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: "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."
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,±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!
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