in this way. The quantity χ = v – e + f is called the Euler characteristic. It is always 2 for spherical tessellations (and for polyhedra), but can actually be different from 2 on other surfaces. For example, a careful count of v, e, and f on a donut made of polygons shows that its Euler characteristic is 0.

Torus.png

The Euler characteristic of this donut is 0.


The number χ can detect the shape of a surface without noticing size or other deformations such as stretching (e.g. a sphere stretched into a football shape). Because of this, it is considered part of a branch of mathematics called topology.

Duality

Every tessellation by polygons has a dual. The duality process works in Euclidean geometry, non-Euclidean geometry, and even with polyhedra. We start with Euclidean geometry first, to get the idea. To find the dual to a tessellation, start with a tessellation by polygons, and put a point inside each polygon of the tessellation. Connect these new points by line segments, connecting two points when their enclosing polygons share an edge. The dual tessellation is made up of these points and the edges connecting them.

Duality.png

From left to right: the original tessallation; Dual points; Connect neighbors; Dual tessellation.

Sometimes the line joining two new points will have to cross through other tiles of the tessellation, but that’s fine. Another question is what to do near the edges of a tessellation, and as you can see in the example we left some polygons open to avoid this. For tessellations that cover the entire plane, and for spherical tessellations, this isn’t an issue so we won’t worry about it. In this process, we got one new vertex for every tile of the original tessellation. We also got one edge in our dual for every edge in the original, since two points got connected by an edge if they had an edge in common. Finally, we got one new tile in the dual for each vertex of the original. This is because the edges leaving an original vertex get turned into edges of a polygon that surrounds the original vertex. Repeating the dual returns to the original tessellation, which means that neither one is particularly more original than the other, so we just say that two tessellations are dual. For example, the tessellation by equilateral triangles and the tessellation by regular hexagons are dual:

Duality2.png

Duality works for spherical tessellations and for polyhedra. With polyhedra, the process forms the dual polyhedron inside the original, but altering the sizes can lead to pleasing “compound” solids. As an example, the octahedron and cube are dual.

Dualplatonics.png

From left to right: Octahedron inside cube; Cube-octahedron compound; Cube inside octahedron


The cube-octahedron compound is visible in the upper left corner of Escher’s Stars.

Symmetries in Spherical Geometry

Spherical rotations involve spinning the sphere around an axis line that goes through the center of the sphere. A spherical rotation has two points that don’t move, where the rotation axis hits the sphere at a pair of antipodal points. For example, the Earth (idealized a bit) rotates on its axis, and the North and South poles don’t move. Translations on the sphere are exactly the same as rotations. A translation should slide along a geodesic. The geodesics are great circles, and if you slide along a great circle the sphere rotates around an axis. Picture the Earth’s equator, and as the world turns it appears that points near the equator are being translated east.

Sliderotate.png

Translation along a great circle is the same as rotation around the corresponding pole.

You can reflect a sphere using a geodesic as your reflection line. The reflection exchanges the two hemispheres. Like Euclidean geometry, the combination of a reflection and a translation is a new kind of symmetry, and could be called a glide reflection or a rotation-reflection.