#### Cynthia Lanius

Thanks to PBS for permission to use the Pyramid photo.

# History of Geometry

Egyptians
c. 2000 - 500 B.C.

Ancient Egyptians demonstrated a practical knowledge of geometry through surveying and construction projects. The Nile River overflowed its banks every year, and the river banks would have to be re-surveyed. See a PBS Nova unit on those big pointy buildings. In the Rhind Papyrus, pi is approximated.

Babylonians
c. 2000 - 500 B.C.

Ancient clay tablets reveal that the Babylonians knew the Pythagorean relationships. One clay tablet reads
4 is the length and 5 the diagonal. What is the breadth? Its size is not known. 4 times 4 is 16. 5 times 5 is 25. You take 16 from 25 and there remains 9. What times what shall I take in order to get 9? 3 times 3 is 9. 3 is the breadth.

Greeks
c. 750-250 B.C.

Ancient Greeks practiced centuries of experimental geometry like Egypt and Babylonia had, and they absorbed the experimental geometry of both of those cultures. Then they created the first formal mathematics of any kind by organizing geometry with rules of logic. Euclid's (400BC) important geometry book The Elements formed the basis for most of the geometry studied in schools ever since.

The Fifth Postulate Controversy
c. 400 B.C. - 1800 A. D.

There are two main types of mathematical (including geometric) rules : postulates (also called axioms), and theorems. Postulates are basic assumptions - rules that seem to be obvious and are therefore accepted without proof. Theorems are rules that must be proved.

Euclid gave five postulates. The fifth postulate reads: Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.

Euclid was not satisfied with accepting the fifth postulate (also known as the parallel postulate) without proof. Many mathematicians throughout the next centuries unsuccessfully attempted to prove Euclid's Fifth.

The Search for pi
??? B.C. - present

It seems to have been known from most ancient of times that the ratio of the circumference and diameter of a circle is a constant, but what is that constant? A search for a better answer to that question has intrigued mathematicians throughout history.
Coordinate Geometry
c. 1600 A.D.

Descartes made one of the greatest advances in geometry by connecting algebra and geometry. A myth is that he was watching a fly on the ceiling when he conceived of locating points on a plane with a pair of numbers. Maybe this has something to do with the fact that he stayed in bed everyday until 11:00 A.M. Fermat also discovered coordinate geometry, but it's Descartes' version that we use today.

Non-Euclidean Geometries
c. early 1800's

Since mathematicians couldn't prove the 5th postulate, they devised new geometries with "strange" notions of parallelism. (A geometry with no parallel lines?!?) Bolyai and Lobachevsky are credited with devising the first non-euclidean geometries.

Differential Geometry
c. late 1800's-1900's

Differential geometry combines geometry with the techniques of calculus to provide a method for studying geometry on curved surfaces. Gauss and Riemann (his student) laid the foundation of this field. Einstein credits Gauss with formulating the mathematical fundamentals of the theory of relativity.

Fractal Geometry
c. late 1800's-1900's

Fractals are geometric figures that model many natural structures like ferns or clouds. The invention of computers has greatly aided the study of fractals since many calculations are required. Mandelbrot is one of the researchers of fractal geometry.

Assignment:
Research a mathematician who has made a contribution to geometry. Write a short biography of the mathematician and summarize his/her contributions to geometry.

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