# Portal:Geometry

## Geometry

**Geometry** arose as the field of knowledge dealing with spatial relationships. Geometry is one of the two fields of pre-modern mathematics, the other being the study of numbers.

In modern times, geometric concepts have been extended. They sometimes show a high level of abstraction and complexity. Geometry now uses methods of calculus and abstract algebra, so that many modern branches of the field are not easily recognizable as the descendants of early geometry. (See areas of mathematics.) A geometer is one who works or is specialized in geometry.

## Selected article

The Pythagorean theorem:
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The **Pythagorean theorem** or **Pythagoras' theorem** is a relation in Euclidean geometry among the three sides of a right triangle. The theorem is named after the Greek mathematician Pythagoras, who is traditionally credited with its discovery, although knowledge of the theorem almost certainly predates him.

The theorem is as follows:

In any right triangle, the area of the square whose side is the hypotenuse (the side of a right triangle opposite the right angle) is equal to the sum of areas of the squares whose sides are the two legs (i.e. the two sides other than the hypotenuse).

This provides a simple relation among the three sides of a right triangle so that if the lengths of any two sides are known, the length of the third side can be found. This theorem may have more known proofs than any other. *The Pythagorean Proposition*, a book published in 1940, contains 370 proofs of Pythagoras' theorem, including one by American President James Garfield.

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## Selected biography

**Euclid** (also referred to as **Euclid of Alexandria**) (Greek: Εὐκλείδης) (c. 325–c. 265 BC), a Greek mathematician, who lived in Alexandria, Hellenistic Egypt, almost certainly during the reign of Ptolemy I (323 BC–283 BC), is often considered to be the "father of geometry". His most popular work, *Elements*, is thought to be one of the most successful textbooks in the history of mathematics. Within it, the properties of geometrical objects are deduced from a small set of axioms, thereby founding the axiomatic method of mathematics.

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## Selected picture

The above shows an example of doubly ruled surface – the hyperboloid of one sheet. Although the wires are straight lines, they are lying within the surface. Through any point on this surface pass two straight lines, so it is doubly ruled.

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## Did you know?

- ...that the hyperboloid of one sheet is a doubly ruled surface?
- ...that as the dimension of a hypersphere tends to infinity, its "volume" (content) tends to 0?
- ...that a nonconvex polygon with three convex vertices is called a pseudotriangle?
- ...that a regular heptagon is the regular polygon with the fewest number of sides which is not constructible with a compass and straightedge?
- ...that it is possible for a three-dimensional figure to have a finite volume but infinite surface area? An example of this is Gabriel's Horn.

## Categories

Algebraic geometry • Classical geometry

Conformal geometry • Convex geometry

Coordinate systems • Differential geometry

Digital geometry • Dimension • Discrete geometry

Duality theories • Figurate numbers

Frames of reference • Geometers

Geometric algorithms • Geometric graph theory

Geometric group theory • Geometric shapes

Homogeneous spaces • Incidence geometry

Integral geometry • Metric geometry

Symmetry • Trigonometry

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*Topics in Geometry*

Basic topics | Trigonometry | Euclidean geometry | Other geometries |
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Differential geometry | Riemannian geometry | Algebraic geometry | Other |

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