Vortex math in other bases · digital root in base b
Base Shifter
In any base b, the digit b − 1 does the job 9 does in base 10: the digital root is the remainder after dividing by b − 1, and doubling on a circle of b − 1 points splits into loops. The famous loop 1 2 4 8 7 5 with 3 and 6 flipping is what base 10 gives, and base 12 gives one long loop of 10 digits with no 3 6 axis at all. Pick a base from 2 to 36 and a multiplier to see the loops, the keeper digit and the digital root of any number.
- Keeper
- Loops
- How many
- One loop
- Axis
- Drains
- Root rule
The same rule in bases 2 to 36, with the number of loops, the keeper counted. Tap one to load it.
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How it works
Nine is just ten minus one
1. The keeper
In base 10 every place is worth 1, 10, 100, 1000, and each of those is a pile of 9s with 1 left over. So adding up the digits keeps the remainder after dividing by 9 and throws the rest away. That is the digital root. In base b the places are 1, b, b × b and so on, and each of those is a pile of b − 1 with 1 left over. So in base b the digit b − 1 plays the part of 9. We call it the keeper. In base 12 it is B, which means eleven: 7 × B = 77, written 65 in base 12, and 6 + 5 = B.
2. Loops on a circle
Put the digits 1 to b − 2 around a circle with the keeper at the top, the way Marko Rodin puts 9 at the top of his circle of nine. Join each digit d to the root of k × d, which is k × d counted around the circle. Every digit has exactly one arrow going out, so if you keep following arrows you must end up going round a loop. In base 10 with k = 2 that gives the loop 1 2 4 8 7 5, the pair 3 6, and 9 standing still. It is the same picture as the times table circle with 9 points.
3. When it is one big loop
If b − 1 is a prime and k takes the longest possible route around it, every digit except the keeper sits on one loop. Doubling does this in bases 4, 6, 12, 14, 20 and 30. In base 10 it cannot happen, because 9 = 3 × 3, and multiplying a number that is not a multiple of 3 never makes one, so 3 and 6 are always cut off from the rest.
4. Drains
When k and b − 1 share a factor, two digits land on the same point. Then some digits have no arrow coming in and are not on any loop. They drain into one. In base 10 times 3 sends 1, 4 and 7 to 3, and 3 goes on to 9. On the circle, drains are dashed lines and the draining digits have dashed rings.
The picture
Main loop in copper, the second loop in violet, the keeper in pale gold, and any other loops in their own colours. Small arrows show the direction. A small ring beside a digit means it goes to itself.
Real vs legend
What is math and what is story
| Claim | Status |
|---|---|
| Doubling on a circle of 9 gives the loop 1 2 4 8 7 5, the pair 3 6 and 9 on its own. This comes from counting in base 10, where the keeper is 9 = 10 − 1. | Real |
| In every base b the digit b − 1 behaves like 9: casting out b − 1 checks sums, and multiples of b − 1 have digit sums that are multiples of b − 1. | Real |
| In base 12 doubling makes one loop of 10 digits, 1 2 4 8 5 A 9 7 3 6, with no axis. | Real |
| In base 16 doubling makes three loops of four and one pair, 5 and A. In base 8 it makes (1 2 4) and (3 6 5). | Real |
| From base 2 to base 36, doubling makes one single loop only in bases 4, 6, 12, 14, 20 and 30. | Real |
| Doubling gives a flipping pair like 3 and 6 exactly when b − 1 is a multiple of 3: bases 4, 7, 10, 13, 16 and so on. | Real |
| 3, 6 and 9 are a universal key or a law of energy. | Legend |
| Rodin's vortex describes the whole universe or leads to free energy. | Legend |
| Tesla said "If you only knew the magnificence of 3, 6 and 9". | No source |
Questions
Other bases FAQ
What is vortex math in other bases?
Vortex math draws doubling on a circle of 9 points in base 10, which gives the loop 1 2 4 8 7 5 with 3 and 6 flipping and 9 standing still. In base b the circle has b − 1 points and the digit b − 1 takes the place of 9. The loops change with the base: base 8 gives (1 2 4) and (3 6 5), base 12 gives one loop of 10 digits, and base 16 gives three loops of four and one pair.
What does vortex math look like in base 12?
The keeper is B, which means eleven. Doubling runs through every other digit in one loop: 1, 2, 4, 8, 5, A, 9, 7, 3, 6 and back to 1. There is no pair like 3 and 6, because 11 is not a multiple of 3.
How do you find the digital root in another base?
Write the number in that base, add its digits, write the sum in the same base and repeat until one digit is left. The answer is the remainder after dividing the number by b − 1, with the keeper b − 1 standing in for a remainder of 0. 2026 is 120A in base 12. 1 + 2 + 0 + A is 13, written 11 in base 12, and 1 + 1 = 2. 2026 divided by 11 leaves 2.
Which bases make one single doubling loop?
From base 2 to base 36, only bases 4, 6, 12, 14, 20 and 30. In each of them b − 1 is a prime (3, 5, 11, 13, 19 and 29) and doubling visits every other digit before it comes back to 1. Base 10 cannot do it, because 9 is not prime: 3 and 6 are always cut off.
What happens in base 2 and base 3?
In base 2 the keeper is 1 and the circle has one point. Every whole number above 0 has digital root 1, so there is nothing to loop. In base 3 the keeper is 2. Doubling always gives an even number, and every even number has root 2 in base 3, so the digit 1 drains straight into the keeper.
What does the multiplier k change?
Each digit d is joined to k × d, counted around the circle. When k and b − 1 share a factor, two digits land on the same point, so some digits have no arrow coming in. They are not on any loop. They drain into one. In base 10, times 3 sends 1, 4 and 7 to 3, and 3 goes on to 9.
Is 3 6 9 a universal key?
No. The 3 6 9 pattern is what doubling does on a circle of 9 points, and the circle has 9 points because we count in tens. Change the base and the pattern moves or disappears. The line "If you only knew the magnificence of 3, 6 and 9" is often credited to Tesla, but nobody has found it in anything he wrote or said.
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