Break the rules of conventional geometry
Introduction
My doubt on the Euclidean geometry that the sum of angles in a triangle is 180°
started from my experience on one birthday party.Many balloons were decorated in
the room and there were some pictures on the balloons,I found that triangles on the
balloons were not the same as what I see in common—they are elliptic
triangles(see in figure1).Then a wonder created in my mind that whether the sum of
angles in those triangles is larger than usual.
Figure.1
The first step of breaking the conventional rule is to prove it.
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek
mathematician Euclid, which he described in his textbook on geometry: the
Elements. Euclid's method consists in assuming a small set of intuitively appealing
axioms, and deducing many other propositions (theorems) from these.
One of the important well-known results is the triangle angle sum—the sum of the
angles of a triangle is equal to a straight angle (180 degrees). This causes an
equilateral triangle to have 3 interior angles of 60 degrees. Also, it causes every
triangle to have at least 2 acute angles and up to 1 obtuse or right angle.
Proving Euclidean geometry
There is a given triangle:
Draw a line parallel to side BC of the triangle that passes through the vertex A.
Label the line as PQ.
∠PAB+∠BAC+∠CAQ＝180º( Because PQ is a straight line which means ∠PAQ is
180º)
∠PAB=∠ABC(congruent as PQ//BC and AB transversal line)
∠CAQ=∠ACB(congruent as PQ//BC and AC transversal line)
so, in the triangle ABC,
∠BAC+∠ABC+∠ACB＝180º
In the above situation, there is precondition that the triangle must follow Euclid’s
fifth postulate which “Through a point not on a line, there exists one and only one
line parallel to this line.”In other words, this rule is true in the plane, but in some
other types of surfaces, it may not be.
In mathematics, we divide various surfaces depending on Constant Gaussian
Curvature, some are surfaces of Constant Gaussian Curvature and others are not.
In common, surfaces of Constant Gaussian Curvature are three below:sphere,
pseudosphere, and Euclidean plane.
* Gaussian Curvature:Gaussian curvature is an intrinsic measure of curvature,
depending only on distances that are measured on the surface, not on the way it is
isometrically embedded in Euclidean space.
To explore the sum of angles in the triangle in the sphere,I designed an
experience .
Experiment
Figure 2
Sphere
Figure 3
Pseudosphere
Figure 4
Euclidean plane
the sum of angles in the triangle in sphere.
Step1
Prepare a ping-pong, a tennis ball and a basketball. (different sizes)
Step2
Measure the radius of three balls.
Use two pieces of board to fix the ball, and then mark the position of board,
measure the shortest distance between two marks(the diameter of the ball) and use
the radius formula
d=2r r=d/2
to calculate the radius of the ball.
Step3
Draw a triangle on the ball.
Learned from the three-dimensio...
