Sometimes you encounter that math student who is simply intrigued by strange numbers. Here are some famous (and not so famous) sets of numbers that have curious properties.

### The Fibonacci Series

The Fibonacci series starts with zero and one. To find the next number, and add the previous two numbers together:

- 0 + 1 = 1
- 1 + 1 = 2
- 1 + 2 = 3
- 2 + 3 = 5

0, 1, 1, 2, 3, 5, 8, 13…

The Fibonacci series has many interesting appearances in nature and is related to our next curiosity: the golden ratio.

### The Golden Ratio

Known as *phi* or φ, the golden ratio is approximately 1.618.

A golden rectangle is a rectangle whose sides have a ratio of φ. Continuing to divide this rectangle by the golden ratio leads to some amazing visual properties:

Note that if you divide a Fibonacci number by its predecessor, you always hover around the golden ratio. And the farther you go in the sequence, the closer to the golden ratio you get:

- 5 ÷ 3 = 1.667
- 8 ÷ 5 = 1.6
- 13 ÷ 8 = 1.625
- 21 ÷ 13 = 1.615

Images that make use of the golden ratio are considered to be pleasing to the human eye. Dali’s *The Sacrament of the Last Supper* is purposefully based around the golden ratio, and many other works of art incorporate the rectangle (although perhaps accidentally!).

### Perfect Numbers

A perfect number’s factors sum to the number itself:

- 6 = 1 + 2 + 3
- 28 = 1 + 2 + 4 + 7 + 14
- 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248

These get large pretty quickly! Here is far more information on the perfect numbers.

### Pythagorean Triples

A Pythagorean Triple consist of three whole numbers that satisfy the equation *a*^{2} + *b*^{2} = *c*^{2}, also known as the *Pythagorean theorem*.

Here are the first three triples:

- 3 + 4 = 5 (9 + 16 = 25)
- 5 + 12 = 13 (25 + 144 = 169)
- 7 + 24 = 25 (49 + 576 = 625)

### Figurate Numbers

Figurate numbers *look* like the shape that they are named after. There are tons of these, but we’ll examine just three.

#### Perfect Squares

These numbers can be arranged to look like squares: 4,9,16

The enemy of a perfect square is a *square-free* number: one who is divisible by no perfect squares.

1, 2, 3, 5, 6, 7, 10, 11, 13…

Note that 8 doesn’t make the cut since it’s divisible by 4, 9 is divisible by 9, and 12 is divisible by 4.

#### Triangular Numbers

These numbers can be arranged into a stacked triangle: 3, 6, 10, 15, 21

Bonus: can you find numbers that are both triangular and perfect squares?

#### Centered Hexagonal Numbers

These numbers can be arranged into a hexagon, starting with one point in the middle:

1, 7, 19, 37

### Further Reading

For more mathematical curiosities, here’s a great BBC series about five quirky numbers, including π, infinity, and *i*.

*Have a favorite set of numbers that has a curious property? Let me know at ian@byrdseed.com or @IanAByrd on Twitter*

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