# Triangles in Wonderlandx ```Triangles in Wonderland
Are there more acute or obtuse
triangles?
Lewis Carroll/Charles Dodgson
Some fun facts:
.. as a mathematician, Dodgson was, in the
words of Peter Heath: &quot;An inveterate publisher
of trifles [who] was forever putting out
pamphlets, papers, broadsheets, and books on
mathematical topics [that] earned him no
reputation beyond that of a crotchety, if
sometimes amusing, controversialist, a compiler
of puzzles and curiosities, and a busy yet
ineffective reformer on elementary points of
computation and instructional method. In the
higher reaches of the subject he made no mark
at all, and has left none since.&quot;
Lewis Carroll/Charles Dodgson
• “Three Points are taken at random on an
infinite plane. Find the chance of their being
the vertices of an obtuse-angled Triangle.”
• Pillow Problems Thought Out During Wakeful
Hours in 1893. Problem #58
Solution
• We assume that the longest edge is from
A=(0,0) to B= (b,0)
• (Why can we do this? Can we do this?)
Third point must occur where?
Right triangle w longest side AB?
Pythogorean Theorem:
__
___
__
||AC || ^2 + || BC || ^2 = || AB ||^2
[Sqrt(x^2+y^2) ]^2 + [sqrt( (x-b)^2 + y^2) ]^2 = b^2
2x^2 + 2xb + 2 y^2 =0
Or [x-(b/2) ]^2 + y^2 = [b/2]^2
Nice precalc/ High school result
Need relative areas
In green circle: obtuse
In orange region: acute
Calc 1!
• Circle = π b
2
4
• Orange region
= 4*
So
• Obtuse/ total =
• ≈0.6393825607119623027857777410193414
1234….
3 dimensions
• 3 points determine a triangle and a plane
• Same issue
3 dimensions
• Same issue w/ pythagorean theorem. Get a
sphere centered at (b/2,0,0)
Inside: obtuse
Outside: acute
3 dimensions
• Sphere = 4/3 
• 1/2 of football:

b 3
2
 b
1
6
3
Go spherical! Calc 3
• 1/2 of ‘football’:
So
• Obtuse/ total =
( ) 2


0
.
4
5
5
24  b
2
3
b 3
2
3
• More acute triangles in 3d than obtuse, unlike
plane
Motivation for project
Larson pg 573 Essential Calc:
n dimensions
• Longest edge again from (0,0,0,0) to (b,0,0,0)
2
2
2
2
2
• In 4d, sphere: x  y  z  w  b
• Pythagorean Theorem:
all pts that form right triangles with AB
– ‘Sphere’ centered at (b/2,0,0,0) with radius b/2
n-dimensional spherical cap formula
n dimensions
Dimension
2
3
4
5
6
7
8
9
10
Obtuse/Total
3
8  6 3
2
5
3
32  36 3
8
53
15
640  864 3
16
289
105
26784 3  17920
128
6413
105
71680  114048 3
Decimal
0.6393825611
0.4
0.2468696971
0.1509433962
0.09165800095
0.05536332180
0.03329943290
0.01995945735
0.01192904991
Open Questions
• What is the exact probability on the unit
square in 2d or n dimensions. (unit disk is
known)-simulation
• Can be Simulated (unlike my problem)
Hyperbolic (Poincare) Plane
• ‘straight lines’ are arcs of circles that are
perpendicular with the boundary
Distance and size
• As one approaches the boundary, ‘measuring
sticks’ get smaller
• Distance formula:
Distance and size
• As one approaches the boundary, ‘measuring
sticks’ get smaller
Triangles
• Between any 2 points there is a unique line
• So we can form triangles. Angles computed
similar to plane (use tangent lines)
Triangles and area
WOLOG
• Longest side is (0,&plusmn;s)
• Disks ? are disks, center moves.
Pythagorean Thm
• If AB is opposite a right angle then:
cosh(AB) = cosh(AC)*cosh(BC)
Pythagorean Thm
• No longer a circle
acute
obtuse
Small edge on left, big edge on right.
Proved
• As side limits to zero
Obtuse/Total limits to
• As side limits to 1 (infinity) then
Obtuse/Total limits to
0
Next stop unit spheres!