Pythagoras

**Pythagoras**, an ancient Greek mathematician and philosopher, is best known for the **Pythagorean Theorem**. This theorem states that, in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this can be expressed as \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse. Pythagorean triples, such as (3, 4, 5) and (5, 12, 13), are sets of three positive integers that satisfy this equation.

Pythagoras founded a philosophical and religious school in **Croton**, a Greek colony in southern Italy, where he and his followers, known as the **Pythagoreans**, explored mathematics, music, and astronomy. Pythagoras is credited with discovering that musical intervals correspond to simple ratios of integers, leading to the concept of the **music of the spheres**, a theory that celestial bodies produce harmonies based on their motions.

Although the theorem attributed to Pythagoras was known to Babylonian mathematicians centuries earlier, Pythagoras is traditionally associated with its geometric proof. His school also developed mathematical and philosophical ideas, including the concept of **irrational numbers**—a discovery that allegedly led to the drowning of his follower **Hippasus**.

### Recurring Clues in Quiz Bowl Questions:

1. **Pythagorean Theorem**: The most frequently cited fact about Pythagoras is his theorem, \(a^2 + b^2 = c^2\), which relates the sides of a right triangle.

2. **Pythagorean Triples**: Specific sets of integers (like 3, 4, 5 or 5, 12, 13) that satisfy the theorem are often referenced.

3. **Music of the Spheres**: Pythagoras’s idea that celestial bodies produce harmonic sounds based on mathematical ratios.

4. **Vegetarianism and Bean Taboo**: Pythagoras and his followers were vegetarians and abstained from eating beans, as they believed beans contained souls.

5. **Hippasus and Irrational Numbers**: The discovery of irrational numbers, attributed to Hippasus, was controversial within the Pythagorean school.

6. **Metempsychosis**: The belief in the transmigration of souls, or reincarnation, which was central to Pythagorean philosophy.

### Related Quizbowl Facts That Appeared In More Than One Toss-up on qbreader.org:

1. **Pythagorean Triples** are sets of three positive integers, such as ___1___ and 5, 12, 13, that satisfy the Pythagorean theorem.

2. The **Pythagorean school** was founded in **Croton**, a colony in southern ___2___.

3. The idea of the **music of the spheres** suggests that celestial bodies produce ___3___ sounds based on their motions.

4. **Hippasus** is said to have drowned for discovering that the square root of ___4___ was irrational.

5. Pythagorean philosophy included **metempsychosis**, or the ___5___ of souls.

Fill-in-the-blanks answers for Quizbowl practice:

1. 3, 4, 5

2. Italy

3. harmonic

4. 2

5. transmigration

### Clues and Frequencies (Sorted in Descending Order):

1. **Pythagorean school located in Croton** - 8

2. **Transmigration of souls (metempsychosis)** - 7

3. **Music of the spheres** - 6

4. **Prohibition against eating beans** - 6

5. **Drowning of Hippasus after discovering irrational numbers** - 5

6. **James Garfield’s proof using a trapezoid** - 3

7. **Pythagorean triples (e.g., 3, 4, 5)** - 3

8. **Law of cosines as a generalization of Pythagorean theorem** - 2

9. **Pythagorean theorem as a special case of Euclidean geometry** - 2