Triangles in Wonderland

Are there more acute or obtuse triangles?

Lewis Carroll/Charles DodgsonSome 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 =0Or [x-(b/2) ]^2 + y^2 = [b/2]^2 Nice precalc/ High school result

Need relative areas

In green circle: obtuseIn orange region: acute

Calc 1!

• Circle = π

• Orange region

= 4*

4

2b

So

• Obtuse/ total =

• ≈0.63938256071196230278577774101934141234….

3 dimensions

• 3 points determine a triangle and a plane

• Same issue

3 dimensions

• Same issue w/ pythagorean theorem. Get asphere centered at (b/2,0,0)

Inside: obtuseOutside: acute

3 dimensions

• Sphere = 4/3

• 1/2 of football:

3613

2 bb

Go spherical! Calc 3

• 1/2 of ‘football’:

So

• Obtuse/ total =

• More acute triangles in 3d than obtuse, unlike plane

4.052)(

3245

323

2

b

b

Motivation for projectLarson pg 573 Essential Calc:

n dimensions

• Longest edge again from (0,0,0,0) to (b,0,0,0)• In 4d, sphere:

• Pythagorean Theorem:all pts that form right triangles with AB

– ‘Sphere’ centered at (b/2,0,0,0) with radius b/2

22222 bwzyx

n-dimensional spherical cap formula

n dimensionsDimension Obtuse/Total Decimal

2 0.6393825611

3 0.4

4 0.2468696971

5 0.1509433962

6 0.09165800095

7 0.05536332180

8 0.03329943290

9 0.01995945735

10 0.01192904991

3683

52

336323

538

386464015

28916

17920326784

105

6413128

311404871680105

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

Small edge on left, big edge on right.

acute

obtuse

Proved• As side limits to zero

Obtuse/Total limits to

• As side limits to 1 (infinity) thenObtuse/Total limits to 0

Next stop unit spheres!

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