Paper Hyperbolic Plane Models -- AMTNYS 2006, Make-it and Take-it

Kristin Camenga – HoughtonCollege

This introduces two paper models you can make of a hyperbolic plane, which we will call the annular model and the soccer ball model. For each, all you need to construct them are copies of the template, scissors and tape.

THIS SHEET, THE TEMPLATES FOR THE MODELS, AND RELATED LINKS CAN BE FOUND AT

Directions

The annular model: Cut out the annuli on the sheet, and tape the inner edge of one annulus to the outer edge of another. Continue matching inner edge of an annulus to outer edge of an annulus, adding one annulus to the end of another if you wish as the model gets bigger. Because you are matching something with a smaller radius to something with a larger radius, the model will start to “wrinkle” or curve.

The soccer ball model: The model consists of heptagons and hexagons; each heptagon should be surrounded by 7 hexagons (just as on a soccer ball, the dark pentagons are surrounded by 5 hexagons). This means that at each vertex, you have more than 360 degrees and the model will curve away from itself (in opposition to the soccer ball which “curves in” on itself to make a sphere). Hint: you don’t need to cut apart all of the hexagons on the template! You can leave many of them attached to each other, removing hexagons and inserting heptagons as needed.

Information on Hyperbolic Planes

The two paper models are approximations of a hyperbolic plane – but they aren’t quite hyperbolic planes! In the first model, to get a hyperbolic plane you would have to make the annuli thinner and thinner in the process until they are just as thick as a line (i.e. no thickness!). In the second, the heptagons and hexagons would have to get smaller and smaller. In both these cases, the model becomes smooth – there are no rough edges where the annuli meet and no points where the angle around the point adds up to more than 360 degrees! (Note: you can also crochet a better approximation; the pattern for this is linked from the website above.)

What makes something a hyperbolic plane?

Hyperbolic planes are geometric surfaces that follow all of Euclid’s postulates except the 5th one; if we have a line and a point not on the line, there are infinitely many lines on the hyperbolic plane that go through the point and do not intersect the original line, while on the Euclidean (standard) plane there is a unique such line. You can draw a line on these models by using the reflective symmetry of a line: it should look like a line on any section of the model that is from one piece of paper and when you cross to another section, the total angle on either side of the line at the crossing should be equal so the area just to one side of the line is a reflection in the line of the area just to the other side. (This is equivalent to folding a piece of paper to get a straight line.) Using this, you can check to make sure that you can make a line and then multiple lines through another point that don’t intersect the first line!

It may seem strange that these are lines since they don’t look straight to us. A good way to think about this is to imagine a bug on the surface of the hyperbolic plane which is so small that it can’t tell that the surface is curving in three dimensions (just like we can’t tell that the surface of the earth is curving as we walk on it!). It will think something is straight when it continues to move ahead in the same direction, which means that its left and right feet must go the same distance – but this is the same as saying that the left and right of the line are reflections of each other.

Why don’t these models look like most pictures of hyperbolic planes I’ve seen?

Since these models curve, it is very hard to draw them on a piece of paper! Therefore, when we draw hyperbolic planes on paper, we represent them in a different way, just as we do to put the surface of the earth on paper. Just like maps of the earth, to put things on flat paper we end up distorting pieces (frequently Greenland looks huge!) – distances on paper don’t match the actual distances on the hyperbolic plane. One popular representation, the Poincaré diskmodel, puts all of the infinite hyperbolic plane into a finite circle. Lines are drawn as portions of circles perpendicular to the outside circle. This keeps all angles the same as on our models, but we can draw an infinite line with just a portion of a small circle, so distances are not the same! Some of Escher’s drawings closely approximate this model, with shapes that are big in the middle of the circle getting smaller and smaller as they get toward the end of the circle – but if they were actually on a hyperbolic plane, all these shapes would be the same size! Other models include the half-plane model and the Klein model.

These representations in the plane don’t seem “real” to us in the same way that the curving paper models do because two copies of the same object frequently don’t look the same size or shape. This is part of the reason that for a long time people had a hard time accepting that hyperbolic geometry was real! One of the final keys to acceptance was Beltrami’s discovery of the pseudosphere, which looks like a long medieval trumpet – infinitely long. It is a hyperbolic plane rolled up into a cone, where it never quite comes to a point at the top – the point is infinitely high – but around each point it is exactly like a hyperbolic plane. On Beltrami’s model, people could measure things and objects that were congruent would actually look that way!

Some interesting properties of hyperbolic planes you can explore:

  • Triangles on hyperbolic planes have angles that sum to less than 180 degrees; in fact, they can sum to as small a number as you wish!
  • The area of a triangle on a hyperbolic plane depends totally on the sum of the angles! If you measure the angle in radians as α, β, and γ and the radius of the annuli in the annular model is r, then the area is (π-(α + β + γ))r2. As a result, triangles on a hyperbolic plane have a maximum area: πr2. (In many books they simplify this by assuming r = 1.)
  • Two triangles are congruent on the hyperbolic plane if and only if all of their angles correspond. That is, AAA works on the hyperbolic plane, so there is no such thing as a pair of similar hyperbolic triangles. All the regular triangle congruence rules work, too!

Some other interesting facts:

  • Spherical triangles have similar properties to those above, but triangles have angles that sum to greater than 180 degrees, the area is ((α + β + γ)-π)r2, and we need to stick to ‘small’ triangles so that the congruence rules work..
  • Scientists think that the shape of our universe could be a three dimensional version of a hyperbolic plane!