- 1 Explorations
- 2 Points and Lines
- 3 Angle Sum and Area
- 4 Spherical Tessellations And Polyhedra
- 5 Platonic Solids
- 6 Euler Characteristic
- 7 Duality
- 8 Symmetries in Spherical Geometry
- 9 Symmetry Groups of the Sphere
- 10 Exercises
- 11 Relevant examples from Escher's work
- 12 Related Sites
- 13 Notes
Begin learning about spherical geometry with:
- Spherical Easel Exploration This exploration uses Spherical Easel (a Java applet) to explore the basics of spherical geometry.
- Spherical Geometry Exploration Using a ball and markers, this is a hands on exploration of spherical geometry.
- Spherical Geometry: Polygons What type of polygons exist on the sphere? Use of Spherical Easel is recommended.
- Spherical Triangles Exploration Explore properties of spherical triangles with Kaleidotile.
- Regular Spherical Tessellations Exploration Find the regular tessellations of the sphere.
- Spherical Geometry: Isometry Exploration Explore the properties of translations, rotations, relfections and glide reflections on the sphere.
- Spherical versus Euclidean Polygons Exploration
For platonic solids and duality turn to:
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 it is not possible to prove”, according to Aristotle. 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. You can try this yourself with Spherical Geometry Exploration.
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:
- 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.
- Walking forwards, without turning, one should follow a straight line.
- Great Circle
- A great circle is a circle on a sphere which divides the sphere into two equal hemispheres.
A person walking on the surface of a sphere without 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 treat geodesics in spherical geometry as we treat straight lines in Euclidean 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:
- Antipodal points
- 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.
- Geodesic segment
- 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
- Spherical Polygon
- 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.
- 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, and find that the fraction of the sphere covered by a triangle is the triangle's defect divided by 720°. As a formula:
Area fraction =
The Spherical Triangles Exploration should help you understand this formula.
To find the actual area covered by a triangle, you need to know the radius of the sphere and then use the fact that the total surface area of a sphere of radius is .
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 Tessellations by Polygons#euclidean-angle-sum that a Euclidean polygon with sides has angle sum , by cutting the polygon into triangles. A spherical polygon with sides can be cut in the same way into spherical triangles, each of which has angle sum more than 180°, and so the angle sum of a spherical -gon is more than . 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. Look for them yourself with Regular Spherical Tessellations Exploration.
Regular Polygons on the Sphere
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 , and this forces the corner angles of a regular -gon to be . This means there is only one shape of Euclidean regular -gon.
In spherical geometry there are many regular -gons. There is a regular -gon with any angle sum larger than (up to a maximum size). So, there is a regular -gon with any choice of corner angle larger than (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|
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, which 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.
Compare the corner angles needed for tessellating and the corner angles of spherical polygons in the table. Most spherical polygons have corner angles too large to fit together at a vertex. Ignoring biangles and 180° corner angles for the moment, there are only five possibilites for regular spherical tessellations:
- 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:
|120° Triangles||90° Triangles||72° Triangles|
|120° Quadrilaterals||120° Pentagons|
Degenerate Regular Tessellations
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, and there is one for any choice of k > 1.
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. The polygons are known as the faces of the polyhedron.
Spherical tessellations and polyhedra are closely related. Starting with a spherical tessellation by polygons, we can often replace the spherical (curved) polygons by flat polygons (that lie inside of the sphere) with the same vertices. The resulting solid is a polyhedron. Doing this for the regular tessellations of the sphere results in five polyhedra known as the Platonic solids:
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.
Make friends with the Platonic solids by doing the Platonic Solids Exploration.
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:
As a first step to understanding this equation, we will calculate v, e, and f for the Platonic solids and check that in these cases.
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*5/2 = 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*3/2 = 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*5/3 = 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*3/4 = 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|
We calculate one more example, the “tetrakis hexahedron”, which is the basis for Escher’s Sphere with Angels and Devils. 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.
The easiest way to count the vertices of the tetrakis hexahedron is to use . 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 .
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, .
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), 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 . 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 .
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 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. A torus is a surface with one hole, for example a donut or an inner tube. A careful count of v, e, and f on a torus made of polygons shows that its Euler characteristic is 0.
The number 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.
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.
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 the dualizing process, we get one new vertex for every tile of the original tessellation. We also get one edge in our dual for every edge in the original, since two points are connected by an edge if their faces had an edge in common. Finally, we get 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:
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.
|Octahedron inside cube; Cube-octahedron compound; Cube inside octahedron.||A Japanese temari ball showing duality of the cube (orange) and octahedron (purple). (D. Abolt)|
Practice with duality by doing the Duality Exploration.
The cube-octahedron compound is visible in the upper left corner of Escher’s Stars. The duality in Escher's Double Planetoid works on two levels. First, there is a mathematical duality between the two tetrahedra, while at the same time there is the natural duality between the urban city and the wild jungle landscape. Duality is a recurring theme in Escher's work. His early works emphasize duality using rotation or reflection symmetry. Scapegoat juxtaposes good and evil with an order 2 rotation, while Paradise depicts man and woman almost as mirror images.
Many of Escher's tessellations feature two opposing figures, and are used in prints where one figure dominates half the image while the other figure dominates in the other half. For example, in Day and Night, black birds flying left dominate the daytime half of the print, while white birds flying right dominate the nighttime half. In the center of the print, both sets of birds fit together and balance the print. Escher takes a similar approach in Sky and Water I and Sky and Water II, though these are less developed than Day and Night.
As a print artist, and particularly a maker of woodcut prints, Escher developed a strong sense of the duality between figure (foreground) and ground (background). When making a woodcut, the artist carves away wood where the block will not print, leaving the printing areas untouched. In other words, to create figure, the artist removes the ground. Sun and Moon is a beautiful example of the duality between figure and ground. One can see this as a picture of grey birds flying and partially obscuring a golden sun. Alternately, it is a picture of brightly colored birds flying against the backdrop of a night sky. Switching back and forth between these interpretations takes some mental effort, but demonstrates that either set of birds functions equally well as the figure or the ground.
Symmetries in Spherical Geometry
Rotations are Translations: 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.
Note that translations of the sphere do differ quite a bit from translations of the plane. In the Euclidean plane translations and rotations are distinct isometries, while on the sphere they can be thought of as the same rigid motion of the sphere. An added peculiarity is that on the sphere translating through a distance greater than the circumference of the sphere would result in the image circling the sphere before reaching its destination.
Reflections: You can reflect a sphere using a geodesic as your reflection line. The reflection exchanges the two hemispheres.
Reflections play an important role in planar geometry. It can be shown that composing reflections across parallel mirror lines results in a translation. Reflections across two intersecting lines results in a rotation about this intersection point.
On the sphere we do not have any parallel lines, and hence the composition of two distinct reflections always results in a rotation about the intersection point of the two mirror lines. But by comments above it follows that we could also interpret this as a spherical translation if we wanted.
Glide-Reflections: Like Euclidean geometry, the combination of a reflection and a translation is a new kind of symmetry. We saw above that translations on the sphere are really rotations, and hence a glide-reflection could also be called a rotation-reflection.
Symmetry Groups of the Sphere
Symmetry groups of plane figures were completely classified into the rosette, frieze, and wallpaper groups. There are infinitely many rosette groups in two types, depending on whether reflection symmetry is present. The frieze and wallpaper groups were finite lists, harder to classify and less structured.
For the sphere, the classification of symmetry groups is closely tied to the platonic solids. In fact:
Every symmetry group of a spherical figure comes about by selecting some of the symmetries of a regular spherical tessellation.
For example, there are two symmetry groups coming from the octahedron: One which has all symmetries of the octahedron and a second which has only the rotations, and none of the reflections. Escher's Sphere with Fish has symmetry of this reflectionless octahedron type. Two more spherical symmetry groups come from the icosahedron, in exactly the same way. Because of duality, the cube and dodecahedron contribute nothing new - they have the same symmetries as their duals. The tetrahedron contributes three: One with no reflections, one with all six possible reflections, and a third with three of the reflections.
The rest of the symmetry groups possible on the sphere come from the degenerate regular tessellations. All symmetries in these groups keep one axis of the sphere fixed in place, much like the rosette groups have a fixed center point. Because there are infinitely many degenerate regular tessellations, there are also infinitely many symmetry groups derived from them. Again, the situation parallels that of rosette groups.
For more details, notation, and a complete list of all these groups, see Wikipedia's list of spherical symmetry groups.
Relevant examples from Escher's work
- Concentric Rinds
- Sphere with Fish, Sphere with Angels and Devils, Sphere with Eight Grotesques, Sphere with Reptiles, Carved Polyhedron with Flowers
- Sphere Surface with Fish, Sphere Spirals
- Double Planetoid, Tetrahedral Planetoid
- Four Regular Solids (Stereometric Figure)
- Crystal, Order and Chaos, Order and Chaos II
- Stars and Study for Stars
- Spherical Easel web applet for spherical geometry.
- The Geometry of the Sphere, by John C. Polking.
- KaleidoTile, by Jeff Weeks.
- Archimedean Kaleidoscope, by Jim Morey.
- Temari, a Japanese textile art, by Deborah Abolt
- Google Planimeter, measures area of triangles on the Earth.
Platonic Solids and Polyhedra
- wikipedia:Platonic solid
- Polyhedra and Art examples through history, by George W. Hart.
- Platonic Solids from Geometry in Art and Architecture by Paul Calter.
- 3D models, by Evgeny Demidov.
- Paper models of polyhedra, by Korthals Altes.
- MatHSoliD Java applet for viewing and unfolding Platonic and Archimedean polyhedra, by E. daSilva and H. Bortolossi.
- Animated Platonic solids, with duals, from http://www.platonicsolids.info
- Applet for Platonic solids with duals, by G.M. Todesco.
- D.E. Smith, History of Mathematics, Vol. II, pg. 280