### Table of Contents

## Foundations of Geometry

### Pythagorean Theorem

The Pythagorean Theorem is one of the most used theorems in geometry. It was discovered by Pythagoras, who went on to establish the Pythagorean Order, later known as the “Brotherhood.” The “Brotherhood was a organization of logic (Fernando and Berlinghoff 16-17). The Theorem states that if we have a right triangle ABC, a and b are the legs(shorter sides) and c is the hypotenuse then a^2+b^2=c^2.This equation has been proven numerous times and ways (Dr. McCrory).

### Regular Polyhedras

One major contribution to geometry discovered by the Pythagoreans was that of the five regular polyhedra. “A polyhedron is solid whose surface consists of polygon(any shape) faces. A polyhedron is regular or Platonic if its faces are congruent regular polygons and if its polyhedral angles are all congruent” (Anglin, 29).

The five discovered regular polyhedra are the cube, tetrahedron, octahedron, icosahedron, and the dodecahedron. Pythagoras only found and knew of four. One of his followers, Hippasus discovered the dodecahedron. He was then expelled from the Pythagorean society for not attributing the discovery to Pythagoras. The fact that only five polyhedron exist was published in Euclid’s Elements (Anglin, 29-30).

### Corruption and Cult

Although the Pythagoreans had many discoveries, they began to develop into a cult. The so called “Brotherhood” began to always dress in white, never use wool, never eat meat and beans, never hunt, and always slept in white linens.

In addition they started using the symbol of the pentagram. The Pythagoreans instated rituals and went through a simple regimen to strengthen the participant’s mind and body everyday. As they studied mathematike, or that which is learned, they did so in secret. The whole society of the “Brotherhood” was secret (Fernando and Berlinghoff 16-17).

### Euclid’s Elements

Euclid’s Elements are a series of books or treatises. They were written by Euclid of Alexandria, who is often confused with Euclid of Megara who did not contribute to mathematics. Besides Euclid’s Elements, he wrote many other works covering optics, astronomy, music, and mechanics.

The series is split into thirteen different books. The first six cover plane geometry, the seventh through ninth cover number theory, the tenth covers incommensurables or items lacking comparison to others, and the final books cover mainly geometries of solids.

These books were some of the first compilations of geometric and mathematical knowledge in the world. Euclid also established many important axioms and postulates or generally accepted truths not proven or proven in all cases. These were defined in the ten postulates:

- “1. To draw a straight line from any point to any point.

- 2. To produce a finite straight line continuously in a straight line.

- 3. To describe a circle with any center and radius.

- 4. That all right angles are equal.

- 5. 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 the angles are less than the two right angles.

- 6. Things which are equal to the same thing are also equal to one another.

- 7. If equals can be added to equals, the wholes are equal.

- 8. If equals be subtracted from equals, the remainders are equal.

- 9. Things which coincide with one another are equal to one another.

- 10. The whole is greater than the part” (Boyer 1-553).

### Measurement of a Circle

Among his other discoveries, Archimedes made one major contribution to geometry. This contribution is narrowing down the value of Pi. He made this work public by publishing Measurement of a Circle. In this book he determined that pi had a value between 3 10/17 and 3 1/7.

He determined this by circumscribing and inscribing a circle with a 96-sided figure. Then he took the total lengths of these figures and determined the range.

### Heron

Heron of Alexandria is known most for Heron’s formula which determines the area of a triangle using the semiperimeter. The formula is defined as area equals the square root of s*(s-a)*(s-b)*(s-c). Where s is the semiperimeter, and a,b,c are the sides of a triangle. Not only was this formula used as a way to determine area with less information, it was also used as another proof of the Pythagorean Theorem (Dr. McCrory).

### Brahmagupta

Brahmagupta also worked with Heron’s Formula and expounded upon it by making Brahmagupta’s Conjecture. This conjecture is very similar to Heron’s Formula, except Brahmagupta’s Conjecture deals with quadrilaterals instead of triangles (Hess, Ver. 12). This conjecture also helped prove the Pythagorean Theorem (Dr. McCrory).

## Geometry of Today

### Riemannian Geometry

Riemannian Geometry is one of the two major types of Non-Euclidean Geometry. This branch of geometry is conducted on a spherical plane or curved plane. It is named after German mathematician Bernhard Riemann who, in 1889, rediscovered and furthered the works of Girolamo Saccheri, an earlier mathematician. These works proved that Euclidean Geometry had some flaws and did not work for all scenarios.

Riemannian Geometry does change some of geometry’s mathematical rules. The first rule it changes is triangle sum. In a flat plane, the angles of a triangle will add up to 180. Due to the curved surface, these angles will now always add up to more than 180 degrees.

Straight lines are in theory non-existent on a sphere. If a line were to be drawn, once it left the starting point it would curve. This curve makes all lines to be not straight. Instead there are geodesics, or the shortest distance between two points. Unlike on a plane, there can be many different geodesics between two points. For example, there are the lines of latitude connecting the North and South poles on a globe.

Other items such as perpendicular lines are also changed. This picture illustrates how perpendicularity occurs in Riemannian Geometry. The concept is still the same with the lines meeting at a 90 degree angle, but the curve changes the diagram (Roberts).

### Lobachevskian/Hyperbolic Geometry

Hyperbolic Geometry is studied on a saddle or hyperbolic shaped surface. It is also called Lobachevskian Geometry after the mathematician Nicholas Lobachevsky who pioneered the field. Similarly to Riemannian Geometry the surface changes Euclidean Geometry rules.

Hyperbolic geometry mainly changes the rules of triangles. All triangles have less than 180 degrees in them and no similar triangles exist. This is caused by the fact that all triangles with equal angles have an equal area, not allowing for similar triangles.

Finally, a perpendicular line can also be drawn in Hyperbolic geometry, along with that of parallel lines. In Hyperbolic geometry, a perpendicular line is drawn like this. (Roberts)

### Current Uses

The two major types of Non-Euclidian Geometry do have some practical uses in our daily lives. These are mainly in the field of sciences.

Riemannian Geometry is mainly used in the studying of our globe. Latitudes and longitudes that we use today to locate cities and places of interest around the globe are a result of this branch of mathematics.

Hyperbolic Geometry has only one major application, the field of astronomy. It has three major uses inside astronomy. Firstly, it can be used to predict orbits involved with intense gradational fields in Space. It was also used in Einstein’s theory of relativity. Hyperbolic Geometry’s third use is in space travel. Due to the fact that space is a curved surface, bent by gravity, this is helpful in using vehicles such as rockets and space shuttles. (Roberts)

Geometry is a vast subject with some of its divisions still being researched today. What started out as a simple quest for the length of a circle’s circumference and the area of triangle has culminated into what we know today. Bernhard Riemann and other mathematicians strived to uncover new fields and new discoveries in Geometry, expanding their bounds from planes to spheres to calculus.
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^{1)}Works Cited Anglin, W. S.. Mathematics, a concise history and philosophy. New York: Springer-Verlag, 1994. Print. Berlinghoff, William P., and Fernando Q. GouveÌ‚a. Math through the ages: a gentle history for teachers and others. Expanded ed. Washington, DC: Mathematical Association of America, 2004. Print. Boyer, Carl B.. A history of mathematics. New York: Wiley, 1991. Print. Cooke, Roger. The history of mathematics: a brief course. New York: Wiley, 1997. Print. Hess, Albrecht. “A Highway from Heron to Brahmagupta.” Forum Geometricorum. Version 12. Forum Geometricorum, n.d. Web. 6 Feb. 2013. <forumgeom.fau.edu/FG2012volume12/FG201215.pdf>. Quickie Math Team. “Geometry History.” ThinkQuest : Library. N.p., n.d. Web. 7 Feb. 2013. <http://library.thinkquest.org/C006354/history.html>. Roberts, Donna. “Euclidean and Non-Euclidean Geometry.” Oswego City School District Regents Exam Prep Center. Oswego City School District, n.d. Web. 7 Feb. 2013. <http://regentsprep.org/Regents/math/geometry/GG1/Euclidean.htm>. Umberger, Shannon. “Final Project.” Jim Wilson's Home Page. University of Georgia, n.d. Web. 7 Feb. 2013. <http://jwilson.coe.uga.edu/emt668/EMAT6680.2000/Umberger/MATH7200/HeronFormulaProject/finalproject.html>. Zebrowski, Ernest. A History of the Circle: Mathematical reasoning and the physical universe. New Brunswick, N.J.: Rutgers University Press, 1999. Print.