We spoke briefly about the Pythagoreans before. Here, we will explore different aspects of their mathematics...

1, 2, 4, 8, 16, 32, 128, 256, 512, ...

HALVING ↔ DOUBLING

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Example: 1 + 2 + 4 + 8 = 15, which is 16 - 1

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Example: 1, 2, 4, 8, 16 → 1 × 16 = 2 × 8 = 4 × 4

2, 6, 10, 14, 18, 22, ...

These are found by multiplying the Odd Numbers by 2; equivalent to every fourth number from 2...

1 × 2 = 2

3 × 2 = 6

5 × 2 = 10

7 × 2 = 14

9 × 2 = 18

11 × 2 = 22

...etc.

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Example: 18 ÷ 2 = 9 and 18 ÷ 3 = 6

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Example: 6 + 14 = 20 and 20 ÷ 2 = 10

Triangular Numbers: 1, 3, 6, 10, 15, ...

Formed from the sum of consecutive Counting Numbers:

1 + 2 = 3

1 + 2 + 3 = 6

1 + 2 + 3 + 4 = 10

1 + 2 + 3 + 4 + 5 = 15

...etc.

Square Numbers: 1, 4, 9, 16, 25, ...

Formed from the sum of consecutive Odd Numbers:

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

1 + 3 + 5 + 7 + 9 = 25

...etc.

The sum of consecutive Triangular Numbers is a Square Number:

1 + 3 = 4

3 + 6 = 9

6 + 10 = 16

10 + 15 = 25

15 + 21 = 36

...etc.

A "Gnomon" is that part which is added to a shape for it to make a larger copy of itself. It is a process of "self-similarity".

a < b = b < c

This means that a is less than b the same amount that b is less than c. (They have a "common difference".)

Example: 6:9 = 9:12 (which can also be written as 6:9:12)

Notice that 9 - 6 = 3 and 12 - 9 = 3.

This relationship can also be found with:

a:b = b:c

Example: 4:6 = 6:9 (which can also be written as 4:6:9)

Notice that 4 is two-thirds of 6 and 6 is two-thirds of 9. (They have a "common ratio".)

This relationship can also be found by multiplying the "means" and "extremes" together. The a and c are "extremes", while the "b's" are "means". Therefore, a × c = b × b.

Example: 4 × 9 = 36 and 6 × 6 = 36

Example: 2:6 = 4:12 (which can also be written as 6:8:12)

Notice that one-third of 6 is 2 and 8 - 2 = 6. Likewise, one-third of 12 is 4 and 12 - 4 = 8.

This relationship can also be found with:

To make an Arithmetic Sequence, simply add the same number repeatedly.

Example: 6, 9, 12, 15, 18, 21, 24, etc. [We are repeatedly adding 3.]

To make a Geometric Sequence, simply multiply the same number repeatedly.

Example: 2, 4, 8, 16, 32, 64, 128, etc. [We are repeatedly multiplying by 2.]

To make a Harmonic Sequence, take the Reciprocal of each number in an Arithmetic Sequence.

Example: 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, etc. [We took the Reciprocals of 1, 2, 3, 4, etc.]

Interestingly, there was a mathematical board game in the Middle Ages that used these sequences as part of its gameplay. It was called "Rithmomachia". It was similar to chess, but each piece had a number on it, and in order to "win" one had to make these kinds of sequences on the other side of the board with their pieces.

A copy of a booklet from the 1500s that describes the rules of this game still exists.

• The section "Pythagorean Mathematics" from the book

• The 3rd Appendix of the book

• The section "Aether" from the book

• The article "Harmony and Proportion" by John Boyd-Brent