Everybody has taken geometry in high school, there we learned that the sum of the three angles of a triangle equals 180 degrees. So, if years laters a guy would come to you and tell you otherwise. What would you do? Would you tell him that he is crazy for saying that. You would think that this guy oviously didnt pay attention in high school. Well, that guy is actually right, but it depends on they type of geometry that he is talking about.
In high school we learned Euclidean geometry, which is mostly a 2-D geomtry because the triangles are drawn in a straight surface. On the other hand in Riemannian geometry the angles are drawn in a sphere, which is a 3-D surface, so the sum of the angles is greater than 180 degrees.
In high school we learned Euclidean geometry, which is mostly a 2-D geomtry because the triangles are drawn in a straight surface. On the other hand in Riemannian geometry the angles are drawn in a sphere, which is a 3-D surface, so the sum of the angles is greater than 180 degrees.
This amazing discovery was brought to us by the German mathematician Bernhard Riemann (1826-1866).here to edit.
Riemann was a student of Gauss, and later continued on the studing that Gauss had started a few years back. Yet, the whole thing started because on his book of elements Euclid (300 B.C.E.) impacted the world with this excellent
collection of geometrical proofs. Nevertheless, his fith postulate caused
conmotion among the mathematical world because there was some flaws regarding this proof.
Euclid’s fifth postulate states that if a straight line falling on two
straight lines makes the interior angles on the same side less than two right
angles, the two straight lines, if produced indefinitely, meet on that side on
which are the angles less than the two right angles. Yet, according to Riemann through a given point not on a given line, there exist no lines parallel to the line through the given point. This idea was the based on the study of curved spaces.
collection of geometrical proofs. Nevertheless, his fith postulate caused
conmotion among the mathematical world because there was some flaws regarding this proof.
Euclid’s fifth postulate states that if a straight line falling on two
straight lines makes the interior angles on the same side less than two right
angles, the two straight lines, if produced indefinitely, meet on that side on
which are the angles less than the two right angles. Yet, according to Riemann through a given point not on a given line, there exist no lines parallel to the line through the given point. This idea was the based on the study of curved spaces.
Riemann's study was based on curved space geometry, which is very different than what we have seen before. Here Riemann tried to construct a triangle inside a sphere. Thus, due to the curvature of the sphere the sides of the sphere will also show certain curvature that allows the angle between two sides to be more than one right angle. The exact same thing will happen with the remaining two angles. So, as a consequence we will have that the sum of the three angles of the triangle add up to more than 180 degrees.
Riemann's work was not publised until two years after his death. Yet, the incredible contributions that he did to the area of geometry are still captivating people who still studing Riemmans work.
Riemann's work was not publised until two years after his death. Yet, the incredible contributions that he did to the area of geometry are still captivating people who still studing Riemmans work.