SOAR Homework Five
These homework problems are meant to expand your understanding of what
goes on during class. Any you turn in will be graded and returned to
you. Answers may or may not be posted on the web, depending on demand.
 When discussing tilings of the hyperbolic plane by regular ngons,
we said that we would need at least 4 ngons at each vertex when
n=5 or 6, and at least 3 ngons when n ≥ 7. Explain why
this is so.

In spherical geometry, the angle excess plays the same role as the
angle defect does in hyperbolic geometry. Let E(Δ) =
S(Δ)  180° be the angle excess, where
S(Δ) is (as usual) the angle sum in degrees.

Explain why the angle excess is additive. That is, if a triangle
Δ_{1} is formed by joining two other triangles
Δ_{2} and Δ_{3} along an edge, show that
E(Δ_{1}) = E(Δ_{2}) + E(Δ_{3}).

Assume that the area of a triangle A(Δ) is given
by a constant multiple of the excess E(Δ); that is,
assume that A(Δ) = C·E(Δ). Find the
constant C. Your answer should involve the radius r of the
sphere.
Hint: Consider the 90°  90°  90°
triangle that is oneeighth of a sphere. What is its
area?

Here is Euclid's proof of Proposition 16 (Book I):
In any triangle, if one of the sides is produced, then the exterior
angle is greater than either of the interior and opposite angles.
We draw a picture to make clear what Euclid is saying (see the pdf file for the picture). The
claim is that both interior angles ∠BAC and ∠ABC are smaller
than the exterior angle ∠ACD.

Euclid's proof begins by drawing the bisector BE of AC, and
extending it to F so that BE = EF. Use the picture to finish this
proof that ∠BAC < ∠ACD.

Explain why ∠ABC < ∠ACD as well.

This argument fails when it is made on the sphere. Explain why.
Hint: What happens when BF is a semicircle. When
BF is more than a semicircle?
These problems are also available as a pdf file.
Solutions
Please email Peter
if you are interested in answers or solutions for the web. Thanks.
SOAR Winter 2003 Course Homepage