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Number Constellations

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Number Constellations

by MinuteLabs.io

Brady Haran (of Numberphile) came to us with an idea of laying out numbers on a grid and highlighting them based on the “family” they belonged to. His video on the Prime Spirals sparked our curiosity and we decided we could take the idea a bit further…

What’s happening?

Here we’re laying out the positive integers (1… 2… 3…) in different ways; a simple square grid, an Ulam Spiral, a Sacks Spiral, and a Vogel (Fibonacci) Spiral. (We also added a triangle spiral and a pyramid). Once the numbers are layed out, we choose some of them based on the Family that is selected in the settings, and we highlight the numbers that are part of that family. This lets you see some interesting patterns!

A very famous way to do this is laying out the numbers in an “Ulam Spiral” configuration and highlighting the prime numbers. Some interesting patterns emerge.

Also, if you share this page with the whole URL intact, it will automatically load with the same settings you see at the time you shared the URL. If you click on a circle, it will show you what number that is in the bottom right.

Did you find anything interesting? See a mistake? Let us know what you think in the comments.

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Click below to see more information about the selected number family!

Prime Numbers

A prime number is a number that can't be exactly divided by any number except for 1, and itself.

See the sequence on OEIS

Numberphile covers Prime Numbers in this video... (starts at: 6:47)

Numberphile has a whole playlist of videos covering prime numbers.

Carmichael Numbers

Carmichael numbers are numbers that pass Fermat's primality test, but they aren't actually prime numbers. They are Fermat Pseudoprimes.

See the sequence on OEIS

Numberphile covers Carmichael Numbers in this video... (starts at: 6:47)

Numberphile covers Carmichael Numbers in this video...

Fibonacci Numbers

Fibonacci numbers are numbers that belong to the Fibonacci sequence which can be generated by repeatedly adding the previous two numbers of the sequence.

See the sequence on OEIS

Numberphile covers Fibonacci Numbers in this video...

Fibonacci Primes

Fibonacci primes are both Fibonacci numbers and prime numbers.

See the sequence on OEIS

Fibonacci Sums

A Fibonacci sum (also called a Zeckendorf representation) is a way of representing any integer as a sum of non-consecutive Fibonacci numbers.

What we're doing here is: for every number, we're counting how many ways we can do this for that number (how many Zeckendorf representations it has) and coloring the circles darker the more ways we find.

If you select the Vogel Spiral (which is related to the Fibonacci sequence), you'll be able to see some interesting "boxy" patterns emerge.

See the sequence on OEIS

Pythagorean Primes

Pythagorean Primes are prime numbers that can be written as the sum of two square integers.

It turns out that all Pythagorean Primes can be generated by the expression 4*n-1 (but not for every n). Try putting that in as a custom highlighting for comparison!

See the sequence on OEIS

Numberphile covers Pythagorean Primes in this video...

Pythagorean Triples

A Pythagorean Triple is a set of three integers (a,b,c) that satisfy the Pythagorean Theorem; a2 + b2 = c2.

We're highlighting the numbers (a, b, or c) that do this.

See the sequence on OEIS: Hypotenuses, Long Legs, Short Legs

Multiplicative Partition

A Multiplicative Partition is a way to write an integer as a product of other integers greater than 1.

For example, we can write 8 as:
8 or 2×2×2 or 2×4. So the number 8 has 3 Multiplicative Partitions.

We're coloring the numbers with more Multiplicative Partitions darker.

See the sequence on OEIS

Vampire Numbers

Vampire Numbers are integers that can be factored into two numbers that have the same digits as the original.

For example, 1260 = 21 × 60, so 1260 is a Vampire Number, with fangs: 21, and 60

See the sequence on OEIS

Numberphile covers Vampire Numbers in this video...

Largest Metadromes of base n

A metadrome is a number that has its digits increase from left to right. For example: 123 is a metadrome. The largest metadrome you can write in base 10 is: 123456789. But you can write numbers in different bases. This sequence contains the largest metadromes you can write in a specific base.

For example, the largest metadrome you can write in base 3 is: 12, which is 5 in base 10.

See the sequence on OEIS

Numberphile covers these numbers in this video... (starts at: 6:47)

Random Numbers

A set of ordered pseudo-randomly generated integers.

Every number has a 50/50 chance of being highlighted. If you wrote this as a custom function it would be: n * number(random() > 0.5). Try it!

You can use this to compare a random pattern with other patterns.

We didn't use radioactive material to generate these random numbers... but Numberphile has done that...

Random Opacity

We're highlighting each number by a random amount (between 0 and 1).

For every number we select a pseudo-random value between 0 and 1, and highlight the number by that value (1 means fully highlighted)

We didn't use radioactive material to generate these random numbers... but Numberphile has done that...

Custom Highlighting

If you'd like, you can specify your own custom function to highlight the numbers. Just select the "custom" family and enter an equation in the "custom" text box. The input parameter "n" will count up from 1, and the numbers returned from the equation will be highlighted.

For example: n^2 + 4*n + 1 will highlight...

6 (n=1), 13 (n=2), 22 (n=3), ...

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