Euclidean vs. Non-Euclidean Geometry
Euclidean Geometry is the study of points, lines, and planes.
Non-Euclidean Geometry is the study of curves.
History
Euclidean Geometry is based on five assumptions, or postulates.
1. A line can be drawn between any two points.
2. Any line segment can be extended to infinity in either direction.
3. A circle can be drawn with any given point as the center and with any given radius.
4. All right angles are equal.
5. For any given line l, and any given point A, there is only one line through A that does not intersect l. For many years mathematicians tried to prove the fifth postulate, commonly known as the Parallel Postulate, and a new system of geometry was developed.
Therefore, Non-Euclidean Geometry was derived from Euclidean Geometry.
Triangles
k = the curvature of the figure
Euclidean Geometry: The sum of the angles in a triangle will always be equal to 180°.
Spherical Geometry: The sum of the angles in a triangle will always be greater than 180°.
Hyperbolic Geometry: The sum of the angles in a triangle will always be less than 180°.
Example:
Based on the information above, what is the maximum number of right angles in a triangle that lies on a hyperbolic curve?
Solution:
There can only be one right angle in a triangle that lies on the plane.
The curves of a triangle that lies on a sphere curve outward, so there can be a maximum of three right angles.
The curves of a triangle that lies on a hyperbolic curve are always less than 90°, therefore, there can be no right angles.
Points and Lines
Euclidean Geometry
Lines are infinite.
Two lines intersect at one point.
Spherical Geometry
Lines, or great circles, are finite.
Two lines, or great circles, intersect at two points.
The points at the opposite ends of a sphere are called antipodal points.
History of Geometric Systems
Geometric Development in Ancient Civilizations
The English word "geometry" is derived from two Greek words which mean "to measure the earth."
Use of geometry can be traced back to the times of ancient Mesopotamia. In Egypt, knowledge of geometry was necessary to build the first pyramid, which was constructed around 2600 BC. Both of these civilizations used geometry for real-world problems, such as finding the area or perimeter of objects.
Starting around 600 BC, the Greeks began developing and organizing geometric principles. Two of the most well-known Greek mathematicians from this time are Pythagoras and Euclid.
Pythagoras's most notable contribution is the Pythagorean theorem. The theorem is used for finding the side lengths of a right triangle. Pythagoras was greatly influence by two other Greek philosopher mathematicians: Thales and Anaximander. Pythagoras ultimately created his own school and society of Pythagoreans on the southern coast of Italy.
The Elements and Euclidean Geometry
Euclid's most important contribution to geometry was a collection of thirteen books titled The Elements. In his works, Euclid ordered earlier mathematical theorems in an organized way and wrote proofs for many of these theorems. The Elements accepted five axioms (statements accepted as fact, though they cannot be proven) which many proofs were based on. Paraphrases of these axioms are shown below.
1. / Any two points can be connected by one and only one straight line.2. / Any line segment is a part of a full line.
3. / Given a point and a line segment starting at that point, there is a circle that has the given point as its center and the given line segment as its radius.
4. / When two lines are perpendicular, all angles formed by their intersection are equal.
5. / Given a line and a point not on that line, there is one and only one line through that point that never meets the original line.
The Elements was translated into several languages and used as a textbook all around the world. Euclid is often referred to as the "Father of Geometry."
The Development of Analytic Geometry
Analytic Geometry is the use of graphing on the coordinate plane to combine geometry and algebra.
The connection of algebra to geometry occurred when the coordinate system was developed. The coordinate system created a way to model important geometric ideas, such as a point and a line. The use of a coordinate system to graph geometric shapes and objects became known as analytic geometry. This process was developed separately by two French mathematicians, Rene Descartes and Pierre de Fermat.
Descartes published his work on analytical geometry in a piece titled La Geometrie, an appendix to a larger work. The Cartesian coordinate system is named after him, as he developed it to plot figures using their horizontal and vertical distances from the origin. The system allows functions to be visualized and geometric figures to be evaluated algebraically.
Non-Euclidean Geometry
Non-Euclidean geometry accepts Euclid's first four axioms, but adjusts the fifth axiom, also known as the parallel postulate. The parallel postulate has been regarded as being less fundamental than the other axioms, and Euclid avoided using it in his proofs if possible. Many mathematicians attempted to prove the postulate, but did not succeed.
In the 19th century, mathematicians began attempting to prove the parallel postulate by assuming the opposite was true, trying to prove it with logical arguments, and coming to a preposterous conclusion. However, no preposterous conclusions were reached. Instead, an entirely new kind of geometry, which accepted a contradiction of the parallel postulate as its fifth axiom, was created. Janos Bolyai and Nikolai Ivanovich Lobachevsky are credited as the first mathematicians to formulate theories of non-Euclidean geometry.
Hyperbolic geometry exists on a curved space and accepts that given a line and a point not on that line, there are infinitely many lines through that point that are parallel to the original line. Elliptic geometry and spherical geometry exist on a sphere and accept that there is no line parallel to a given line. However, elliptic geometry and spherical geometry differ in the number of intersections that exist between two lines.
More Considerable Contributions to Geometry
Archimedes's greatest contribution to geometry was his work Measurement of a Circle. In it, he used the perimeters of inscribed and circumscribed regular polygons to approximate to be a number between and . Archimedes is also known for proving that the volume of a sphere is two-thirds the volume of a circumscribed cylinder
Heron is most well-known for his work with triangles and his book Metrica. Heron's formula (shown below) can be used to find the area of a triangle, , with side lengths , , and , and .
Euler discovered that for any convex polyhedron, the sum of its vertices and faces is two more than its edges. Therefore, Euler's formula is V - E + F = 2. Euler also completed the first theorem of planar graph theory.
Gauss was also involved in the creation of non-Euclidean geometry. He studied curves and surfaces in a three-dimensional Euclidean space, known as differential geometry. Gauss also came up with the Gaussean curvature, another important element of differential geometry.
Hilbert published his book The Foundations of Geometry in 1899. In it he revealed a fresh set of axioms that avoided the issues of Euclid's five axioms. Later, in 1900, Hilbert called on mathematicians all over the world to solve 23 mathematical problems that had not yet been solved.
Hyperbolic Geometry
Hyperbolic geometry exists on a curved space and accepts that given a line and a point not on that line, there exists at least two lines through that point that are parallel to the original line.
Hyperbolic geometry is a non-Euclidean geometry.
The following statements apply in hyperbolic geometry.
· The sum of the angles of a triangle is less than 180°.
· The sum of the angles of a quadrilateral is less than 360°.
· Squares and rectangles do not exist.
· Triangles that have the same angles have the same areas and are therefore congruent.
· Similar triangles do not exist.
· The area of any triangle is less than .
· The defect of a triangle is equal to the difference between 180° and the sum of the angle measures of the triangle.
Example:
A triangle in hyperbolic geometry has angle measures of 32°, 125°, and 16°. What is the defect of the triangle?
Solution:
The defect of a triangle is equal to the difference between 180° and the sum of the angle measures of the triangle.
First, find the sum of the angles of the triangle.
32° + 125° + 16° = 173°
Next, subtract the sum from 180°.
180° - 173° = 7°
Therefore, the defect of the triangle is 7°.
Taxicab Geometry
Taxicab geometry is a form of geometry based on the taxicab metric where the distance between two distinct points P(x1, y1) and Q(x2, y2) is the sum of the absolute differences of their coordinates.
Example 1:
Determine what kind of triangle is represented in the given figure using the taxicab metric.
Solution:
The triangle is made of the three points (2, 6), (6, 6), and (4, 4) represented by A, B, and C repectively.
AB = |6 - 2| + |6 - 6| = 4
BC = |6 - 4| + |6 - 4| = 4
CA = |4 - 2| + |4 - 6| = 4
Thus, the triangle is an equilateral triangle.
*Notice that the formula is adding the horizontal and vertical distances between two points. Thus, an alternate way to calculate the distance is to start at one point and count the number of horizontal units and the number of vertical units to the next point and then add these values. For instance, starting at point A, point C is 2 units to the right and 2 units down. Thus, using the taxicab metric, it is 2 + 2 = 4 units away.