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 radix 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 cycle 1 2 4 8 7 5 with 3 and 6 flipping is what counting in tens gives, and twelve gives one long loop of 10 numerals with no 3 6 axis at all. Pick a radix from 2 to 36 and a multiplier to see the circuits, 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
Every place value in base b (1, b, b × b) is a pile of b − 1 with 1 left over, so adding digits keeps the remainder after dividing by b − 1. The digit b − 1 plays the part of 9 and is the keeper. In radix 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, and join each digit d to the root of k × d. Every digit has one arrow going out, so following arrows always ends in a cycle. Ten with k = 2 gives 1 2 4 8 7 5, the pair 3 6 and 9 standing still, 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 circuit. Doubling does this at 4, 6, 12, 14, 20 and 30. Ten cannot do it, 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 numerals land on the same point. Then some digits have no arrow coming in and are not on any cycle. They drain into one. In tens, 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 symbols have dashed rings.
The picture
Main loop in copper, the second cycle 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.
Computed for this page
Where the cycle of 1 stops in every radix
Nobody seems to have published the full table, so I ran doubling on the circle for every radix from 3 to 36 and counted two things. One is how many digits sit on the cycle that holds the digit 1. The other is how many digits there are apart from the keeper, which is b − 2. Binary has none and is left off.
Six rows are a single point because the hollow dot sits on the solid one: 4, 6, 12, 14, 20 and 30. Each has a prime b − 1.
All 17 odd radixes from 3 to 35 do the opposite. Their b − 1 is even, so doubling cannot be undone and the digit 1 never lands on a loop. The hollow dot rests on zero and the line runs the full width of the row. At nine, 1 goes to 2, then 4, then straight into the keeper 8.
The powers of two are a pattern of their own. At 4, 8, 16 and 32 the circuit of 1 has exactly 2, 3, 4 and 5 numerals, because 2 to the power k is 1 more than b − 1 when b = 2 to the power k, so doubling comes home after k steps. At 16 the cycle holds only 4 of 14 digits and at 32 only 5 of 30.
Twenty-six has a tidy leftover too. Its loop of 1 has 20 symbols, and the 4 outside are 5, A, F and K, the multiples of 5 in a circle of 25 points. In the same way 3 and 6 are the multiples of 3 in a circle of 9.
Ten does well on this chart, though less than the legend suggests. It puts 6 of its 8 digits on the cycle of 1 and leaves out only 3 and 6, the smallest leftover of any radix from 3 to 36 apart from the six clean ones. As a share of its digits, twenty-six does better with 4 of 24. The famous axis is simply the two numerals that 9 = 3 × 3 keeps off the loop.
Real vs legend
What is math and what is story
| Claim | Status |
|---|---|
| Doubling on a circle of 9 gives the circuit 1 2 4 8 7 5, the pair 3 6 and 9 on its own. This comes from counting in tens, where the keeper is 9 = 10 − 1. | Real |
| For any radix 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 |
| At twelve, doubling makes one cycle of 10 digits, 1 2 4 8 5 A 9 7 3 6, with no axis. | Real |
| At sixteen, doubling makes three loops of four and one pair, 5 and A. At eight it makes (1 2 4) and (3 6 5). | Real |
| Between 2 and 36, doubling makes one single cycle only at 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: 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 |
Questions
Other bases FAQ
What is vortex math in other bases?
Vortex math draws doubling on a circle of 9 points in radix 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 circuits change with it: eight gives (1 2 4) and (3 6 5), twelve gives one cycle of 10 numerals, and sixteen gives three loops of four and one pair.
What does vortex math look like at twelve?
The keeper is B, which means eleven. Doubling runs through every other digit in one cycle: 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 radix?
Write the number in that base, add its symbols, 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 radix 12. 1 + 2 + 0 + A is 13, written 11 in base 12, and 1 + 1 = 2. 2026 divided by 11 leaves 2.
Which radixes make one single doubling loop?
Between 2 and 36, only 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. Ten cannot do it, because 9 is not prime: 3 and 6 are always cut off.
What happens in binary and ternary?
In binary the keeper is 1 and the circle has one point. Every whole number above 0 has digital root 1, so there is nothing to cycle through. At three the keeper is 2. Doubling always gives an even number, and every even number has root 2 there, 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 symbols land on the same point, so some digits have no arrow coming in. They are not on any cycle. They drain into one. In tens, 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 radix and the pattern moves or disappears.
What does the chart of every base show?
It counts, for each radix from 3 to 36, how many digits sit on the doubling loop that holds the digit 1 and how many digits exist besides the keeper. Six rows leave none out (4, 6, 12, 14, 20 and 30). Every odd radix leaves every symbol out, because 1 never lands on a cycle. Ten puts 6 of 8 digits on the loop and leaves out 3 and 6.
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Sources
- Digital root, and how it depends on the radix: en.wikipedia.org
- Casting out nines: en.wikipedia.org
- OEIS A010888, digital root of n: oeis.org
- OEIS A002326, multiplicative order of 2 mod 2n+1, the loop length for even bases: oeis.org
- The every-base table above was computed for this page with a short Python script that follows x to 2x mod b − 1 for each base 3 to 36 and counts the cycles.